Generating function

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convergence"> <div class="vector-toc-text"> 2.1 Convergence </div> </a> <ul id="toc-Convergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limitations"> <div class="vector-toc-text"> 2.2 Limitations </div> </a> <ul id="toc-Limitations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Types" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> 3 Types </div> </a> <button aria-controls="toc-Types-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Types subsection </button> <ul id="toc-Types-sublist" class="vector-toc-list"> <li id="toc-Ordinary_generating_function_(OGF)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordinary_generating_function_(OGF)"> <div class="vector-toc-text"> 3.1 Ordinary generating function (OGF) </div> </a> <ul id="toc-Ordinary_generating_function_(OGF)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exponential_generating_function_(EGF)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exponential_generating_function_(EGF)"> <div class="vector-toc-text"> 3.2 Exponential generating function (EGF) </div> </a> <ul id="toc-Exponential_generating_function_(EGF)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_generating_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_generating_function"> <div class="vector-toc-text"> 3.3 Poisson generating function </div> </a> <ul id="toc-Poisson_generating_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lambert_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lambert_series"> <div class="vector-toc-text"> 3.4 Lambert series </div> </a> <ul id="toc-Lambert_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bell_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bell_series"> <div class="vector-toc-text"> 3.5 Bell series </div> </a> <ul id="toc-Bell_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dirichlet_series_generating_functions_(DGFs)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dirichlet_series_generating_functions_(DGFs)"> <div class="vector-toc-text"> 3.6 Dirichlet series generating functions (DGFs) </div> </a> <ul id="toc-Dirichlet_series_generating_functions_(DGFs)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomial_sequence_generating_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomial_sequence_generating_functions"> <div class="vector-toc-text"> 3.7 Polynomial sequence generating functions </div> </a> <ul id="toc-Polynomial_sequence_generating_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_generating_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_generating_functions"> <div class="vector-toc-text"> 3.8 Other generating functions </div> </a> <ul id="toc-Other_generating_functions-sublist" class="vector-toc-list"> <li id="toc-Convolution_polynomials" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Convolution_polynomials"> <div class="vector-toc-text"> 3.8.1 Convolution polynomials </div> </a> <ul id="toc-Convolution_polynomials-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Ordinary_generating_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ordinary_generating_functions"> <div class="vector-toc-text"> 4 Ordinary generating functions </div> </a> <button aria-controls="toc-Ordinary_generating_functions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Ordinary generating functions subsection </button> <ul id="toc-Ordinary_generating_functions-sublist" class="vector-toc-list"> <li id="toc-Examples_for_simple_sequences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_for_simple_sequences"> <div class="vector-toc-text"> 4.1 Examples for simple sequences </div> </a> <ul id="toc-Examples_for_simple_sequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rational_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rational_functions"> <div class="vector-toc-text"> 4.2 Rational functions </div> </a> <ul id="toc-Rational_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operations_on_generating_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operations_on_generating_functions"> <div class="vector-toc-text"> 4.3 Operations on generating functions </div> </a> <ul id="toc-Operations_on_generating_functions-sublist" class="vector-toc-list"> <li id="toc-Multiplication_yields_convolution" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multiplication_yields_convolution"> <div class="vector-toc-text"> 4.3.1 Multiplication yields convolution </div> </a> <ul id="toc-Multiplication_yields_convolution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shifting_sequence_indices" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Shifting_sequence_indices"> <div class="vector-toc-text"> 4.3.2 Shifting sequence indices </div> </a> <ul id="toc-Shifting_sequence_indices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differentiation_and_integration_of_generating_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Differentiation_and_integration_of_generating_functions"> <div class="vector-toc-text"> 4.3.3 Differentiation and integration of generating functions </div> </a> <ul id="toc-Differentiation_and_integration_of_generating_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enumerating_arithmetic_progressions_of_sequences" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Enumerating_arithmetic_progressions_of_sequences"> <div class="vector-toc-text"> 4.3.4 Enumerating arithmetic progressions of sequences </div> </a> <ul id="toc-Enumerating_arithmetic_progressions_of_sequences-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-P-recursive_sequences_and_holonomic_generating_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#P-recursive_sequences_and_holonomic_generating_functions"> <div class="vector-toc-text"> 4.4 P-recursive sequences and holonomic generating functions </div> </a> <ul id="toc-P-recursive_sequences_and_holonomic_generating_functions-sublist" class="vector-toc-list"> <li id="toc-Definitions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> 4.4.1 Definitions </div> </a> <ul id="toc-Definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 4.4.2 Examples </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Software_for_working_with_P-recursive_sequences_and_holonomic_generating_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Software_for_working_with_P-recursive_sequences_and_holonomic_generating_functions"> <div class="vector-toc-text"> 4.4.3 Software for working with P-recursive sequences and holonomic generating functions </div> </a> <ul id="toc-Software_for_working_with_P-recursive_sequences_and_holonomic_generating_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_to_discrete-time_Fourier_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_discrete-time_Fourier_transform"> <div class="vector-toc-text"> 4.5 Relation to discrete-time Fourier transform </div> </a> <ul id="toc-Relation_to_discrete-time_Fourier_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asymptotic_growth_of_a_sequence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asymptotic_growth_of_a_sequence"> <div class="vector-toc-text"> 4.6 Asymptotic growth of a sequence </div> </a> <ul id="toc-Asymptotic_growth_of_a_sequence-sublist" class="vector-toc-list"> <li id="toc-Asymptotic_growth_of_the_sequence_of_squares" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Asymptotic_growth_of_the_sequence_of_squares"> <div class="vector-toc-text"> 4.6.1 Asymptotic growth of the sequence of squares </div> </a> <ul id="toc-Asymptotic_growth_of_the_sequence_of_squares-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asymptotic_growth_of_the_Catalan_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Asymptotic_growth_of_the_Catalan_numbers"> <div class="vector-toc-text"> 4.6.2 Asymptotic growth of the Catalan numbers </div> </a> <ul id="toc-Asymptotic_growth_of_the_Catalan_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bivariate_and_multivariate_generating_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bivariate_and_multivariate_generating_functions"> <div class="vector-toc-text"> 4.7 Bivariate and multivariate generating functions </div> </a> <ul id="toc-Bivariate_and_multivariate_generating_functions-sublist" class="vector-toc-list"> <li id="toc-Bivariate_case" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bivariate_case"> <div class="vector-toc-text"> 4.7.1 Bivariate case </div> </a> <ul id="toc-Bivariate_case-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multivariate_case" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multivariate_case"> <div class="vector-toc-text"> 4.7.2 Multivariate case </div> </a> <ul id="toc-Multivariate_case-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representation_by_continued_fractions_(Jacobi-type_J-fractions)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representation_by_continued_fractions_(Jacobi-type_J-fractions)"> <div class="vector-toc-text"> 4.8 Representation by continued fractions (Jacobi-type J-fractions) </div> </a> <ul id="toc-Representation_by_continued_fractions_(Jacobi-type_J-fractions)-sublist" class="vector-toc-list"> <li id="toc-Definitions_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Definitions_2"> <div class="vector-toc-text"> 4.8.1 Definitions </div> </a> <ul id="toc-Definitions_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_the_hth_convergent_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Properties_of_the_hth_convergent_functions"> <div class="vector-toc-text"> 4.8.2 Properties of the hth convergent functions </div> </a> <ul id="toc-Properties_of_the_hth_convergent_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> 4.8.3 Examples </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Examples_3" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples_3"> <div class="vector-toc-text"> 5 Examples </div> </a> <button aria-controls="toc-Examples_3-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples subsection </button> <ul id="toc-Examples_3-sublist" class="vector-toc-list"> <li id="toc-Square_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Square_numbers"> <div class="vector-toc-text"> 5.1 Square numbers </div> </a> <ul id="toc-Square_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 6 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Various_techniques:_Evaluating_sums_and_tackling_other_problems_with_generating_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Various_techniques:_Evaluating_sums_and_tackling_other_problems_with_generating_functions"> <div class="vector-toc-text"> 6.1 Various techniques: Evaluating sums and tackling other problems with generating functions </div> </a> <ul id="toc-Various_techniques:_Evaluating_sums_and_tackling_other_problems_with_generating_functions-sublist" class="vector-toc-list"> <li id="toc-Example_1:_Formula_for_sums_of_harmonic_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_1:_Formula_for_sums_of_harmonic_numbers"> <div class="vector-toc-text"> 6.1.1 Example 1: Formula for sums of harmonic numbers </div> </a> <ul id="toc-Example_1:_Formula_for_sums_of_harmonic_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_2:_Modified_binomial_coefficient_sums_and_the_binomial_transform" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_2:_Modified_binomial_coefficient_sums_and_the_binomial_transform"> <div class="vector-toc-text"> 6.1.2 Example 2: Modified binomial coefficient sums and the binomial transform </div> </a> <ul id="toc-Example_2:_Modified_binomial_coefficient_sums_and_the_binomial_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_3:_Generating_functions_for_mutually_recursive_sequences" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_3:_Generating_functions_for_mutually_recursive_sequences"> <div class="vector-toc-text"> 6.1.3 Example 3: Generating functions for mutually recursive sequences </div> </a> <ul id="toc-Example_3:_Generating_functions_for_mutually_recursive_sequences-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Convolution_(Cauchy_products)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convolution_(Cauchy_products)"> <div class="vector-toc-text"> 6.2 Convolution (Cauchy products) </div> </a> <ul id="toc-Convolution_(Cauchy_products)-sublist" class="vector-toc-list"> <li id="toc-Example:_Generating_function_for_the_Catalan_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example:_Generating_function_for_the_Catalan_numbers"> <div class="vector-toc-text"> 6.2.1 Example: Generating function for the Catalan numbers </div> </a> <ul id="toc-Example:_Generating_function_for_the_Catalan_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example:_Spanning_trees_of_fans_and_convolutions_of_convolutions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example:_Spanning_trees_of_fans_and_convolutions_of_convolutions"> <div class="vector-toc-text"> 6.2.2 Example: Spanning trees of fans and convolutions of convolutions </div> </a> <ul id="toc-Example:_Spanning_trees_of_fans_and_convolutions_of_convolutions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Implicit_generating_functions_and_the_Lagrange_inversion_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Implicit_generating_functions_and_the_Lagrange_inversion_formula"> <div class="vector-toc-text"> 6.3 Implicit generating functions and the Lagrange inversion formula </div> </a> <ul id="toc-Implicit_generating_functions_and_the_Lagrange_inversion_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Introducing_a_free_parameter_(snake_oil_method)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Introducing_a_free_parameter_(snake_oil_method)"> <div class="vector-toc-text"> 6.4 Introducing a free parameter (snake oil method) </div> </a> <ul id="toc-Introducing_a_free_parameter_(snake_oil_method)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generating_functions_prove_congruences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generating_functions_prove_congruences"> <div class="vector-toc-text"> 6.5 Generating functions prove congruences </div> </a> <ul id="toc-Generating_functions_prove_congruences-sublist" class="vector-toc-list"> <li id="toc-The_Stirling_numbers_modulo_small_integers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_Stirling_numbers_modulo_small_integers"> <div class="vector-toc-text"> 6.5.1 The Stirling numbers modulo small integers </div> </a> <ul id="toc-The_Stirling_numbers_modulo_small_integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congruences_for_the_partition_function" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Congruences_for_the_partition_function"> <div class="vector-toc-text"> 6.5.2 Congruences for the partition function </div> </a> <ul id="toc-Congruences_for_the_partition_function-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Transformations_of_generating_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transformations_of_generating_functions"> <div class="vector-toc-text"> 6.6 Transformations of generating functions </div> </a> <ul id="toc-Transformations_of_generating_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tables_of_special_generating_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Tables_of_special_generating_functions"> <div class="vector-toc-text"> 7 Tables of special generating functions </div> </a> <ul id="toc-Tables_of_special_generating_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 10 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 10.1 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 11 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Generating function</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 26 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-26" aria-hidden="true" > 26 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9_%D9%85%D9%88%D9%84%D8%AF%D8%A9" title="دالة مولدة – Arabic" lang="ar" hreflang="ar" data-title="دالة مولدة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_generatriu" title="Funció generatriu – Catalan" lang="ca" hreflang="ca" data-title="Funció generatriu" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vytvo%C5%99uj%C3%ADc%C3%AD_funkce_(posloupnost)" title="Vytvořující funkce (posloupnost) – Czech" lang="cs" hreflang="cs" data-title="Vytvořující funkce (posloupnost)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Erzeugende_Funktion" title="Erzeugende Funktion – German" lang="de" hreflang="de" data-title="Erzeugende Funktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_generatriz" title="Función generatriz – Spanish" lang="es" hreflang="es" data-title="Función generatriz" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D9%85%D9%88%D9%84%D8%AF" title="تابع مولد – Persian" lang="fa" hreflang="fa" data-title="تابع مولد" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/S%C3%A9rie_g%C3%A9n%C3%A9ratrice" title="Série génératrice – French" lang="fr" hreflang="fr" data-title="Série génératrice" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_xeradora" title="Función xeradora – Galician" lang="gl" hreflang="gl" data-title="Función xeradora" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%83%9D%EC%84%B1%ED%95%A8%EC%88%98_(%EC%88%98%ED%95%99)" title="생성함수 (수학) – Korean" lang="ko" hreflang="ko" data-title="생성함수 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_pembangkit" title="Fungsi pembangkit – Indonesian" lang="id" hreflang="id" data-title="Fungsi pembangkit" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_generatrice" title="Funzione generatrice – Italian" lang="it" hreflang="it" data-title="Funzione generatrice" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%99%D7%95%D7%A6%D7%A8%D7%AA" title="פונקציה יוצרת – Hebrew" lang="he" hreflang="he" data-title="פונקציה יוצרת" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Voortbrengende_functie" title="Voortbrengende functie – Dutch" lang="nl" hreflang="nl" data-title="Voortbrengende functie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Formal power series; coefficients encode information about a sequence indexed by natural numbers</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about generating functions in mathematics. For generating functions in classical mechanics, see <a href="/wiki/Generating_function_(physics)" title="Generating function (physics)">Generating function (physics)</a>. For generators in computer programming, see <a href="/wiki/Generator_(computer_programming)" title="Generator (computer programming)">Generator (computer programming)</a>. For the moment generating function in statistics, see <a href="/wiki/Moment_generating_function" class="mw-redirect" title="Moment generating function">Moment generating function</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Very_long plainlinks metadata ambox ambox-style ambox-very_long" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This article may be <a href="/wiki/Wikipedia:Article_size" title="Wikipedia:Article size">too long</a> to read and navigate comfortably. Consider <a href="/wiki/Wikipedia:Splitting" title="Wikipedia:Splitting">splitting</a> content into sub-articles, <a href="/wiki/Wikipedia:Summary_style" title="Wikipedia:Summary style">condensing</a> it, or adding <a href="/wiki/Help:Section#Subsections" title="Help:Section">subheadings</a>. Please discuss this issue on the article's <a href="/wiki/Talk:Generating_function" title="Talk:Generating function">talk page</a>. (July 2022)</div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a generating function is a representation of an <a href="/wiki/Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequence</a> of numbers as the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> of a <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a>. Generating functions are often expressed in <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed form</a> (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are sometimes called generating series,<a href="#cite_note-1">[1]</a> in that a series of terms can be said to be the generator of its sequence of term coefficients. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=1" title="Edit section: History">edit</a>]</div> Generating functions were first introduced by <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> in 1730, in order to solve the general linear recurrence problem.<a href="#cite_note-2">[2]</a> <a href="/wiki/George_P%C3%B3lya" title="George Pólya">George Pólya</a> writes in <a href="/wiki/Mathematics_and_plausible_reasoning" class="mw-redirect" title="Mathematics and plausible reasoning">Mathematics and plausible reasoning</a>: <blockquote>The name "generating function" is due to <a href="/wiki/Laplace" class="mw-redirect" title="Laplace">Laplace</a>. Yet, without giving it a name, <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a> used the device of generating functions long before Laplace [..]. He applied this mathematical tool to several problems in Combinatory Analysis and the <a href="/wiki/Number_theory" title="Number theory">Theory of Numbers</a>.</blockquote> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=2" title="Edit section: Definition">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.<div class="templatequotecite">— <cite><a href="/wiki/George_P%C3%B3lya" title="George Pólya">George Pólya</a>, <a href="/wiki/Mathematics_and_plausible_reasoning" class="mw-redirect" title="Mathematics and plausible reasoning">Mathematics and plausible reasoning</a> (1954)</cite></div></blockquote> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">A generating function is a clothesline on which we hang up a sequence of numbers for display.<div class="templatequotecite">— <cite><a href="/wiki/Herbert_Wilf" title="Herbert Wilf">Herbert Wilf</a>, <a rel="nofollow" class="external text" href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a> (1994)</cite></div></blockquote> <div class="mw-heading mw-heading3"><h3 id="Convergence">Convergence</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=3" title="Edit section: Convergence">edit</a>]</div> Unlike an ordinary series, the formal <a href="/wiki/Power_series" title="Power series">power series</a> is not required to <a href="/wiki/Convergent_series" title="Convergent series">converge</a>: in fact, the generating function is not actually regarded as a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, and the "variable" remains an <a href="/wiki/Indeterminate_(variable)" title="Indeterminate (variable)">indeterminate</a>. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. Thus generating functions are not functions in the formal sense of a mapping from a <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> to a <a href="/wiki/Codomain" title="Codomain">codomain</a>. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal series as its <a href="/wiki/Series_expansion" title="Series expansion">series expansion</a>; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a <a href="/wiki/Convergent_series" title="Convergent series">convergent series</a> when a nonzero numeric value is substituted for x. <div class="mw-heading mw-heading3"><h3 id="Limitations">Limitations</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=4" title="Edit section: Limitations">edit</a>]</div> Not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative and fractional powers of x are examples of functions that do not have a corresponding formal power series. <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=5" title="Edit section: Types">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Ordinary_generating_function_(OGF)">Ordinary generating function (OGF)</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=6" title="Edit section: Ordinary generating function (OGF)">edit</a>]</div> When the term generating function is used without qualification, it is usually taken to mean an ordinary generating function. The ordinary generating function of a sequence an is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(a_{n};x)=\sum _{n=0}^{\infty }a_{n}x^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(a_{n};x)=\sum _{n=0}^{\infty }a_{n}x^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad639191e7ebb6af511920f2e7a6b02eed511a9e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.932ex; height:6.843ex;" alt="{\displaystyle G(a_{n};x)=\sum _{n=0}^{\infty }a_{n}x^{n}.}"> If an is the <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> of a <a href="/wiki/Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variable</a>, then its ordinary generating function is called a <a href="/wiki/Probability-generating_function" title="Probability-generating function">probability-generating function</a>. <div class="mw-heading mw-heading3"><h3 id="Exponential_generating_function_(EGF)">Exponential generating function (EGF)</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=7" title="Edit section: Exponential generating function (EGF)">edit</a>]</div> The exponential generating function of a sequence an is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {EG} (a_{n};x)=\sum _{n=0}^{\infty }a_{n}{\frac {x^{n}}{n!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>EG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {EG} (a_{n};x)=\sum _{n=0}^{\infty }a_{n}{\frac {x^{n}}{n!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c31459ce0cc949e236f60e905944908c12cb2f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.348ex; height:6.843ex;" alt="{\displaystyle \operatorname {EG} (a_{n};x)=\sum _{n=0}^{\infty }a_{n}{\frac {x^{n}}{n!}}.}"> Exponential generating functions are generally more convenient than ordinary generating functions for <a href="/wiki/Combinatorial_enumeration" class="mw-redirect" title="Combinatorial enumeration">combinatorial enumeration</a> problems that involve labelled objects.<a href="#cite_note-3">[3]</a> Another benefit of exponential generating functions is that they are useful in transferring linear <a href="/wiki/Recurrence_relations" class="mw-redirect" title="Recurrence relations">recurrence relations</a> to the realm of <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>. For example, take the <a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a> {fn} that satisfies the linear recurrence relation fn+2 = fn+1 + fn. The corresponding exponential generating function has the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {EF} (x)=\sum _{n=0}^{\infty }{\frac {f_{n}}{n!}}x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>EF</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {EF} (x)=\sum _{n=0}^{\infty }{\frac {f_{n}}{n!}}x^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c142709069e77fe1c0a4959b2a943f70fadf1814" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.822ex; height:6.843ex;" alt="{\displaystyle \operatorname {EF} (x)=\sum _{n=0}^{\infty }{\frac {f_{n}}{n!}}x^{n}}"> and its derivatives can readily be shown to satisfy the differential equation EF″(x) = EF′(x) + EF(x) as a direct analogue with the recurrence relation above. In this view, the factorial term n! is merely a counter-term to normalise the derivative operator acting on xn. <div class="mw-heading mw-heading3"><h3 id="Poisson_generating_function">Poisson generating function</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=8" title="Edit section: Poisson generating function">edit</a>]</div> The Poisson generating function of a sequence an is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {PG} (a_{n};x)=\sum _{n=0}^{\infty }a_{n}e^{-x}{\frac {x^{n}}{n!}}=e^{-x}\,\operatorname {EG} (a_{n};x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>PG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>EG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {PG} (a_{n};x)=\sum _{n=0}^{\infty }a_{n}e^{-x}{\frac {x^{n}}{n!}}=e^{-x}\,\operatorname {EG} (a_{n};x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4982ac1f8a9f5981ead45ad651e8b4f30c11c651" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.319ex; height:6.843ex;" alt="{\displaystyle \operatorname {PG} (a_{n};x)=\sum _{n=0}^{\infty }a_{n}e^{-x}{\frac {x^{n}}{n!}}=e^{-x}\,\operatorname {EG} (a_{n};x).}"> <div class="mw-heading mw-heading3"><h3 id="Lambert_series">Lambert series</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=9" title="Edit section: Lambert series">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lambert_series" title="Lambert series">Lambert series</a></div> The Lambert series of a sequence an is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {LG} (a_{n};x)=\sum _{n=1}^{\infty }a_{n}{\frac {x^{n}}{1-x^{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>LG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {LG} (a_{n};x)=\sum _{n=1}^{\infty }a_{n}{\frac {x^{n}}{1-x^{n}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb4c80af50da16cff898e70c70323057ead525f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.221ex; height:6.843ex;" alt="{\displaystyle \operatorname {LG} (a_{n};x)=\sum _{n=1}^{\infty }a_{n}{\frac {x^{n}}{1-x^{n}}}.}">Note that in a Lambert series the index n starts at 1, not at 0, as the first term would otherwise be undefined. The Lambert series coefficients in the power series expansions <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}:=[x^{n}]\operatorname {LG} (a_{n};x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:=</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>LG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}:=[x^{n}]\operatorname {LG} (a_{n};x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545c302a68d78c5b29be0ee431163be4aa0e726a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.089ex; height:2.843ex;" alt="{\displaystyle b_{n}:=[x^{n}]\operatorname {LG} (a_{n};x)}">for integers n ≥ 1 are related by the <a href="/wiki/Divisor_sum_identities" title="Divisor sum identities">divisor sum</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}=\sum _{d|n}a_{d}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}=\sum _{d|n}a_{d}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2deefbe19d8b05cd8c9d0df0e31947794749310d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:12.025ex; height:6.009ex;" alt="{\displaystyle b_{n}=\sum _{d|n}a_{d}.}">The <a href="/wiki/Lambert_series" title="Lambert series">main article</a> provides several more classical, or at least well-known examples related to special <a href="/wiki/Arithmetic_functions" class="mw-redirect" title="Arithmetic functions">arithmetic functions</a> in <a href="/wiki/Number_theory" title="Number theory">number theory</a>. As an example of a Lambert series identity not given in the main article, we can show that for |x|, |xq| < 1 we have that <a href="#cite_note-4">[4]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {q^{n}x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {q^{n}x^{n^{2}}}{1-qx^{n}}}+\sum _{n=1}^{\infty }{\frac {q^{n}x^{n(n+1)}}{1-x^{n}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>q</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {q^{n}x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {q^{n}x^{n^{2}}}{1-qx^{n}}}+\sum _{n=1}^{\infty }{\frac {q^{n}x^{n(n+1)}}{1-x^{n}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6700dbee898cecb0f0f6f39f42d8b11956f5ec9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.704ex; height:7.176ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {q^{n}x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {q^{n}x^{n^{2}}}{1-qx^{n}}}+\sum _{n=1}^{\infty }{\frac {q^{n}x^{n(n+1)}}{1-x^{n}}},}"> where we have the special case identity for the generating function of the <a href="/wiki/Divisor_function" title="Divisor function">divisor function</a>, d(n) ≡ σ0(n), given by<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {x^{n^{2}}\left(1+x^{n}\right)}{1-x^{n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {x^{n^{2}}\left(1+x^{n}\right)}{1-x^{n}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5767e9318741cdf8c74a03e47499fbc82b360c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.58ex; height:7.343ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {x^{n^{2}}\left(1+x^{n}\right)}{1-x^{n}}}.}"> <div class="mw-heading mw-heading3"><h3 id="Bell_series">Bell series</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=10" title="Edit section: Bell series">edit</a>]</div> The <a href="/wiki/Bell_series" title="Bell series">Bell series</a> of a sequence an is an expression in terms of both an indeterminate x and a prime p and is given by:<a href="#cite_note-5">[5]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {BG} _{p}(a_{n};x)=\sum _{n=0}^{\infty }a_{p^{n}}x^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>BG</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {BG} _{p}(a_{n};x)=\sum _{n=0}^{\infty }a_{p^{n}}x^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae998d48783f3a1168b375afc43d1529a7f76a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.44ex; height:6.843ex;" alt="{\displaystyle \operatorname {BG} _{p}(a_{n};x)=\sum _{n=0}^{\infty }a_{p^{n}}x^{n}.}"> <div class="mw-heading mw-heading3"><h3 id="Dirichlet_series_generating_functions_(DGFs)">Dirichlet series generating functions (DGFs)</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=11" title="Edit section: Dirichlet series generating functions (DGFs)">edit</a>]</div> <a href="/wiki/Formal_Dirichlet_series" class="mw-redirect" title="Formal Dirichlet series">Formal Dirichlet series</a> are often classified as generating functions, although they are not strictly formal power series. The Dirichlet series generating function of a sequence an is:<a href="#cite_note-W56-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DG} (a_{n};s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} (a_{n};s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95187281459d56dafbf8a18ca4b883f22b168977" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.754ex; height:6.843ex;" alt="{\displaystyle \operatorname {DG} (a_{n};s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}"> The Dirichlet series generating function is especially useful when an is a <a href="/wiki/Multiplicative_function" title="Multiplicative function">multiplicative function</a>, in which case it has an <a href="/wiki/Euler_product" title="Euler product">Euler product</a> expression<a href="#cite_note-W59-7">[7]</a> in terms of the function's Bell series: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DG} (a_{n};s)=\prod _{p}\operatorname {BG} _{p}(a_{n};p^{-s})\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <msub> <mi>BG</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} (a_{n};s)=\prod _{p}\operatorname {BG} _{p}(a_{n};p^{-s})\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbe85816a3e3df510d65ad55f280b6e9920a199" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.743ex; height:5.676ex;" alt="{\displaystyle \operatorname {DG} (a_{n};s)=\prod _{p}\operatorname {BG} _{p}(a_{n};p^{-s})\,.}"> If an is a <a href="/wiki/Dirichlet_character" title="Dirichlet character">Dirichlet character</a> then its Dirichlet series generating function is called a <a href="/wiki/Dirichlet_L-series" class="mw-redirect" title="Dirichlet L-series">Dirichlet L-series</a>. We also have a relation between the pair of coefficients in the <a href="/wiki/Lambert_series" title="Lambert series">Lambert series</a> expansions above and their DGFs. Namely, we can prove that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x^{n}]\operatorname {LG} (a_{n};x)=b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>LG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x^{n}]\operatorname {LG} (a_{n};x)=b_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f54db76593f74193454360c4f9eb5f8d6e5ed3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.442ex; height:2.843ex;" alt="{\displaystyle [x^{n}]\operatorname {LG} (a_{n};x)=b_{n}}">if and only if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DG} (a_{n};s)\zeta (s)=\operatorname {DG} (b_{n};s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>DG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} (a_{n};s)\zeta (s)=\operatorname {DG} (b_{n};s),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cebfbbead7f85dc6aabb8392f60596d70493d5fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.472ex; height:2.843ex;" alt="{\displaystyle \operatorname {DG} (a_{n};s)\zeta (s)=\operatorname {DG} (b_{n};s),}">where ζ(s) is the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>.<a href="#cite_note-8">[8]</a> The sequence ak generated by a <a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a> generating function (DGF) corresponding to:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DG} (a_{k};s)=\zeta (s)^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>;</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} (a_{k};s)=\zeta (s)^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b720a01d0d9c498ea9879cf2abd0ad534dee88aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.621ex; height:2.843ex;" alt="{\displaystyle \operatorname {DG} (a_{k};s)=\zeta (s)^{m}}">has the ordinary generating function:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{k=n}a_{k}x^{k}=x+{\binom {m}{1}}\sum _{2\leq a\leq n}x^{a}+{\binom {m}{2}}{\underset {ab\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }}}x^{ab}+{\binom {m}{3}}{\underset {abc\leq n}{\sum _{a=2}^{\infty }\sum _{c=2}^{\infty }\sum _{b=2}^{\infty }}}x^{abc}+{\binom {m}{4}}{\underset {abcd\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }\sum _{c=2}^{\infty }\sum _{d=2}^{\infty }}}x^{abcd}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>m</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>≤</mo> <mi>a</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>m</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> </mrow> <mrow> <mi>a</mi> <mi>b</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>m</mi> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> </mrow> <mrow> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>m</mi> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> </mrow> <mrow> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{k=n}a_{k}x^{k}=x+{\binom {m}{1}}\sum _{2\leq a\leq n}x^{a}+{\binom {m}{2}}{\underset {ab\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }}}x^{ab}+{\binom {m}{3}}{\underset {abc\leq n}{\sum _{a=2}^{\infty }\sum _{c=2}^{\infty }\sum _{b=2}^{\infty }}}x^{abc}+{\binom {m}{4}}{\underset {abcd\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }\sum _{c=2}^{\infty }\sum _{d=2}^{\infty }}}x^{abcd}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640519030fa3531384a9a2809a572cf8d7e14634" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:104.331ex; height:9.343ex;" alt="{\displaystyle \sum _{k=1}^{k=n}a_{k}x^{k}=x+{\binom {m}{1}}\sum _{2\leq a\leq n}x^{a}+{\binom {m}{2}}{\underset {ab\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }}}x^{ab}+{\binom {m}{3}}{\underset {abc\leq n}{\sum _{a=2}^{\infty }\sum _{c=2}^{\infty }\sum _{b=2}^{\infty }}}x^{abc}+{\binom {m}{4}}{\underset {abcd\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }\sum _{c=2}^{\infty }\sum _{d=2}^{\infty }}}x^{abcd}+\cdots }"> <div class="mw-heading mw-heading3"><h3 id="Polynomial_sequence_generating_functions">Polynomial sequence generating functions</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=12" title="Edit section: Polynomial sequence generating functions">edit</a>]</div> The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of <a href="/wiki/Binomial_type" title="Binomial type">binomial type</a> are generated by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{xf(t)}=\sum _{n=0}^{\infty }{\frac {p_{n}(x)}{n!}}t^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{xf(t)}=\sum _{n=0}^{\infty }{\frac {p_{n}(x)}{n!}}t^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bfb3ec67484ece87ec341a3ad667e9cf0adaa6e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.295ex; height:6.843ex;" alt="{\displaystyle e^{xf(t)}=\sum _{n=0}^{\infty }{\frac {p_{n}(x)}{n!}}t^{n}}">where pn(x) is a sequence of polynomials and f(t) is a function of a certain form. <a href="/wiki/Sheffer_sequence" title="Sheffer sequence">Sheffer sequences</a> are generated in a similar way. See the main article <a href="/wiki/Generalized_Appell_polynomials" title="Generalized Appell polynomials">generalized Appell polynomials</a> for more information. Examples of <a href="/wiki/Polynomial_sequence" title="Polynomial sequence">polynomial sequences</a> generated by more complex generating functions include: <ul><li><a href="/wiki/Appell_polynomials" class="mw-redirect" title="Appell polynomials">Appell polynomials</a></li> <li><a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a></li> <li><a href="/wiki/Difference_polynomials" title="Difference polynomials">Difference polynomials</a></li> <li><a href="/wiki/Generalized_Appell_polynomials" title="Generalized Appell polynomials">Generalized Appell polynomials</a></li> <li><a href="/wiki/Q-difference_polynomial" title="Q-difference polynomial">q-difference polynomials</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_generating_functions">Other generating functions</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=13" title="Edit section: Other generating functions">edit</a>]</div> Other sequences generated by more complex generating functions include: <ul><li>Double exponential generating functions. For example: <a rel="nofollow" class="external text" href="https://oeis.org/search?q=1%2C1%2C2%2C2%2C3%2C5%2C5%2C7%2C10%2C15%2C15&sort=&language=&go=Search">Aitken's Array: Triangle of Numbers</a></li> <li>Hadamard products of generating functions and diagonal generating functions, and their corresponding <a href="/wiki/Generating_function_transformation#Hadamard_products_and_diagonal_generating_functions" title="Generating function transformation">integral transformations</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Convolution_polynomials">Convolution polynomials</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=14" title="Edit section: Convolution polynomials">edit</a>]</div> Knuth's article titled "Convolution Polynomials"<a href="#cite_note-9">[9]</a> defines a generalized class of convolution polynomial sequences by their special generating functions of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z)^{x}=\exp {\bigl (}x\log F(z){\bigr )}=\sum _{n=0}^{\infty }f_{n}(x)z^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mi>log</mi> <mo>⁡</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(z)^{x}=\exp {\bigl (}x\log F(z){\bigr )}=\sum _{n=0}^{\infty }f_{n}(x)z^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2481c36af50cdc09be616ec8947076a3f6a885fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.985ex; height:6.843ex;" alt="{\displaystyle F(z)^{x}=\exp {\bigl (}x\log F(z){\bigr )}=\sum _{n=0}^{\infty }f_{n}(x)z^{n},}"> for some analytic function F with a power series expansion such that F(0) = 1. We say that a family of polynomials, f0, f1, f2, ..., forms a convolution family if <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">deg</a> fn ≤ n and if the following convolution condition holds for all x, y and for all n ≥ 0: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}(x+y)=f_{n}(x)f_{0}(y)+f_{n-1}(x)f_{1}(y)+\cdots +f_{1}(x)f_{n-1}(y)+f_{0}(x)f_{n}(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}(x+y)=f_{n}(x)f_{0}(y)+f_{n-1}(x)f_{1}(y)+\cdots +f_{1}(x)f_{n-1}(y)+f_{0}(x)f_{n}(y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6a76cd6a1880f4f10fbb551e263f6537718776" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:74.143ex; height:2.843ex;" alt="{\displaystyle f_{n}(x+y)=f_{n}(x)f_{0}(y)+f_{n-1}(x)f_{1}(y)+\cdots +f_{1}(x)f_{n-1}(y)+f_{0}(x)f_{n}(y).}"> We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above. A sequence of convolution polynomials defined in the notation above has the following properties: <ul><li>The sequence n! · fn(x) is of <a href="/wiki/Binomial_type" title="Binomial type">binomial type</a></li> <li>Special values of the sequence include fn(1) = [zn] F(z) and fn(0) = δn,0, and</li> <li>For arbitrary (fixed) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,t\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,t\in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2604c2e32ced47b462417730dab22fa26a2c3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.509ex;" alt="{\displaystyle x,y,t\in \mathbb {C} }">, these polynomials satisfy convolution formulas of the form</li></ul> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f_{n}(x+y)&=\sum _{k=0}^{n}f_{k}(x)f_{n-k}(y)\\f_{n}(2x)&=\sum _{k=0}^{n}f_{k}(x)f_{n-k}(x)\\xnf_{n}(x+y)&=(x+y)\sum _{k=0}^{n}kf_{k}(x)f_{n-k}(y)\\{\frac {(x+y)f_{n}(x+y+tn)}{x+y+tn}}&=\sum _{k=0}^{n}{\frac {xf_{k}(x+tk)}{x+tk}}{\frac {yf_{n-k}(y+t(n-k))}{y+t(n-k)}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mi>n</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>k</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>t</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>t</mi> <mi>n</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>y</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{n}(x+y)&=\sum _{k=0}^{n}f_{k}(x)f_{n-k}(y)\\f_{n}(2x)&=\sum _{k=0}^{n}f_{k}(x)f_{n-k}(x)\\xnf_{n}(x+y)&=(x+y)\sum _{k=0}^{n}kf_{k}(x)f_{n-k}(y)\\{\frac {(x+y)f_{n}(x+y+tn)}{x+y+tn}}&=\sum _{k=0}^{n}{\frac {xf_{k}(x+tk)}{x+tk}}{\frac {yf_{n-k}(y+t(n-k))}{y+t(n-k)}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77980c0a505e361441c0a6620d3ee8ab9521e40f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:63.586ex; height:28.509ex;" alt="{\displaystyle {\begin{aligned}f_{n}(x+y)&=\sum _{k=0}^{n}f_{k}(x)f_{n-k}(y)\\f_{n}(2x)&=\sum _{k=0}^{n}f_{k}(x)f_{n-k}(x)\\xnf_{n}(x+y)&=(x+y)\sum _{k=0}^{n}kf_{k}(x)f_{n-k}(y)\\{\frac {(x+y)f_{n}(x+y+tn)}{x+y+tn}}&=\sum _{k=0}^{n}{\frac {xf_{k}(x+tk)}{x+tk}}{\frac {yf_{n-k}(y+t(n-k))}{y+t(n-k)}}.\end{aligned}}}"> For a fixed non-zero parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d53146084bd2654adeb7cb529c2e96ad39e6b62f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {C} }">, we have modified generating functions for these convolution polynomial sequences given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {zF_{n}(x+tn)}{(x+tn)}}=\left[z^{n}\right]{\mathcal {F}}_{t}(z)^{x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>]</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {zF_{n}(x+tn)}{(x+tn)}}=\left[z^{n}\right]{\mathcal {F}}_{t}(z)^{x},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e50addec72a9b314c4331c3358f64b766c379f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.153ex; height:6.509ex;" alt="{\displaystyle {\frac {zF_{n}(x+tn)}{(x+tn)}}=\left[z^{n}\right]{\mathcal {F}}_{t}(z)^{x},}"> where 𝓕t(z) is implicitly defined by a <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> of the form 𝓕t(z) = F(x𝓕t(z)t). Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, ⟨ fn(x) ⟩ and ⟨ gn(x) ⟩, with respective corresponding generating functions, F(z)x and G(z)x, then for arbitrary t we have the identity <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[z^{n}\right]\left(G(z)F\left(zG(z)^{t}\right)\right)^{x}=\sum _{k=0}^{n}F_{k}(x)G_{n-k}(x+tk).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>]</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[z^{n}\right]\left(G(z)F\left(zG(z)^{t}\right)\right)^{x}=\sum _{k=0}^{n}F_{k}(x)G_{n-k}(x+tk).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efe9b05190bae62e45913c208a17e369136f2ec2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.332ex; height:7.009ex;" alt="{\displaystyle \left[z^{n}\right]\left(G(z)F\left(zG(z)^{t}\right)\right)^{x}=\sum _{k=0}^{n}F_{k}(x)G_{n-k}(x+tk).}"> Examples of convolution polynomial sequences include the binomial power series, 𝓑t(z) = 1 + z𝓑t(z)t, so-termed tree polynomials, the <a href="/wiki/Bell_numbers" class="mw-redirect" title="Bell numbers">Bell numbers</a>, B(n), the <a href="/wiki/Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a>, and the <a href="/wiki/Stirling_polynomial" class="mw-redirect" title="Stirling polynomial">Stirling convolution polynomials</a>. <div class="mw-heading mw-heading2"><h2 id="Ordinary_generating_functions">Ordinary generating functions</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=15" title="Edit section: Ordinary generating functions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Examples_for_simple_sequences">Examples for simple sequences</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=16" title="Edit section: Examples for simple sequences">edit</a>]</div> Polynomials are a special case of ordinary generating functions, corresponding to finite sequences, or equivalently sequences that vanish after a certain point. These are important in that many finite sequences can usefully be interpreted as generating functions, such as the <a href="/wiki/Poincar%C3%A9_polynomial" class="mw-redirect" title="Poincaré polynomial">Poincaré polynomial</a> and others. A fundamental generating function is that of the constant sequence 1, 1, 1, 1, 1, 1, 1, 1, 1, ..., whose ordinary generating function is the <a href="/wiki/Geometric_series#Closed-form_formula" title="Geometric series">geometric series</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9d143f5d0e6857123138824f18759a42f1463ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.204ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}}.}"> The left-hand side is the <a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin series</a> expansion of the right-hand side. Alternatively, the equality can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of x0 are equal to 0). Moreover, there can be no other power series with this property. The left-hand side therefore designates the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of 1 − x in the ring of power series. Expressions for the ordinary generating function of other sequences are easily derived from this one. For instance, the substitution x → ax gives the generating function for the <a href="/wiki/Geometric_progression" title="Geometric progression">geometric sequence</a> 1, a, a2, a3, ... for any constant a: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }(ax)^{n}={\frac {1}{1-ax}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }(ax)^{n}={\frac {1}{1-ax}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b52389c241c7b900e3e8090c0bf7b99127c59e5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.086ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }(ax)^{n}={\frac {1}{1-ax}}.}"> (The equality also follows directly from the fact that the left-hand side is the Maclaurin series expansion of the right-hand side.) In particular, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}x^{n}={\frac {1}{1+x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}x^{n}={\frac {1}{1+x}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/befaebd3c29304d1bf2c068348daf1cc77babf53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.815ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}x^{n}={\frac {1}{1+x}}.}"> One can also introduce regular gaps in the sequence by replacing x by some power of x, so for instance for the sequence 1, 0, 1, 0, 1, 0, 1, 0, ... (which skips over x, x3, x5, ...) one gets the generating function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }x^{2n}={\frac {1}{1-x^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }x^{2n}={\frac {1}{1-x^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/553a587d76d4d1251d57d483d0a0623f32c436ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.08ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }x^{2n}={\frac {1}{1-x^{2}}}.}"> By squaring the initial generating function, or by finding the derivative of both sides with respect to x and making a change of running variable n → n + 1, one sees that the coefficients form the sequence 1, 2, 3, 4, 5, ..., so one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }(n+1)x^{n}={\frac {1}{(1-x)^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }(n+1)x^{n}={\frac {1}{(1-x)^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6547efb25f9c90fc7e8c53997edf78e84fa2acd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.888ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }(n+1)x^{n}={\frac {1}{(1-x)^{2}}},}"> and the third power has as coefficients the <a href="/wiki/Triangular_number" title="Triangular number">triangular numbers</a> 1, 3, 6, 10, 15, 21, ... whose term n is the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> (n + 2 2), so that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\binom {n+2}{2}}x^{n}={\frac {1}{(1-x)^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\binom {n+2}{2}}x^{n}={\frac {1}{(1-x)^{3}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15ed69282f8adecdd63796c49e97eafdf9340e49" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.886ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\binom {n+2}{2}}x^{n}={\frac {1}{(1-x)^{3}}}.}"> More generally, for any non-negative integer k and non-zero real value a, it is true that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }a^{n}{\binom {n+k}{k}}x^{n}={\frac {1}{(1-ax)^{k+1}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }a^{n}{\binom {n+k}{k}}x^{n}={\frac {1}{(1-ax)^{k+1}}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8953f45d6feba70cce8284beea2c8a20ea98cb2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.135ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }a^{n}{\binom {n+k}{k}}x^{n}={\frac {1}{(1-ax)^{k+1}}}\,.}"> Since <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\binom {n+2}{2}}-3{\binom {n+1}{1}}+{\binom {n}{0}}=2{\frac {(n+1)(n+2)}{2}}-3(n+1)+1=n^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\binom {n+2}{2}}-3{\binom {n+1}{1}}+{\binom {n}{0}}=2{\frac {(n+1)(n+2)}{2}}-3(n+1)+1=n^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb6bf2afcd6b873a190bc1bfcdd00a6a3a458cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.377ex; height:6.343ex;" alt="{\displaystyle 2{\binom {n+2}{2}}-3{\binom {n+1}{1}}+{\binom {n}{0}}=2{\frac {(n+1)(n+2)}{2}}-3(n+1)+1=n^{2},}"> one can find the ordinary generating function for the sequence 0, 1, 4, 9, 16, ... of <a href="/wiki/Square_number" title="Square number">square numbers</a> by linear combination of binomial-coefficient generating sequences: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n^{2};x)=\sum _{n=0}^{\infty }n^{2}x^{n}={\frac {2}{(1-x)^{3}}}-{\frac {3}{(1-x)^{2}}}+{\frac {1}{1-x}}={\frac {x(x+1)}{(1-x)^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n^{2};x)=\sum _{n=0}^{\infty }n^{2}x^{n}={\frac {2}{(1-x)^{3}}}-{\frac {3}{(1-x)^{2}}}+{\frac {1}{1-x}}={\frac {x(x+1)}{(1-x)^{3}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36417cb42242f8e3fab602b4ab8c69eda92bc301" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:66.352ex; height:6.843ex;" alt="{\displaystyle G(n^{2};x)=\sum _{n=0}^{\infty }n^{2}x^{n}={\frac {2}{(1-x)^{3}}}-{\frac {3}{(1-x)^{2}}}+{\frac {1}{1-x}}={\frac {x(x+1)}{(1-x)^{3}}}.}"> We may also expand alternately to generate this same sequence of squares as a sum of derivatives of the <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a> in the following form: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}G(n^{2};x)&=\sum _{n=0}^{\infty }n^{2}x^{n}\\[4px]&=\sum _{n=0}^{\infty }n(n-1)x^{n}+\sum _{n=0}^{\infty }nx^{n}\\[4px]&=x^{2}D^{2}\left[{\frac {1}{1-x}}\right]+xD\left[{\frac {1}{1-x}}\right]\\[4px]&={\frac {2x^{2}}{(1-x)^{3}}}+{\frac {x}{(1-x)^{2}}}={\frac {x(x+1)}{(1-x)^{3}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>G</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>x</mi> <mi>D</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G(n^{2};x)&=\sum _{n=0}^{\infty }n^{2}x^{n}\\[4px]&=\sum _{n=0}^{\infty }n(n-1)x^{n}+\sum _{n=0}^{\infty }nx^{n}\\[4px]&=x^{2}D^{2}\left[{\frac {1}{1-x}}\right]+xD\left[{\frac {1}{1-x}}\right]\\[4px]&={\frac {2x^{2}}{(1-x)^{3}}}+{\frac {x}{(1-x)^{2}}}={\frac {x(x+1)}{(1-x)^{3}}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129d178c08eecd1ea5ada33fc9e4cb3ec6b5a15c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.043ex; margin-bottom: -0.294ex; width:46.256ex; height:29.843ex;" alt="{\displaystyle {\begin{aligned}G(n^{2};x)&=\sum _{n=0}^{\infty }n^{2}x^{n}\\[4px]&=\sum _{n=0}^{\infty }n(n-1)x^{n}+\sum _{n=0}^{\infty }nx^{n}\\[4px]&=x^{2}D^{2}\left[{\frac {1}{1-x}}\right]+xD\left[{\frac {1}{1-x}}\right]\\[4px]&={\frac {2x^{2}}{(1-x)^{3}}}+{\frac {x}{(1-x)^{2}}}={\frac {x(x+1)}{(1-x)^{3}}}.\end{aligned}}}"> By induction, we can similarly show for positive integers m ≥ 1 that<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{m}=\sum _{j=0}^{m}{\begin{Bmatrix}m\\j\end{Bmatrix}}{\frac {n!}{(n-j)!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{m}=\sum _{j=0}^{m}{\begin{Bmatrix}m\\j\end{Bmatrix}}{\frac {n!}{(n-j)!}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb1e20e43c098072809cd6495b1b6b71b4a414da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.321ex; height:7.176ex;" alt="{\displaystyle n^{m}=\sum _{j=0}^{m}{\begin{Bmatrix}m\\j\end{Bmatrix}}{\frac {n!}{(n-j)!}},}"> where {n k} denote the <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a> and where the generating function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\frac {n!}{(n-j)!}}\,z^{n}={\frac {j!\cdot z^{j}}{(1-z)^{j+1}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>j</mi> <mo>!</mo> <mo>⋅</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {n!}{(n-j)!}}\,z^{n}={\frac {j!\cdot z^{j}}{(1-z)^{j+1}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852c0846ecf6bb81bcdb14fc0342ff8f15a53141" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.415ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {n!}{(n-j)!}}\,z^{n}={\frac {j!\cdot z^{j}}{(1-z)^{j+1}}},}"> so that we can form the analogous generating functions over the integral mth powers generalizing the result in the square case above. In particular, since we can write <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z^{k}}{(1-z)^{k+1}}}=\sum _{i=0}^{k}{\binom {k}{i}}{\frac {(-1)^{k-i}}{(1-z)^{i+1}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {z^{k}}{(1-z)^{k+1}}}=\sum _{i=0}^{k}{\binom {k}{i}}{\frac {(-1)^{k-i}}{(1-z)^{i+1}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/825af953595768e745bfba3f67ac588d8d40fd37" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.682ex; height:7.343ex;" alt="{\displaystyle {\frac {z^{k}}{(1-z)^{k+1}}}=\sum _{i=0}^{k}{\binom {k}{i}}{\frac {(-1)^{k-i}}{(1-z)^{i+1}}},}"> we can apply a well-known finite sum identity involving the <a href="/wiki/Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a> to obtain that<a href="#cite_note-12">[12]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }n^{m}z^{n}=\sum _{j=0}^{m}{\begin{Bmatrix}m+1\\j+1\end{Bmatrix}}{\frac {(-1)^{m-j}j!}{(1-z)^{j+1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mi>j</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }n^{m}z^{n}=\sum _{j=0}^{m}{\begin{Bmatrix}m+1\\j+1\end{Bmatrix}}{\frac {(-1)^{m-j}j!}{(1-z)^{j+1}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba66146f01928774970de89b80c7534da263d93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.741ex; height:7.343ex;" alt="{\displaystyle \sum _{n=0}^{\infty }n^{m}z^{n}=\sum _{j=0}^{m}{\begin{Bmatrix}m+1\\j+1\end{Bmatrix}}{\frac {(-1)^{m-j}j!}{(1-z)^{j+1}}}.}"> <div class="mw-heading mw-heading3"><h3 id="Rational_functions">Rational functions</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=17" title="Edit section: Rational functions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Linear_recursive_sequence" class="mw-redirect" title="Linear recursive sequence">Linear recursive sequence</a></div> The ordinary generating function of a sequence can be expressed as a <a href="/wiki/Rational_function" title="Rational function">rational function</a> (the ratio of two finite-degree polynomials) if and only if the sequence is a <a href="/wiki/Linear_recursive_sequence" class="mw-redirect" title="Linear recursive sequence">linear recursive sequence</a> with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear recurrence with constant coefficients; these coefficients are identical to the coefficients of the fraction denominator polynomial (so they can be directly read off). This observation shows it is easy to solve for generating functions of sequences defined by a linear <a href="/wiki/Finite_difference_equation" class="mw-redirect" title="Finite difference equation">finite difference equation</a> with constant coefficients, and then hence, for explicit closed-form formulas for the coefficients of these generating functions. The prototypical example here is to derive <a href="/wiki/Binet%27s_formula" class="mw-redirect" title="Binet's formula">Binet's formula</a> for the <a href="/wiki/Fibonacci_numbers" class="mw-redirect" title="Fibonacci numbers">Fibonacci numbers</a> via generating function techniques. We also notice that the class of rational generating functions precisely corresponds to the generating functions that enumerate quasi-polynomial sequences of the form <a href="#cite_note-GFLECT-13">[13]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}=p_{1}(n)\rho _{1}^{n}+\cdots +p_{\ell }(n)\rho _{\ell }^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}=p_{1}(n)\rho _{1}^{n}+\cdots +p_{\ell }(n)\rho _{\ell }^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/172e304e6871008f84da0ad54561d3cef4bd5eb3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.067ex; height:3.009ex;" alt="{\displaystyle f_{n}=p_{1}(n)\rho _{1}^{n}+\cdots +p_{\ell }(n)\rho _{\ell }^{n},}"> where the reciprocal roots, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{i}\in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb19b21a719205aa55c43020a9c2c68c396c1997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.52ex; height:2.676ex;" alt="{\displaystyle \rho _{i}\in \mathbb {C} }">, are fixed scalars and where pi(n) is a polynomial in n for all 1 ≤ i ≤ ℓ. In general, <a href="/wiki/Generating_function_transformation#Hadamard_products_and_diagonal_generating_functions" title="Generating function transformation">Hadamard products</a> of rational functions produce rational generating functions. Similarly, if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s,t):=\sum _{m,n\geq 0}f(m,n)w^{m}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s,t):=\sum _{m,n\geq 0}f(m,n)w^{m}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31111d18f8063c39294ac92b51903d33fd33a08c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.838ex; height:5.843ex;" alt="{\displaystyle F(s,t):=\sum _{m,n\geq 0}f(m,n)w^{m}z^{n}}"> is a bivariate rational generating function, then its corresponding diagonal generating function, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {diag} (F):=\sum _{n=0}^{\infty }f(n,n)z^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {diag} (F):=\sum _{n=0}^{\infty }f(n,n)z^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ecc6213ace8df853de50d8d3778c365db860e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.169ex; height:6.843ex;" alt="{\displaystyle \operatorname {diag} (F):=\sum _{n=0}^{\infty }f(n,n)z^{n},}"> is algebraic. For example, if we let<a href="#cite_note-14">[14]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s,t):=\sum _{i,j\geq 0}{\binom {i+j}{i}}s^{i}t^{j}={\frac {1}{1-s-t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>s</mi> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s,t):=\sum _{i,j\geq 0}{\binom {i+j}{i}}s^{i}t^{j}={\frac {1}{1-s-t}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6487c363345ffae360afb6a3169c6cb91344b6d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.466ex; height:7.009ex;" alt="{\displaystyle F(s,t):=\sum _{i,j\geq 0}{\binom {i+j}{i}}s^{i}t^{j}={\frac {1}{1-s-t}},}"> then this generating function's diagonal coefficient generating function is given by the well-known OGF formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {diag} (F)=\sum _{n=0}^{\infty }{\binom {2n}{n}}z^{n}={\frac {1}{\sqrt {1-4z}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mi>z</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {diag} (F)=\sum _{n=0}^{\infty }{\binom {2n}{n}}z^{n}={\frac {1}{\sqrt {1-4z}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce21c49b1b4627213da5e89e4a012884d00c777" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.713ex; height:6.843ex;" alt="{\displaystyle \operatorname {diag} (F)=\sum _{n=0}^{\infty }{\binom {2n}{n}}z^{n}={\frac {1}{\sqrt {1-4z}}}.}"> This result is computed in many ways, including <a href="/wiki/Cauchy%27s_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a> or <a href="/wiki/Contour_integration" title="Contour integration">contour integration</a>, taking complex <a href="/wiki/Residue_(complex_analysis)" title="Residue (complex analysis)">residues</a>, or by direct manipulations of <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a> in two variables. <div class="mw-heading mw-heading3"><h3 id="Operations_on_generating_functions">Operations on generating functions</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=18" title="Edit section: Operations on generating functions">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Multiplication_yields_convolution">Multiplication yields convolution</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=19" title="Edit section: Multiplication yields convolution">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cauchy_product" title="Cauchy product">Cauchy product</a></div> Multiplication of ordinary generating functions yields a discrete <a href="/wiki/Convolution" title="Convolution">convolution</a> (the <a href="/wiki/Cauchy_product" title="Cauchy product">Cauchy product</a>) of the sequences. For example, the sequence of cumulative sums (compare to the slightly more general <a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a>) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{0},a_{0}+a_{1},a_{0}+a_{1}+a_{2},\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{0},a_{0}+a_{1},a_{0}+a_{1}+a_{2},\ldots )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa4fc34c24d6d16750b0f4ac69171d50759922d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.86ex; height:2.843ex;" alt="{\displaystyle (a_{0},a_{0}+a_{1},a_{0}+a_{1}+a_{2},\ldots )}"> of a sequence with ordinary generating function G(an; x) has the generating function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(a_{n};x)\cdot {\frac {1}{1-x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(a_{n};x)\cdot {\frac {1}{1-x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17160c67440e1160873f0fdc8bc86385640c18b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.296ex; height:5.343ex;" alt="{\displaystyle G(a_{n};x)\cdot {\frac {1}{1-x}}}"> because <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/1 − x⁠ is the ordinary generating function for the sequence (1, 1, ...). See also the <a href="#Convolution_(Cauchy_products)">section on convolutions</a> in the applications section of this article below for further examples of problem solving with convolutions of generating functions and interpretations. <div class="mw-heading mw-heading4"><h4 id="Shifting_sequence_indices">Shifting sequence indices</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=20" title="Edit section: Shifting sequence indices">edit</a>]</div> For integers m ≥ 1, we have the following two analogous identities for the modified generating functions enumerating the shifted sequence variants of ⟨ gn − m ⟩ and ⟨ gn + m ⟩, respectively: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&z^{m}G(z)=\sum _{n=m}^{\infty }g_{n-m}z^{n}\\[4px]&{\frac {G(z)-g_{0}-g_{1}z-\cdots -g_{m-1}z^{m-1}}{z^{m}}}=\sum _{n=0}^{\infty }g_{n+m}z^{n}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>−</mo> <mo>⋯</mo> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&z^{m}G(z)=\sum _{n=m}^{\infty }g_{n-m}z^{n}\\[4px]&{\frac {G(z)-g_{0}-g_{1}z-\cdots -g_{m-1}z^{m-1}}{z^{m}}}=\sum _{n=0}^{\infty }g_{n+m}z^{n}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3052ce87c17f67792fcf7588a5cc631e826b027c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:50.407ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}&z^{m}G(z)=\sum _{n=m}^{\infty }g_{n-m}z^{n}\\[4px]&{\frac {G(z)-g_{0}-g_{1}z-\cdots -g_{m-1}z^{m-1}}{z^{m}}}=\sum _{n=0}^{\infty }g_{n+m}z^{n}.\end{aligned}}}"> <div class="mw-heading mw-heading4"><h4 id="Differentiation_and_integration_of_generating_functions">Differentiation and integration of generating functions</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=21" title="Edit section: Differentiation and integration of generating functions">edit</a>]</div> We have the following respective power series expansions for the first derivative of a generating function and its integral: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}G'(z)&=\sum _{n=0}^{\infty }(n+1)g_{n+1}z^{n}\\[4px]z\cdot G'(z)&=\sum _{n=0}^{\infty }ng_{n}z^{n}\\[4px]\int _{0}^{z}G(t)\,dt&=\sum _{n=1}^{\infty }{\frac {g_{n-1}}{n}}z^{n}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo>⋅</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msubsup> <mi>G</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G'(z)&=\sum _{n=0}^{\infty }(n+1)g_{n+1}z^{n}\\[4px]z\cdot G'(z)&=\sum _{n=0}^{\infty }ng_{n}z^{n}\\[4px]\int _{0}^{z}G(t)\,dt&=\sum _{n=1}^{\infty }{\frac {g_{n-1}}{n}}z^{n}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86d9f7073663fa6d47194c8c55e90f81418204a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:31.92ex; height:23.176ex;" alt="{\displaystyle {\begin{aligned}G'(z)&=\sum _{n=0}^{\infty }(n+1)g_{n+1}z^{n}\\[4px]z\cdot G'(z)&=\sum _{n=0}^{\infty }ng_{n}z^{n}\\[4px]\int _{0}^{z}G(t)\,dt&=\sum _{n=1}^{\infty }{\frac {g_{n-1}}{n}}z^{n}.\end{aligned}}}"> The differentiation–multiplication operation of the second identity can be repeated k times to multiply the sequence by nk, but that requires alternating between differentiation and multiplication. If instead doing k differentiations in sequence, the effect is to multiply by the kth <a href="/wiki/Falling_factorial" class="mw-redirect" title="Falling factorial">falling factorial</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{k}G^{(k)}(z)=\sum _{n=0}^{\infty }n^{\underline {k}}g_{n}z^{n}=\sum _{n=0}^{\infty }n(n-1)\dotsb (n-k+1)g_{n}z^{n}\quad {\text{for all }}k\in \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>k</mi> <mo>_</mo> </munder> </mrow> </msup> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{k}G^{(k)}(z)=\sum _{n=0}^{\infty }n^{\underline {k}}g_{n}z^{n}=\sum _{n=0}^{\infty }n(n-1)\dotsb (n-k+1)g_{n}z^{n}\quad {\text{for all }}k\in \mathbb {N} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e12fbe1df3f9a1d03fbf34afe7cd9ca5ce2135" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:73.17ex; height:6.843ex;" alt="{\displaystyle z^{k}G^{(k)}(z)=\sum _{n=0}^{\infty }n^{\underline {k}}g_{n}z^{n}=\sum _{n=0}^{\infty }n(n-1)\dotsb (n-k+1)g_{n}z^{n}\quad {\text{for all }}k\in \mathbb {N} .}"> Using the <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a>, that can be turned into another formula for multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53af70c07e0932d85d2fdb70c56360544c3a0b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.483ex; height:2.676ex;" alt="{\displaystyle n^{k}}"> as follows (see the main article on <a href="/wiki/Generating_function_transformation#Derivative_transformations" title="Generating function transformation">generating function transformations</a>): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=0}^{k}{\begin{Bmatrix}k\\j\end{Bmatrix}}z^{j}F^{(j)}(z)=\sum _{n=0}^{\infty }n^{k}f_{n}z^{n}\quad {\text{for all }}k\in \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{k}{\begin{Bmatrix}k\\j\end{Bmatrix}}z^{j}F^{(j)}(z)=\sum _{n=0}^{\infty }n^{k}f_{n}z^{n}\quad {\text{for all }}k\in \mathbb {N} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe488b92d166efc74cd993f27d62b4ba69e461db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.186ex; height:7.676ex;" alt="{\displaystyle \sum _{j=0}^{k}{\begin{Bmatrix}k\\j\end{Bmatrix}}z^{j}F^{(j)}(z)=\sum _{n=0}^{\infty }n^{k}f_{n}z^{n}\quad {\text{for all }}k\in \mathbb {N} .}"> A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the <a href="/wiki/Generating_function_transformation#Derivative_transformations" title="Generating function transformation">zeta series transformation</a> and its generalizations defined as a derivative-based <a href="/wiki/Generating_function_transformation" title="Generating function transformation">transformation of generating functions</a>, or alternately termwise by and performing an <a href="/wiki/Generating_function_transformation#Polylogarithm_series_transformations" title="Generating function transformation">integral transformation</a> on the sequence generating function. Related operations of performing <a href="/wiki/Fractional_calculus" title="Fractional calculus">fractional integration</a> on a sequence generating function are discussed <a href="/wiki/Generating_function_transformation#Fractional_integrals_and_derivatives" title="Generating function transformation">here</a>. <div class="mw-heading mw-heading4"><h4 id="Enumerating_arithmetic_progressions_of_sequences">Enumerating arithmetic progressions of sequences</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=22" title="Edit section: Enumerating arithmetic progressions of sequences">edit</a>]</div> In this section we give formulas for generating functions enumerating the sequence {fan + b} given an ordinary generating function F(z), where a ≥ 2, 0 ≤ b < a, and a and b are integers (see the <a href="/wiki/Generating_function_transformation" title="Generating function transformation">main article on transformations</a>). For a = 2, this is simply the familiar decomposition of a function into <a href="/wiki/Even_and_odd_functions" title="Even and odd functions">even and odd parts</a> (i.e., even and odd powers): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }f_{2n}z^{2n}&={\frac {F(z)+F(-z)}{2}}\\[4px]\sum _{n=0}^{\infty }f_{2n+1}z^{2n+1}&={\frac {F(z)-F(-z)}{2}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }f_{2n}z^{2n}&={\frac {F(z)+F(-z)}{2}}\\[4px]\sum _{n=0}^{\infty }f_{2n+1}z^{2n+1}&={\frac {F(z)-F(-z)}{2}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed064327749ed440e9262e3744d84a84403a977" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:33.511ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }f_{2n}z^{2n}&={\frac {F(z)+F(-z)}{2}}\\[4px]\sum _{n=0}^{\infty }f_{2n+1}z^{2n+1}&={\frac {F(z)-F(-z)}{2}}.\end{aligned}}}"> More generally, suppose that a ≥ 3 and that ωa = exp <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2πi/a⁠ denotes the ath <a href="/wiki/Root_of_unity" title="Root of unity">primitive root of unity</a>. Then, as an application of the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a>, we have the formula<a href="#cite_note-TAOCPV1-15">[15]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }f_{an+b}z^{an+b}={\frac {1}{a}}\sum _{m=0}^{a-1}\omega _{a}^{-mb}F\left(\omega _{a}^{m}z\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>n</mi> <mo>+</mo> <mi>b</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>n</mi> <mo>+</mo> <mi>b</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>m</mi> <mi>b</mi> </mrow> </msubsup> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }f_{an+b}z^{an+b}={\frac {1}{a}}\sum _{m=0}^{a-1}\omega _{a}^{-mb}F\left(\omega _{a}^{m}z\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40fc59cef8f37522b539383585a132e989cf6ff6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.882ex; height:7.343ex;" alt="{\displaystyle \sum _{n=0}^{\infty }f_{an+b}z^{an+b}={\frac {1}{a}}\sum _{m=0}^{a-1}\omega _{a}^{-mb}F\left(\omega _{a}^{m}z\right).}"> For integers m ≥ 1, another useful formula providing somewhat reversed floored arithmetic progressions — effectively repeating each coefficient m times — are generated by the identity<a href="#cite_note-16">[16]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }f_{\left\lfloor {\frac {n}{m}}\right\rfloor }z^{n}={\frac {1-z^{m}}{1-z}}F(z^{m})=\left(1+z+\cdots +z^{m-2}+z^{m-1}\right)F(z^{m}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>m</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }f_{\left\lfloor {\frac {n}{m}}\right\rfloor }z^{n}={\frac {1-z^{m}}{1-z}}F(z^{m})=\left(1+z+\cdots +z^{m-2}+z^{m-1}\right)F(z^{m}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/483d7963f8d49682384da1ec44e4f99adc22159a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:66.645ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }f_{\left\lfloor {\frac {n}{m}}\right\rfloor }z^{n}={\frac {1-z^{m}}{1-z}}F(z^{m})=\left(1+z+\cdots +z^{m-2}+z^{m-1}\right)F(z^{m}).}"> <div class="mw-heading mw-heading3"><h3 id="P-recursive_sequences_and_holonomic_generating_functions">P-recursive sequences and holonomic generating functions</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=23" title="Edit section: P-recursive sequences and holonomic generating functions">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Definitions">Definitions</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=24" title="Edit section: Definitions">edit</a>]</div> A formal power series (or function) F(z) is said to be holonomic if it satisfies a linear differential equation of the form<a href="#cite_note-17">[17]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}(z)F^{(r)}(z)+c_{1}(z)F^{(r-1)}(z)+\cdots +c_{r}(z)F(z)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}(z)F^{(r)}(z)+c_{1}(z)F^{(r-1)}(z)+\cdots +c_{r}(z)F(z)=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155cb0976d0fe6bdbc32969be546ea7294b26cb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.617ex; height:3.343ex;" alt="{\displaystyle c_{0}(z)F^{(r)}(z)+c_{1}(z)F^{(r-1)}(z)+\cdots +c_{r}(z)F(z)=0,}"> where the coefficients ci(z) are in the field of rational functions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} (z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5917f17c3d7c66273521f830ef30a8950703188c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} (z)}">. Equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf298d777120df944559c5e985b88a824debb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(z)}"> is holonomic if the vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} (z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5917f17c3d7c66273521f830ef30a8950703188c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} (z)}"> spanned by the set of all of its derivatives is finite dimensional. Since we can clear denominators if need be in the previous equation, we may assume that the functions, ci(z) are polynomials in z. Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a P-recurrence of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {c}}_{s}(n)f_{n+s}+{\widehat {c}}_{s-1}(n)f_{n+s-1}+\cdots +{\widehat {c}}_{0}(n)f_{n}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {c}}_{s}(n)f_{n+s}+{\widehat {c}}_{s-1}(n)f_{n+s-1}+\cdots +{\widehat {c}}_{0}(n)f_{n}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e26d9066f79d39db7fa1775d17db3ba4bcf2329a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.073ex; height:2.843ex;" alt="{\displaystyle {\widehat {c}}_{s}(n)f_{n+s}+{\widehat {c}}_{s-1}(n)f_{n+s-1}+\cdots +{\widehat {c}}_{0}(n)f_{n}=0,}"> for all large enough n ≥ n0 and where the ĉi(n) are fixed finite-degree polynomials in n. In other words, the properties that a sequence be P-recursive and have a holonomic generating function are equivalent. Holonomic functions are closed under the <a href="/wiki/Generating_function_transformation#Hadamard_products_and_diagonal_generating_functions" title="Generating function transformation">Hadamard product</a> operation ⊙ on generating functions. <div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=25" title="Edit section: Examples">edit</a>]</div> The functions ez, log z, cos z, arcsin z, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>z</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15eb2cb08e00208010c8524c7d374483b1f534dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.027ex; height:3.009ex;" alt="{\displaystyle {\sqrt {1+z}}}">, the <a href="/wiki/Dilogarithm" title="Dilogarithm">dilogarithm</a> function Li2(z), the <a href="/wiki/Generalized_hypergeometric_function" title="Generalized hypergeometric function">generalized hypergeometric functions</a> pFq(...; ...; z) and the functions defined by the power series <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\frac {z^{n}}{(n!)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {z^{n}}{(n!)^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a09bc0aa51e32d0c43eb5c82e9d7002f6a4237a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.483ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {z^{n}}{(n!)^{2}}}}"> and the non-convergent <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }n!\cdot z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> <mo>!</mo> <mo>⋅</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }n!\cdot z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764ccb6dc6468600949bc37f3270d22a733201b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.772ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }n!\cdot z^{n}}"> are all holonomic. Examples of P-recursive sequences with holonomic generating functions include fn ≔ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/n + 1⁠ (2n n) and fn ≔ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2n/n2 + 1⁠, where sequences such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2994734eae382ce30100fb17b9447fd8e99f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}"> and log n are not P-recursive due to the nature of singularities in their corresponding generating functions. Similarly, functions with infinitely many singularities such as tan z, sec z, and <a href="/wiki/Gamma_function" title="Gamma function">Γ(z)</a> are not holonomic functions. <div class="mw-heading mw-heading4"><h4 id="Software_for_working_with_P-recursive_sequences_and_holonomic_generating_functions">Software for working with P-recursive sequences and holonomic generating functions</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=26" title="Edit section: Software for working with P-recursive sequences and holonomic generating functions">edit</a>]</div> Tools for processing and working with P-recursive sequences in <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> include the software packages provided for non-commercial use on the <a rel="nofollow" class="external text" href="https://www.risc.jku.at/research/combinat/software/">RISC Combinatorics Group algorithmic combinatorics software</a> site. Despite being mostly closed-source, particularly powerful tools in this software suite are provided by the <code>Guess</code> package for guessing P-recurrences for arbitrary input sequences (useful for <a href="/wiki/Experimental_mathematics" title="Experimental mathematics">experimental mathematics</a> and exploration) and the <code>Sigma</code> package which is able to find P-recurrences for many sums and solve for closed-form solutions to P-recurrences involving generalized <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic numbers</a>.<a href="#cite_note-18">[18]</a> Other packages listed on this particular RISC site are targeted at working with holonomic generating functions specifically. <div class="mw-heading mw-heading3"><h3 id="Relation_to_discrete-time_Fourier_transform">Relation to discrete-time Fourier transform</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=27" title="Edit section: Relation to discrete-time Fourier transform">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a></div> When the series <a href="/wiki/Absolute_convergence" title="Absolute convergence">converges absolutely</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\left(a_{n};e^{-i\omega }\right)=\sum _{n=0}^{\infty }a_{n}e^{-i\omega n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\left(a_{n};e^{-i\omega }\right)=\sum _{n=0}^{\infty }a_{n}e^{-i\omega n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da35b9e349447bf4c47c322a87be0a66e5911227" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.469ex; height:6.843ex;" alt="{\displaystyle G\left(a_{n};e^{-i\omega }\right)=\sum _{n=0}^{\infty }a_{n}e^{-i\omega n}}"> is the discrete-time Fourier transform of the sequence a0, a1, .... <div class="mw-heading mw-heading3"><h3 id="Asymptotic_growth_of_a_sequence">Asymptotic growth of a sequence</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=28" title="Edit section: Asymptotic growth of a sequence">edit</a>]</div> In calculus, often the growth rate of the coefficients of a power series can be used to deduce a <a href="/wiki/Radius_of_convergence" title="Radius of convergence">radius of convergence</a> for the power series. The reverse can also hold; often the radius of convergence for a generating function can be used to deduce the <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotic growth</a> of the underlying sequence. For instance, if an ordinary generating function G(an; x) that has a finite radius of convergence of r can be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(a_{n};x)={\frac {A(x)+B(x)\left(1-{\frac {x}{r}}\right)^{-\beta }}{x^{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> </mrow> </msup> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(a_{n};x)={\frac {A(x)+B(x)\left(1-{\frac {x}{r}}\right)^{-\beta }}{x^{\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a30eb91ec0d4814313a36bb19af6c35ef63b6ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.37ex; height:6.676ex;" alt="{\displaystyle G(a_{n};x)={\frac {A(x)+B(x)\left(1-{\frac {x}{r}}\right)^{-\beta }}{x^{\alpha }}}}"> where each of A(x) and B(x) is a function that is <a href="/wiki/Analytic_function" title="Analytic function">analytic</a> to a radius of convergence greater than r (or is <a href="/wiki/Entire_function" title="Entire function">entire</a>), and where B(r) ≠ 0 then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}\sim {\frac {B(r)}{r^{\alpha }}}{\binom {n+\beta -1}{n}}\left({\frac {1}{r}}\right)^{n}={\frac {B(r)}{r^{\alpha }}}\left(\!\!{\binom {\beta }{n}}\!\!\right)\left({\frac {1}{r}}\right)^{n}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>β</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}\sim {\frac {B(r)}{r^{\alpha }}}{\binom {n+\beta -1}{n}}\left({\frac {1}{r}}\right)^{n}={\frac {B(r)}{r^{\alpha }}}\left(\!\!{\binom {\beta }{n}}\!\!\right)\left({\frac {1}{r}}\right)^{n}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c816bb453097307a923b6d02e9d7b5af75d26278" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:76.882ex; height:6.509ex;" alt="{\displaystyle a_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}\sim {\frac {B(r)}{r^{\alpha }}}{\binom {n+\beta -1}{n}}\left({\frac {1}{r}}\right)^{n}={\frac {B(r)}{r^{\alpha }}}\left(\!\!{\binom {\beta }{n}}\!\!\right)\left({\frac {1}{r}}\right)^{n}\,,}"> using the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>, a <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>, or a <a href="/wiki/Multiset_coefficient" class="mw-redirect" title="Multiset coefficient">multiset coefficient</a>. Note that limit as n goes to infinity of the ratio of an to any of these expressions is guaranteed to be 1; not merely that an is proportional to them. Often this approach can be iterated to generate several terms in an asymptotic series for an. In particular, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\left(a_{n}-{\frac {B(r)}{r^{\alpha }}}{\binom {n+\beta -1}{n}}\left({\frac {1}{r}}\right)^{n};x\right)=G(a_{n};x)-{\frac {B(r)}{r^{\alpha }}}\left(1-{\frac {x}{r}}\right)^{-\beta }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>;</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\left(a_{n}-{\frac {B(r)}{r^{\alpha }}}{\binom {n+\beta -1}{n}}\left({\frac {1}{r}}\right)^{n};x\right)=G(a_{n};x)-{\frac {B(r)}{r^{\alpha }}}\left(1-{\frac {x}{r}}\right)^{-\beta }\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1ad0233f1c19a7f22837bc7e0b2b3d9c294d19" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:70.65ex; height:6.343ex;" alt="{\displaystyle G\left(a_{n}-{\frac {B(r)}{r^{\alpha }}}{\binom {n+\beta -1}{n}}\left({\frac {1}{r}}\right)^{n};x\right)=G(a_{n};x)-{\frac {B(r)}{r^{\alpha }}}\left(1-{\frac {x}{r}}\right)^{-\beta }\,.}"> The asymptotic growth of the coefficients of this generating function can then be sought via the finding of A, B, α, β, and r to describe the generating function, as above. Similar asymptotic analysis is possible for exponential generating functions; with an exponential generating function, it is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠an/n!⁠ that grows according to these asymptotic formulae. Generally, if the generating function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than the radius of convergence of the individual generating functions then the two sequences have the same asymptotic growth. <div class="mw-heading mw-heading4"><h4 id="Asymptotic_growth_of_the_sequence_of_squares">Asymptotic growth of the sequence of squares</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=29" title="Edit section: Asymptotic growth of the sequence of squares">edit</a>]</div> As derived above, the ordinary generating function for the sequence of squares is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n^{2};x)={\frac {x(x+1)}{(1-x)^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n^{2};x)={\frac {x(x+1)}{(1-x)^{3}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf96abc7a2a94389f4b3c26b9049d5da2935a24b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.502ex; height:6.509ex;" alt="{\displaystyle G(n^{2};x)={\frac {x(x+1)}{(1-x)^{3}}}.}"> With r = 1, α = −1, β = 3, A(x) = 0, and B(x) = x + 1, we can verify that the squares grow as expected, like the squares: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}={\frac {1+1}{1^{-1}\,\Gamma (3)}}\,n^{3-1}\left({\frac {1}{1}}\right)^{n}=n^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}={\frac {1+1}{1^{-1}\,\Gamma (3)}}\,n^{3-1}\left({\frac {1}{1}}\right)^{n}=n^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1022d3eb36909770648d1a32f7034ff2b6c70589" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:55.015ex; height:6.676ex;" alt="{\displaystyle a_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}={\frac {1+1}{1^{-1}\,\Gamma (3)}}\,n^{3-1}\left({\frac {1}{1}}\right)^{n}=n^{2}.}"> <div class="mw-heading mw-heading4"><h4 id="Asymptotic_growth_of_the_Catalan_numbers">Asymptotic growth of the Catalan numbers</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=30" title="Edit section: Asymptotic growth of the Catalan numbers">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Catalan_number" title="Catalan number">Catalan number</a></div> The ordinary generating function for the <a href="/wiki/Catalan_number" title="Catalan number">Catalan numbers</a> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(C_{n};x)={\frac {1-{\sqrt {1-4x}}}{2x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mi>x</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(C_{n};x)={\frac {1-{\sqrt {1-4x}}}{2x}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0b91599ffac9641cd1488d4799ca7e3d42c548" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.895ex; height:5.843ex;" alt="{\displaystyle G(C_{n};x)={\frac {1-{\sqrt {1-4x}}}{2x}}.}"> With r = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠, α = 1, β = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, A(x) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, and B(x) = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, we can conclude that, for the Catalan numbers: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}={\frac {-{\frac {1}{2}}}{\left({\frac {1}{4}}\right)^{1}\Gamma \left(-{\frac {1}{2}}\right)}}\,n^{-{\frac {1}{2}}-1}\left({\frac {1}{\,{\frac {1}{4}}\,}}\right)^{n}={\frac {4^{n}}{n^{\frac {3}{2}}{\sqrt {\pi }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}={\frac {-{\frac {1}{2}}}{\left({\frac {1}{4}}\right)^{1}\Gamma \left(-{\frac {1}{2}}\right)}}\,n^{-{\frac {1}{2}}-1}\left({\frac {1}{\,{\frac {1}{4}}\,}}\right)^{n}={\frac {4^{n}}{n^{\frac {3}{2}}{\sqrt {\pi }}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b3fa5c4131baedbf626ae81842721b9508a43a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:68.339ex; height:9.509ex;" alt="{\displaystyle C_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}={\frac {-{\frac {1}{2}}}{\left({\frac {1}{4}}\right)^{1}\Gamma \left(-{\frac {1}{2}}\right)}}\,n^{-{\frac {1}{2}}-1}\left({\frac {1}{\,{\frac {1}{4}}\,}}\right)^{n}={\frac {4^{n}}{n^{\frac {3}{2}}{\sqrt {\pi }}}}.}"> <div class="mw-heading mw-heading3"><h3 id="Bivariate_and_multivariate_generating_functions">Bivariate and multivariate generating functions</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=31" title="Edit section: Bivariate and multivariate generating functions">edit</a>]</div> The generating function in several variables can be generalized to arrays with multiple indices. These non-polynomial double sum examples are called multivariate generating functions, or super generating functions. For two variables, these are often called bivariate generating functions. <div class="mw-heading mw-heading4"><h4 id="Bivariate_case">Bivariate case</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=32" title="Edit section: Bivariate case">edit</a>]</div> The ordinary generating function of a two-dimensional array am,n (where n and m are natural numbers) is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(a_{m,n};x,y)=\sum _{m,n=0}^{\infty }a_{m,n}x^{m}y^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>;</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(a_{m,n};x,y)=\sum _{m,n=0}^{\infty }a_{m,n}x^{m}y^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/638d06a7509760b056b52e61ec94d8dd3419a626" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.389ex; height:7.176ex;" alt="{\displaystyle G(a_{m,n};x,y)=\sum _{m,n=0}^{\infty }a_{m,n}x^{m}y^{n}.}">For instance, since (1 + x)n is the ordinary generating function for <a href="/wiki/Binomial_coefficients" class="mw-redirect" title="Binomial coefficients">binomial coefficients</a> for a fixed n, one may ask for a bivariate generating function that generates the binomial coefficients (n k) for all k and n. To do this, consider (1 + x)n itself as a sequence in n, and find the generating function in y that has these sequence values as coefficients. Since the generating function for an is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1-ay}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-ay}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c516e61c0df8af7d46d7cd944de8fa96b7eab0a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.871ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{1-ay}},}">the generating function for the binomial coefficients is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n,k}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-(1+x)y}}={\frac {1}{1-y-xy}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n,k}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-(1+x)y}}={\frac {1}{1-y-xy}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5095a9c833f4210d7df8250efb1eaa2eff8e907e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.655ex; height:7.009ex;" alt="{\displaystyle \sum _{n,k}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-(1+x)y}}={\frac {1}{1-y-xy}}.}">Other examples of such include the following two-variable generating functions for the <a href="/wiki/Binomial_coefficients" class="mw-redirect" title="Binomial coefficients">binomial coefficients</a>, the <a href="/wiki/Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a>, and the <a href="/wiki/Eulerian_numbers" class="mw-redirect" title="Eulerian numbers">Eulerian numbers</a>, where ω and z denote the two variables:<a href="#cite_note-19">[19]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}e^{z+wz}&=\sum _{m,n\geq 0}{\binom {n}{m}}w^{m}{\frac {z^{n}}{n!}}\\[4px]e^{w(e^{z}-1)}&=\sum _{m,n\geq 0}{\begin{Bmatrix}n\\m\end{Bmatrix}}w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {1}{(1-z)^{w}}}&=\sum _{m,n\geq 0}{\begin{bmatrix}n\\m\end{bmatrix}}w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {1-w}{e^{(w-1)z}-w}}&=\sum _{m,n\geq 0}\left\langle {\begin{matrix}n\\m\end{matrix}}\right\rangle w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {e^{w}-e^{z}}{we^{z}-ze^{w}}}&=\sum _{m,n\geq 0}\left\langle {\begin{matrix}m+n+1\\m\end{matrix}}\right\rangle {\frac {w^{m}z^{n}}{(m+n+1)!}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mi>z</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>w</mi> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>w</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>z</mi> </mrow> </msup> <mo>−</mo> <mi>w</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> <mo>⟩</mo> </mrow> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mrow> <mrow> <mi>w</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo>−</mo> <mi>z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{z+wz}&=\sum _{m,n\geq 0}{\binom {n}{m}}w^{m}{\frac {z^{n}}{n!}}\\[4px]e^{w(e^{z}-1)}&=\sum _{m,n\geq 0}{\begin{Bmatrix}n\\m\end{Bmatrix}}w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {1}{(1-z)^{w}}}&=\sum _{m,n\geq 0}{\begin{bmatrix}n\\m\end{bmatrix}}w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {1-w}{e^{(w-1)z}-w}}&=\sum _{m,n\geq 0}\left\langle {\begin{matrix}n\\m\end{matrix}}\right\rangle w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {e^{w}-e^{z}}{we^{z}-ze^{w}}}&=\sum _{m,n\geq 0}\left\langle {\begin{matrix}m+n+1\\m\end{matrix}}\right\rangle {\frac {w^{m}z^{n}}{(m+n+1)!}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e60055d1b40ff6690b0555231f439c736fc54a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.805ex; margin-bottom: -0.2ex; width:50.327ex; height:39.176ex;" alt="{\displaystyle {\begin{aligned}e^{z+wz}&=\sum _{m,n\geq 0}{\binom {n}{m}}w^{m}{\frac {z^{n}}{n!}}\\[4px]e^{w(e^{z}-1)}&=\sum _{m,n\geq 0}{\begin{Bmatrix}n\\m\end{Bmatrix}}w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {1}{(1-z)^{w}}}&=\sum _{m,n\geq 0}{\begin{bmatrix}n\\m\end{bmatrix}}w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {1-w}{e^{(w-1)z}-w}}&=\sum _{m,n\geq 0}\left\langle {\begin{matrix}n\\m\end{matrix}}\right\rangle w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {e^{w}-e^{z}}{we^{z}-ze^{w}}}&=\sum _{m,n\geq 0}\left\langle {\begin{matrix}m+n+1\\m\end{matrix}}\right\rangle {\frac {w^{m}z^{n}}{(m+n+1)!}}.\end{aligned}}}"> <div class="mw-heading mw-heading4"><h4 id="Multivariate_case">Multivariate case</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=33" title="Edit section: Multivariate case">edit</a>]</div> Multivariate generating functions arise in practice when calculating the number of <a href="/wiki/Contingency_tables" class="mw-redirect" title="Contingency tables">contingency tables</a> of non-negative integers with specified row and column totals. Suppose the table has r rows and c columns; the row sums are t1, t2 ... tr and the column sums are s1, s2 ... sc. Then, according to <a href="/wiki/I._J._Good" title="I. J. Good">I. J. Good</a>,<a href="#cite_note-Good_1986-20">[20]</a> the number of such tables is the coefficient of: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{t_{1}}\cdots x_{r}^{t_{r}}y_{1}^{s_{1}}\cdots y_{c}^{s_{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}^{t_{1}}\cdots x_{r}^{t_{r}}y_{1}^{s_{1}}\cdots y_{c}^{s_{c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb7b8a4da4dd1cf673ed4b278667382712f13a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.805ex; height:3.343ex;" alt="{\displaystyle x_{1}^{t_{1}}\cdots x_{r}^{t_{r}}y_{1}^{s_{1}}\cdots y_{c}^{s_{c}}}">in:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{i=1}^{r}\prod _{j=1}^{c}{\frac {1}{1-x_{i}y_{j}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munderover> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{i=1}^{r}\prod _{j=1}^{c}{\frac {1}{1-x_{i}y_{j}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb418a2a75c63ffda303d0bbc2648b1758bf3a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.377ex; height:7.176ex;" alt="{\displaystyle \prod _{i=1}^{r}\prod _{j=1}^{c}{\frac {1}{1-x_{i}y_{j}}}.}"> <div class="mw-heading mw-heading3"><h3 id="Representation_by_continued_fractions_(Jacobi-type_J-fractions)">Representation by continued fractions (Jacobi-type J-fractions)</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=34" title="Edit section: Representation by continued fractions (Jacobi-type J-fractions)">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Definitions_2">Definitions</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=35" title="Edit section: Definitions">edit</a>]</div> Expansions of (formal) Jacobi-type and Stieltjes-type <a href="/wiki/Generalized_continued_fraction" class="mw-redirect" title="Generalized continued fraction">continued fractions</a> (J-fractions and S-fractions, respectively) whose hth rational convergents represent <a href="/wiki/Order_of_accuracy" title="Order of accuracy">2h-order accurate</a> power series are another way to express the typically divergent ordinary generating functions for many special one and two-variate sequences. The particular form of the <a href="/w/index.php?title=Jacobi-type_continued_fraction&action=edit&redlink=1" class="new" title="Jacobi-type continued fraction (page does not exist)">Jacobi-type continued fractions</a> (J-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to z for some specific, application-dependent component sequences, {abi} and {ci}, where z ≠ 0 denotes the formal variable in the second power series expansion given below:<a href="#cite_note-21">[21]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}J^{[\infty ]}(z)&={\cfrac {1}{1-c_{1}z-{\cfrac {{\text{ab}}_{2}z^{2}}{1-c_{2}z-{\cfrac {{\text{ab}}_{3}z^{2}}{\ddots }}}}}}\\[4px]&=1+c_{1}z+\left({\text{ab}}_{2}+c_{1}^{2}\right)z^{2}+\left(2{\text{ab}}_{2}c_{1}+c_{1}^{3}+{\text{ab}}_{2}c_{2}\right)z^{3}+\cdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">]</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J^{[\infty ]}(z)&={\cfrac {1}{1-c_{1}z-{\cfrac {{\text{ab}}_{2}z^{2}}{1-c_{2}z-{\cfrac {{\text{ab}}_{3}z^{2}}{\ddots }}}}}}\\[4px]&=1+c_{1}z+\left({\text{ab}}_{2}+c_{1}^{2}\right)z^{2}+\left(2{\text{ab}}_{2}c_{1}+c_{1}^{3}+{\text{ab}}_{2}c_{2}\right)z^{3}+\cdots \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f869868261347bd95267a6102ca604e32b15749" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.671ex; width:67.238ex; height:20.509ex;" alt="{\displaystyle {\begin{aligned}J^{[\infty ]}(z)&={\cfrac {1}{1-c_{1}z-{\cfrac {{\text{ab}}_{2}z^{2}}{1-c_{2}z-{\cfrac {{\text{ab}}_{3}z^{2}}{\ddots }}}}}}\\[4px]&=1+c_{1}z+\left({\text{ab}}_{2}+c_{1}^{2}\right)z^{2}+\left(2{\text{ab}}_{2}c_{1}+c_{1}^{3}+{\text{ab}}_{2}c_{2}\right)z^{3}+\cdots \end{aligned}}}"> The coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1a8cdd7ee39054e510deeb38ee551cc7616ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.309ex; height:2.343ex;" alt="{\displaystyle z^{n}}">, denoted in shorthand by jn ≔ [zn] J[∞](z), in the previous equations correspond to matrix solutions of the equations: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}k_{0,1}&k_{1,1}&0&0&\cdots \\k_{0,2}&k_{1,2}&k_{2,2}&0&\cdots \\k_{0,3}&k_{1,3}&k_{2,3}&k_{3,3}&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}}={\begin{bmatrix}k_{0,0}&0&0&0&\cdots \\k_{0,1}&k_{1,1}&0&0&\cdots \\k_{0,2}&k_{1,2}&k_{2,2}&0&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}}\cdot {\begin{bmatrix}c_{1}&1&0&0&\cdots \\{\text{ab}}_{2}&c_{2}&1&0&\cdots \\0&{\text{ab}}_{3}&c_{3}&1&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}k_{0,1}&k_{1,1}&0&0&\cdots \\k_{0,2}&k_{1,2}&k_{2,2}&0&\cdots \\k_{0,3}&k_{1,3}&k_{2,3}&k_{3,3}&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}}={\begin{bmatrix}k_{0,0}&0&0&0&\cdots \\k_{0,1}&k_{1,1}&0&0&\cdots \\k_{0,2}&k_{1,2}&k_{2,2}&0&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}}\cdot {\begin{bmatrix}c_{1}&1&0&0&\cdots \\{\text{ab}}_{2}&c_{2}&1&0&\cdots \\0&{\text{ab}}_{3}&c_{3}&1&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec568544855697bd3ec9c34a068cb81edd3ce0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:89.24ex; height:14.509ex;" alt="{\displaystyle {\begin{bmatrix}k_{0,1}&k_{1,1}&0&0&\cdots \\k_{0,2}&k_{1,2}&k_{2,2}&0&\cdots \\k_{0,3}&k_{1,3}&k_{2,3}&k_{3,3}&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}}={\begin{bmatrix}k_{0,0}&0&0&0&\cdots \\k_{0,1}&k_{1,1}&0&0&\cdots \\k_{0,2}&k_{1,2}&k_{2,2}&0&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}}\cdot {\begin{bmatrix}c_{1}&1&0&0&\cdots \\{\text{ab}}_{2}&c_{2}&1&0&\cdots \\0&{\text{ab}}_{3}&c_{3}&1&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}},}"> where j0 ≡ k0,0 = 1, jn = k0,n for n ≥ 1, kr,s = 0 if r > s, and where for all integers p, q ≥ 0, we have an addition formula relation given by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p+q}=k_{0,p}\cdot k_{0,q}+\sum _{i=1}^{\min(p,q)}{\text{ab}}_{2}\cdots {\text{ab}}_{i+1}\times k_{i,p}\cdot k_{i,q}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>q</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{p+q}=k_{0,p}\cdot k_{0,q}+\sum _{i=1}^{\min(p,q)}{\text{ab}}_{2}\cdots {\text{ab}}_{i+1}\times k_{i,p}\cdot k_{i,q}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78edb316c29c9bd533cba5edf52653b75360584" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.027ex; width:49.221ex; height:7.676ex;" alt="{\displaystyle j_{p+q}=k_{0,p}\cdot k_{0,q}+\sum _{i=1}^{\min(p,q)}{\text{ab}}_{2}\cdots {\text{ab}}_{i+1}\times k_{i,p}\cdot k_{i,q}.}"> <div class="mw-heading mw-heading4"><h4 id="Properties_of_the_hth_convergent_functions">Properties of the hth convergent functions</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=36" title="Edit section: Properties of the hth convergent functions">edit</a>]</div> For h ≥ 0 (though in practice when h ≥ 2), we can define the rational hth convergents to the infinite J-fraction, J[∞](z), expanded by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Conv} _{h}(z):={\frac {P_{h}(z)}{Q_{h}(z)}}=j_{0}+j_{1}z+\cdots +j_{2h-1}z^{2h-1}+\sum _{n=2h}^{\infty }{\widetilde {j}}_{h,n}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Conv</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo>~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Conv} _{h}(z):={\frac {P_{h}(z)}{Q_{h}(z)}}=j_{0}+j_{1}z+\cdots +j_{2h-1}z^{2h-1}+\sum _{n=2h}^{\infty }{\widetilde {j}}_{h,n}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5828712300158edd4137f0c3c26d470da107763d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.995ex; height:6.843ex;" alt="{\displaystyle \operatorname {Conv} _{h}(z):={\frac {P_{h}(z)}{Q_{h}(z)}}=j_{0}+j_{1}z+\cdots +j_{2h-1}z^{2h-1}+\sum _{n=2h}^{\infty }{\widetilde {j}}_{h,n}z^{n}}"> component-wise through the sequences, Ph(z) and Qh(z), defined recursively by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P_{h}(z)&=(1-c_{h}z)P_{h-1}(z)-{\text{ab}}_{h}z^{2}P_{h-2}(z)+\delta _{h,1}\\Q_{h}(z)&=(1-c_{h}z)Q_{h-1}(z)-{\text{ab}}_{h}z^{2}Q_{h-2}(z)+(1-c_{1}z)\delta _{h,1}+\delta _{0,1}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>ab</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{h}(z)&=(1-c_{h}z)P_{h-1}(z)-{\text{ab}}_{h}z^{2}P_{h-2}(z)+\delta _{h,1}\\Q_{h}(z)&=(1-c_{h}z)Q_{h-1}(z)-{\text{ab}}_{h}z^{2}Q_{h-2}(z)+(1-c_{1}z)\delta _{h,1}+\delta _{0,1}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6075afe592c8226ac56aff08723a9c9c71176212" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:65.646ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}P_{h}(z)&=(1-c_{h}z)P_{h-1}(z)-{\text{ab}}_{h}z^{2}P_{h-2}(z)+\delta _{h,1}\\Q_{h}(z)&=(1-c_{h}z)Q_{h-1}(z)-{\text{ab}}_{h}z^{2}Q_{h-2}(z)+(1-c_{1}z)\delta _{h,1}+\delta _{0,1}.\end{aligned}}}"> Moreover, the rationality of the convergent function Convh(z) for all h ≥ 2 implies additional finite difference equations and congruence properties satisfied by the sequence of jn, and for Mh ≔ ab2 ⋯ abh + 1 if h ‖ Mh then we have the congruence <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{n}\equiv [z^{n}]\operatorname {Conv} _{h}(z){\pmod {h}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <msub> <mi>Conv</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{n}\equiv [z^{n}]\operatorname {Conv} _{h}(z){\pmod {h}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b86ab45619fafcc3d417f21200fb680a2a79f08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:30.398ex; height:2.843ex;" alt="{\displaystyle j_{n}\equiv [z^{n}]\operatorname {Conv} _{h}(z){\pmod {h}},}"> for non-symbolic, determinate choices of the parameter sequences {abi} and {ci} when h ≥ 2, that is, when these sequences do not implicitly depend on an auxiliary parameter such as q, x, or R as in the examples contained in the table below. <div class="mw-heading mw-heading4"><h4 id="Examples_2">Examples</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=37" title="Edit section: Examples">edit</a>]</div> The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references<a href="#cite_note-22">[22]</a>) in several special cases of the prescribed sequences, jn, generated by the general expansions of the J-fractions defined in the first subsection. Here we define 0 < |a|, |b|, |q| < 1 and the parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,\alpha \in \mathbb {Z} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>α</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,\alpha \in \mathbb {Z} ^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcd7d07bc0704a132024cbb675ce79aa7d292f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.187ex; height:2.843ex;" alt="{\displaystyle R,\alpha \in \mathbb {Z} ^{+}}"> and x to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these J-fractions are defined in terms of the <a href="/wiki/Q-Pochhammer_symbol" title="Q-Pochhammer symbol">q-Pochhammer symbol</a>, <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a>, and the <a href="/wiki/Binomial_coefficients" class="mw-redirect" title="Binomial coefficients">binomial coefficients</a>. <dl><dd><table class="wikitable"> <tbody><tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59cafe09467c3b2a6aa18cbf0218f2c8f2af233e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.203ex; height:2.509ex;" alt="{\displaystyle j_{n}}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7dc6d279091d354e0b90889b463bfa7eb7247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{1}}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}(i\geq 2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo>≥</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}(i\geq 2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce515bcdb92513a1fb01a473ac42f56bf866fa9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.679ex; height:2.843ex;" alt="{\displaystyle c_{i}(i\geq 2)}"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ab} _{i}(i\geq 2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo>≥</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ab} _{i}(i\geq 2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b00de46c913d194d0bd49e51b6cc8d218b76d79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.127ex; height:2.843ex;" alt="{\displaystyle \mathrm {ab} _{i}(i\geq 2)}"> </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{n^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{n^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94913eda8b7f2758179098c83a5e58055b8deaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.129ex; height:3.343ex;" alt="{\displaystyle q^{n^{2}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>h</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>h</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>h</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b61e79b425446d3ea1b6e1e9db4b3a738c22d371" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.803ex; height:3.343ex;" alt="{\displaystyle q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{6h-10}\left(q^{2h-2}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mi>h</mi> <mo>−</mo> <mn>10</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>h</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{6h-10}\left(q^{2h-2}-1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86e9ba3a296b05a3bfff95437d92a1fceee3e32a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.704ex; height:3.343ex;" alt="{\displaystyle q^{6h-10}\left(q^{2h-2}-1\right)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a;q)_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>;</mo> <mi>q</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a;q)_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a132fadd647f6ffcf79efb2ec0534e64aaca3e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.843ex;" alt="{\displaystyle (a;q)_{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c342073d5dde33970fc5ae7c6f06a908a991a6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\displaystyle 1-a}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{h-1}-aq^{h-2}\left(q^{h}+q^{h-1}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{h-1}-aq^{h-2}\left(q^{h}+q^{h-1}-1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5edd5e057fc6b6c0593448004a47e9525cd2cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.766ex; height:3.343ex;" alt="{\displaystyle q^{h-1}-aq^{h-2}\left(q^{h}+q^{h-1}-1\right)}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aq^{2h-4}\left(aq^{h-2}-1\right)\left(q^{h-1}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>h</mi> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aq^{2h-4}\left(aq^{h-2}-1\right)\left(q^{h-1}-1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da03e96f9500bfd3a1fe6110c79ad732a70bb002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.398ex; height:3.343ex;" alt="{\displaystyle aq^{2h-4}\left(aq^{h-2}-1\right)\left(q^{h-1}-1\right)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(zq^{-n};q\right)_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>;</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(zq^{-n};q\right)_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e9ec90acce2a176a1bb82a2129d9837f97ca5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.116ex; height:3.343ex;" alt="{\displaystyle \left(zq^{-n};q\right)_{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q-z}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−</mo> <mi>z</mi> </mrow> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q-z}{q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7501bc38e1cde9ad383527365778e000555a5fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.834ex; height:5.676ex;" alt="{\displaystyle {\frac {q-z}{q}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q^{h}-z-qz+q^{h}z}{q^{2h-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>q</mi> <mi>z</mi> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> <mi>z</mi> </mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q^{h}-z-qz+q^{h}z}{q^{2h-1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abcc2f7b950c8cbdb75a2719d9b6e67654422a76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.208ex; height:6.343ex;" alt="{\displaystyle {\frac {q^{h}-z-qz+q^{h}z}{q^{2h-1}}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\left(q^{h-1}-1\right)\left(q^{h-1}-z\right)\cdot z}{q^{4h-5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>z</mi> </mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>h</mi> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left(q^{h-1}-1\right)\left(q^{h-1}-z\right)\cdot z}{q^{4h-5}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f54919399b8b61b180deba5b1825ffd50ac510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.899ex; height:6.843ex;" alt="{\displaystyle {\frac {\left(q^{h-1}-1\right)\left(q^{h-1}-z\right)\cdot z}{q^{4h-5}}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(a;q)_{n}}{(b;q)_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>;</mo> <mi>q</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>;</mo> <mi>q</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(a;q)_{n}}{(b;q)_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c338d85ad257558c20d84dbee31928d186a9841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:7.197ex; height:6.509ex;" alt="{\displaystyle {\frac {(a;q)_{n}}{(b;q)_{n}}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1-a}{1-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1-a}{1-b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e015a034fc9f267b2968986b4449894fa27c2b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:6.069ex; height:5.509ex;" alt="{\displaystyle {\frac {1-a}{1-b}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q^{i-2}\left(q+abq^{2i-3}+a(1-q^{i-1}-q^{i})+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mo>+</mo> <mi>a</mi> <mi>b</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>−</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>b</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>b</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q^{i-2}\left(q+abq^{2i-3}+a(1-q^{i-1}-q^{i})+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98184553eaf23e32a03eb6093fbac6c798e77404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.292ex; height:6.843ex;" alt="{\displaystyle {\frac {q^{i-2}\left(q+abq^{2i-3}+a(1-q^{i-1}-q^{i})+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^{2}\left(1-bq^{2i-3}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>b</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>b</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>b</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>b</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^{2}\left(1-bq^{2i-3}\right)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409ba47f4c63bba7d277766676139e1de3c3f73d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.928ex; height:7.343ex;" alt="{\displaystyle {\frac {q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^{2}\left(1-bq^{2i-3}\right)}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{n}\cdot \left({\frac {R}{\alpha }}\right)_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mi>α</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{n}\cdot \left({\frac {R}{\alpha }}\right)_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf56d57113459e78d34f68bfaf96abdd51f2176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.625ex; height:6.176ex;" alt="{\displaystyle \alpha ^{n}\cdot \left({\frac {R}{\alpha }}\right)_{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R+2\alpha (i-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>+</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R+2\alpha (i-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f639e71f1277799a5d310a322e4a317f7c1f2220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.869ex; height:2.843ex;" alt="{\displaystyle R+2\alpha (i-1)}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i-1)\alpha {\bigl (}R+(i-2)\alpha {\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>R</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i-1)\alpha {\bigl (}R+(i-2)\alpha {\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa3d1e134ae5c3b9cd832e5f641d764cf118d1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.939ex; height:3.176ex;" alt="{\displaystyle (i-1)\alpha {\bigl (}R+(i-2)\alpha {\bigr )}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{n}{\binom {x}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{n}{\binom {x}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329ef6d4ad4cfe4989264a7fc6cb446a2a46c901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.814ex; height:6.176ex;" alt="{\displaystyle (-1)^{n}{\binom {x}{n}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae55e66aeffc525917eed885b4b753ba5a7f8b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle -x}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {(x+2(i-1)^{2})}{(2i-1)(2i-3)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {(x+2(i-1)^{2})}{(2i-1)(2i-3)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0c192b09031df6047478c06054dd82ead898bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.198ex; height:6.676ex;" alt="{\displaystyle -{\frac {(x+2(i-1)^{2})}{(2i-1)(2i-3)}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}-{\dfrac {(x-i+2)(x+i-1)}{4\cdot (2i-3)^{2}}}&{\text{for }}i\geq 3;\\[4px]-{\frac {1}{2}}x(x+1)&{\text{for }}i=2.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>i</mi> <mo>≥</mo> <mn>3</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>2.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}-{\dfrac {(x-i+2)(x+i-1)}{4\cdot (2i-3)^{2}}}&{\text{for }}i\geq 3;\\[4px]-{\frac {1}{2}}x(x+1)&{\text{for }}i=2.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2863efa002e10d228d6012c3c7dcf9d9e5c03f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:38.43ex; height:10.843ex;" alt="{\displaystyle {\begin{cases}-{\dfrac {(x-i+2)(x+i-1)}{4\cdot (2i-3)^{2}}}&{\text{for }}i\geq 3;\\[4px]-{\frac {1}{2}}x(x+1)&{\text{for }}i=2.\end{cases}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{n}{\binom {x+n}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>x</mi> <mo>+</mo> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{n}{\binom {x+n}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a415011f9f1b166e83e8673a4314e21836bf89b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.984ex; height:6.176ex;" alt="{\displaystyle (-1)^{n}{\binom {x+n}{n}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(x+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(x+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e2106075c786e8568003a498bad678028bdead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.95ex; height:2.843ex;" alt="{\displaystyle -(x+1)}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\bigl (}x-2i(i-2)-1{\bigr )}}{(2i-1)(2i-3)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\bigl (}x-2i(i-2)-1{\bigr )}}{(2i-1)(2i-3)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2afd6cf2f612d4fb739355e314fedf1780344c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.718ex; height:6.843ex;" alt="{\displaystyle {\frac {{\bigl (}x-2i(i-2)-1{\bigr )}}{(2i-1)(2i-3)}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}-{\dfrac {(x-i+2)(x+i-1)}{4\cdot (2i-3)^{2}}}&{\text{for }}i\geq 3;\\[4px]-{\frac {1}{2}}x(x+1)&{\text{for }}i=2.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>i</mi> <mo>≥</mo> <mn>3</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>2.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}-{\dfrac {(x-i+2)(x+i-1)}{4\cdot (2i-3)^{2}}}&{\text{for }}i\geq 3;\\[4px]-{\frac {1}{2}}x(x+1)&{\text{for }}i=2.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2863efa002e10d228d6012c3c7dcf9d9e5c03f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:38.43ex; height:10.843ex;" alt="{\displaystyle {\begin{cases}-{\dfrac {(x-i+2)(x+i-1)}{4\cdot (2i-3)^{2}}}&{\text{for }}i\geq 3;\\[4px]-{\frac {1}{2}}x(x+1)&{\text{for }}i=2.\end{cases}}}"> </td></tr></tbody></table></dd></dl> The radii of convergence of these series corresponding to the definition of the Jacobi-type J-fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences. <div class="mw-heading mw-heading2"><h2 id="Examples_3">Examples</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=38" title="Edit section: Examples">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Square_numbers">Square numbers</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=39" title="Edit section: Square numbers">edit</a>]</div> Generating functions for the sequence of <a href="/wiki/Square_number" title="Square number">square numbers</a> an = n2 are: <table class="wikitable"> <tbody><tr> <th>Generating function type</th> <th>Equation </th></tr> <tr> <td>Ordinary generating function</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n^{2};x)=\sum _{n=0}^{\infty }n^{2}x^{n}={\frac {x(x+1)}{(1-x)^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n^{2};x)=\sum _{n=0}^{\infty }n^{2}x^{n}={\frac {x(x+1)}{(1-x)^{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c69b1bcee224e3c6009e9192f09c47b431f8bf61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.692ex; height:6.843ex;" alt="{\displaystyle G(n^{2};x)=\sum _{n=0}^{\infty }n^{2}x^{n}={\frac {x(x+1)}{(1-x)^{3}}}}"> </td></tr> <tr> <td>Exponential generating function</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {EG} (n^{2};x)=\sum _{n=0}^{\infty }{\frac {n^{2}x^{n}}{n!}}=x(x+1)e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>EG</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {EG} (n^{2};x)=\sum _{n=0}^{\infty }{\frac {n^{2}x^{n}}{n!}}=x(x+1)e^{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cc6f6056f7ee3631e0edb68f3fcc3dd4237fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.529ex; height:6.843ex;" alt="{\displaystyle \operatorname {EG} (n^{2};x)=\sum _{n=0}^{\infty }{\frac {n^{2}x^{n}}{n!}}=x(x+1)e^{x}}"> </td></tr> <tr> <td>Bell series</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {BG} _{p}\left(n^{2};x\right)=\sum _{n=0}^{\infty }\left(p^{n}\right)^{2}x^{n}={\frac {1}{1-p^{2}x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>BG</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {BG} _{p}\left(n^{2};x\right)=\sum _{n=0}^{\infty }\left(p^{n}\right)^{2}x^{n}={\frac {1}{1-p^{2}x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b47657d57c3595f32e31cedf231b9c9e364fbbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.602ex; height:6.843ex;" alt="{\displaystyle \operatorname {BG} _{p}\left(n^{2};x\right)=\sum _{n=0}^{\infty }\left(p^{n}\right)^{2}x^{n}={\frac {1}{1-p^{2}x}}}"> </td></tr> <tr> <td>Dirichlet series</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DG} \left(n^{2};s\right)=\sum _{n=1}^{\infty }{\frac {n^{2}}{n^{s}}}=\zeta (s-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DG</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} \left(n^{2};s\right)=\sum _{n=1}^{\infty }{\frac {n^{2}}{n^{s}}}=\zeta (s-2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34277b6b18a85810a8f3c87cb6f56954ae9a9fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.525ex; height:6.843ex;" alt="{\displaystyle \operatorname {DG} \left(n^{2};s\right)=\sum _{n=1}^{\infty }{\frac {n^{2}}{n^{s}}}=\zeta (s-2)}"> </td></tr></tbody></table> where ζ(s) is the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=40" title="Edit section: Applications">edit</a>]</div> Generating functions are used to: <ul><li>Find a <a href="/wiki/Closed_formula" class="mw-redirect" title="Closed formula">closed formula</a> for a sequence given in a recurrence relation, for example, <a href="/wiki/Fibonacci_number#Generating_function" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a>.</li> <li>Find <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relations</a> for sequences—the form of a generating function may suggest a recurrence formula.</li> <li>Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related.</li> <li>Explore the asymptotic behaviour of sequences.</li> <li>Prove identities involving sequences.</li> <li>Solve <a href="/wiki/Enumeration" title="Enumeration">enumeration</a> problems in <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> and encoding their solutions. <a href="/wiki/Rook_polynomial" title="Rook polynomial">Rook polynomials</a> are an example of an application in combinatorics.</li> <li>Evaluate infinite sums.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Various_techniques:_Evaluating_sums_and_tackling_other_problems_with_generating_functions">Various techniques: Evaluating sums and tackling other problems with generating functions</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=41" title="Edit section: Various techniques: Evaluating sums and tackling other problems with generating functions">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Example_1:_Formula_for_sums_of_harmonic_numbers">Example 1: Formula for sums of harmonic numbers</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=42" title="Edit section: Example 1: Formula for sums of harmonic numbers">edit</a>]</div> Generating functions give us several methods to manipulate sums and to establish identities between sums. The simplest case occurs when sn = Σn k = 0 ak. We then know that S(z) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠A(z)/1 − z⁠ for the corresponding ordinary generating functions. For example, we can manipulate <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}=\sum _{k=1}^{n}H_{k}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}=\sum _{k=1}^{n}H_{k}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a856687382493ffc9d9c0fd3ec6066ce369e49d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.203ex; height:6.843ex;" alt="{\displaystyle s_{n}=\sum _{k=1}^{n}H_{k}\,,}"> where Hk = 1 + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + ⋯ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k⁠ are the <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic numbers</a>. Let <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(z)=\sum _{n=1}^{\infty }{H_{n}z^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(z)=\sum _{n=1}^{\infty }{H_{n}z^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ff198701e3824ed9527814486349f16a486db2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.26ex; height:6.843ex;" alt="{\displaystyle H(z)=\sum _{n=1}^{\infty }{H_{n}z^{n}}}"> be the ordinary generating function of the harmonic numbers. Then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(z)={\frac {1}{1-z}}\sum _{n=1}^{\infty }{\frac {z^{n}}{n}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(z)={\frac {1}{1-z}}\sum _{n=1}^{\infty }{\frac {z^{n}}{n}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19423bb1b021ca2085be8e2ddfc705f172d807f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.295ex; height:6.843ex;" alt="{\displaystyle H(z)={\frac {1}{1-z}}\sum _{n=1}^{\infty }{\frac {z^{n}}{n}}\,,}"> and thus <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(z)=\sum _{n=1}^{\infty }{s_{n}z^{n}}={\frac {1}{(1-z)^{2}}}\sum _{n=1}^{\infty }{\frac {z^{n}}{n}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(z)=\sum _{n=1}^{\infty }{s_{n}z^{n}}={\frac {1}{(1-z)^{2}}}\sum _{n=1}^{\infty }{\frac {z^{n}}{n}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7e408273c6ec143b9aab97afe11eed31a7b6b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.052ex; height:6.843ex;" alt="{\displaystyle S(z)=\sum _{n=1}^{\infty }{s_{n}z^{n}}={\frac {1}{(1-z)^{2}}}\sum _{n=1}^{\infty }{\frac {z^{n}}{n}}\,.}"> Using <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(1-z)^{2}}}=\sum _{n=0}^{\infty }(n+1)z^{n}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1-z)^{2}}}=\sum _{n=0}^{\infty }(n+1)z^{n}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee32b6e85facd8b78ee0f0917b718d5a057241a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.794ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{(1-z)^{2}}}=\sum _{n=0}^{\infty }(n+1)z^{n}\,,}"> <a href="#Convolution_(Cauchy_products)">convolution</a> with the numerator yields <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}=\sum _{k=1}^{n}{\frac {n+1-k}{k}}=(n+1)H_{n}-n\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}=\sum _{k=1}^{n}{\frac {n+1-k}{k}}=(n+1)H_{n}-n\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0497cb91c08cd76ff96ef09706ca63a71daf4b0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.159ex; height:6.843ex;" alt="{\displaystyle s_{n}=\sum _{k=1}^{n}{\frac {n+1-k}{k}}=(n+1)H_{n}-n\,,}"> which can also be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}{H_{k}}=(n+1)(H_{n+1}-1)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}{H_{k}}=(n+1)(H_{n+1}-1)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a85c350400680d1d420f41ac4cfd39ef186577" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.164ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}{H_{k}}=(n+1)(H_{n+1}-1)\,.}"> <div class="mw-heading mw-heading4"><h4 id="Example_2:_Modified_binomial_coefficient_sums_and_the_binomial_transform">Example 2: Modified binomial coefficient sums and the binomial transform</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=43" title="Edit section: Example 2: Modified binomial coefficient sums and the binomial transform">edit</a>]</div> As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence ⟨ fn ⟩ we define the two sequences of sums <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{n}&:=\sum _{m=0}^{n}{\binom {n}{m}}f_{m}3^{n-m}\\[4px]{\tilde {s}}_{n}&:=\sum _{m=0}^{n}{\binom {n}{m}}(m+1)(m+2)(m+3)f_{m}3^{n-m}\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{n}&:=\sum _{m=0}^{n}{\binom {n}{m}}f_{m}3^{n-m}\\[4px]{\tilde {s}}_{n}&:=\sum _{m=0}^{n}{\binom {n}{m}}(m+1)(m+2)(m+3)f_{m}3^{n-m}\,,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7852de896b296a657bbef93c0fe2cbc26f8a849f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:48.907ex; height:15.009ex;" alt="{\displaystyle {\begin{aligned}s_{n}&:=\sum _{m=0}^{n}{\binom {n}{m}}f_{m}3^{n-m}\\[4px]{\tilde {s}}_{n}&:=\sum _{m=0}^{n}{\binom {n}{m}}(m+1)(m+2)(m+3)f_{m}3^{n-m}\,,\end{aligned}}}"> for all n ≥ 0, and seek to express the second sums in terms of the first. We suggest an approach by generating functions. First, we use the <a href="/wiki/Binomial_transform" title="Binomial transform">binomial transform</a> to write the generating function for the first sum as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(z)={\frac {1}{1-3z}}F\left({\frac {z}{1-3z}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>F</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(z)={\frac {1}{1-3z}}F\left({\frac {z}{1-3z}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9e7f563db1b915e656e8be0c4cf25f6ebf43ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.257ex; height:6.176ex;" alt="{\displaystyle S(z)={\frac {1}{1-3z}}F\left({\frac {z}{1-3z}}\right).}"> Since the generating function for the sequence ⟨ (n + 1)(n + 2)(n + 3) fn ⟩ is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6F(z)+18zF'(z)+9z^{2}F''(z)+z^{3}F'''(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>18</mn> <mi>z</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>9</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>F</mi> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6F(z)+18zF'(z)+9z^{2}F''(z)+z^{3}F'''(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89ee9a99b91d1f59c7b14abd0b529ea992c445f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.735ex; height:3.176ex;" alt="{\displaystyle 6F(z)+18zF'(z)+9z^{2}F''(z)+z^{3}F'''(z)}"> we may write the generating function for the second sum defined above in the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {S}}(z)={\frac {6}{(1-3z)}}F\left({\frac {z}{1-3z}}\right)+{\frac {18z}{(1-3z)^{2}}}F'\left({\frac {z}{1-3z}}\right)+{\frac {9z^{2}}{(1-3z)^{3}}}F''\left({\frac {z}{1-3z}}\right)+{\frac {z^{3}}{(1-3z)^{4}}}F'''\left({\frac {z}{1-3z}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>F</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>18</mn> <mi>z</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>F</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>F</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>F</mi> <mo>‴</mo> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {S}}(z)={\frac {6}{(1-3z)}}F\left({\frac {z}{1-3z}}\right)+{\frac {18z}{(1-3z)^{2}}}F'\left({\frac {z}{1-3z}}\right)+{\frac {9z^{2}}{(1-3z)^{3}}}F''\left({\frac {z}{1-3z}}\right)+{\frac {z^{3}}{(1-3z)^{4}}}F'''\left({\frac {z}{1-3z}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba39ff111fda623a34f7297a18fe65d851f83d44" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:110.07ex; height:6.509ex;" alt="{\displaystyle {\tilde {S}}(z)={\frac {6}{(1-3z)}}F\left({\frac {z}{1-3z}}\right)+{\frac {18z}{(1-3z)^{2}}}F'\left({\frac {z}{1-3z}}\right)+{\frac {9z^{2}}{(1-3z)^{3}}}F''\left({\frac {z}{1-3z}}\right)+{\frac {z^{3}}{(1-3z)^{4}}}F'''\left({\frac {z}{1-3z}}\right).}"> In particular, we may write this modified sum generating function in the form of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(z)\cdot S(z)+b(z)\cdot zS'(z)+c(z)\cdot z^{2}S''(z)+d(z)\cdot z^{3}S'''(z),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>z</mi> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>S</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>S</mi> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(z)\cdot S(z)+b(z)\cdot zS'(z)+c(z)\cdot z^{2}S''(z)+d(z)\cdot z^{3}S'''(z),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bde2cdc5b4e8354c1a002e14c14379f35504bfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.367ex; height:3.176ex;" alt="{\displaystyle a(z)\cdot S(z)+b(z)\cdot zS'(z)+c(z)\cdot z^{2}S''(z)+d(z)\cdot z^{3}S'''(z),}"> for a(z) = 6(1 − 3z)3, b(z) = 18(1 − 3z)3, c(z) = 9(1 − 3z)3, and d(z) = (1 − 3z)3, where (1 − 3z)3 = 1 − 9z + 27z2 − 27z3. Finally, it follows that we may express the second sums through the first sums in the following form: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\tilde {s}}_{n}&=[z^{n}]\left(6(1-3z)^{3}\sum _{n=0}^{\infty }s_{n}z^{n}+18(1-3z)^{3}\sum _{n=0}^{\infty }ns_{n}z^{n}+9(1-3z)^{3}\sum _{n=0}^{\infty }n(n-1)s_{n}z^{n}+(1-3z)^{3}\sum _{n=0}^{\infty }n(n-1)(n-2)s_{n}z^{n}\right)\\[4px]&=(n+1)(n+2)(n+3)s_{n}-9n(n+1)(n+2)s_{n-1}+27(n-1)n(n+1)s_{n-2}-(n-2)(n-1)ns_{n-3}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mrow> <mo>(</mo> <mrow> <mn>6</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>18</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>9</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>9</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>27</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>n</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\tilde {s}}_{n}&=[z^{n}]\left(6(1-3z)^{3}\sum _{n=0}^{\infty }s_{n}z^{n}+18(1-3z)^{3}\sum _{n=0}^{\infty }ns_{n}z^{n}+9(1-3z)^{3}\sum _{n=0}^{\infty }n(n-1)s_{n}z^{n}+(1-3z)^{3}\sum _{n=0}^{\infty }n(n-1)(n-2)s_{n}z^{n}\right)\\[4px]&=(n+1)(n+2)(n+3)s_{n}-9n(n+1)(n+2)s_{n-1}+27(n-1)n(n+1)s_{n-2}-(n-2)(n-1)ns_{n-3}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcbe4cfe982ab74c3c90a066d323b24ba9ae0c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:124.462ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}{\tilde {s}}_{n}&=[z^{n}]\left(6(1-3z)^{3}\sum _{n=0}^{\infty }s_{n}z^{n}+18(1-3z)^{3}\sum _{n=0}^{\infty }ns_{n}z^{n}+9(1-3z)^{3}\sum _{n=0}^{\infty }n(n-1)s_{n}z^{n}+(1-3z)^{3}\sum _{n=0}^{\infty }n(n-1)(n-2)s_{n}z^{n}\right)\\[4px]&=(n+1)(n+2)(n+3)s_{n}-9n(n+1)(n+2)s_{n-1}+27(n-1)n(n+1)s_{n-2}-(n-2)(n-1)ns_{n-3}.\end{aligned}}}"> <div class="mw-heading mw-heading4"><h4 id="Example_3:_Generating_functions_for_mutually_recursive_sequences">Example 3: Generating functions for mutually recursive sequences</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=44" title="Edit section: Example 3: Generating functions for mutually recursive sequences">edit</a>]</div> In this example, we reformulate a generating function example given in Section 7.3 of Concrete Mathematics (see also Section 7.1 of the same reference for pretty pictures of generating function series). In particular, suppose that we seek the total number of ways (denoted Un) to tile a 3-by-n rectangle with unmarked 2-by-1 domino pieces. Let the auxiliary sequence, Vn, be defined as the number of ways to cover a 3-by-n rectangle-minus-corner section of the full rectangle. We seek to use these definitions to give a <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed form</a> formula for Un without breaking down this definition further to handle the cases of vertical versus horizontal dominoes. Notice that the ordinary generating functions for our two sequences correspond to the series: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}U(z)=1+3z^{2}+11z^{4}+41z^{6}+\cdots ,\\V(z)=z+4z^{3}+15z^{5}+56z^{7}+\cdots .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>U</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>11</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>41</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mo>+</mo> <mn>4</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>15</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>56</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U(z)=1+3z^{2}+11z^{4}+41z^{6}+\cdots ,\\V(z)=z+4z^{3}+15z^{5}+56z^{7}+\cdots .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91367c12b6ec5e29d14961fff70c7a7ff77b112f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.057ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}U(z)=1+3z^{2}+11z^{4}+41z^{6}+\cdots ,\\V(z)=z+4z^{3}+15z^{5}+56z^{7}+\cdots .\end{aligned}}}"> If we consider the possible configurations that can be given starting from the left edge of the 3-by-n rectangle, we are able to express the following mutually dependent, or mutually recursive, recurrence relations for our two sequences when n ≥ 2 defined as above where U0 = 1, U1 = 0, V0 = 0, and V1 = 1: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}U_{n}&=2V_{n-1}+U_{n-2}\\V_{n}&=U_{n-1}+V_{n-2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U_{n}&=2V_{n-1}+U_{n-2}\\V_{n}&=U_{n-1}+V_{n-2}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c118dc085b44a84173337c5fd0924f31209a8173" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.111ex; margin-bottom: -0.227ex; width:20.239ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}U_{n}&=2V_{n-1}+U_{n-2}\\V_{n}&=U_{n-1}+V_{n-2}.\end{aligned}}}"> Since we have that for all integers m ≥ 0, the index-shifted generating functions satisfy<a href="#cite_note-23">[note 1]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{m}G(z)=\sum _{n=m}^{\infty }g_{n-m}z^{n}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{m}G(z)=\sum _{n=m}^{\infty }g_{n-m}z^{n}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78a4d56b72744922ea23b72f9eeab8dcf703ae8a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.074ex; height:6.843ex;" alt="{\displaystyle z^{m}G(z)=\sum _{n=m}^{\infty }g_{n-m}z^{n}\,,}"> we can use the initial conditions specified above and the previous two recurrence relations to see that we have the next two equations relating the generating functions for these sequences given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}U(z)&=2zV(z)+z^{2}U(z)+1\\V(z)&=zU(z)+z^{2}V(z)={\frac {z}{1-z^{2}}}U(z),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>U</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>z</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>U</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U(z)&=2zV(z)+z^{2}U(z)+1\\V(z)&=zU(z)+z^{2}V(z)={\frac {z}{1-z^{2}}}U(z),\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56b37ffc2dcfe30c7363725fc6a5bc7119a78c9a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:39.381ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}U(z)&=2zV(z)+z^{2}U(z)+1\\V(z)&=zU(z)+z^{2}V(z)={\frac {z}{1-z^{2}}}U(z),\end{aligned}}}"> which then implies by solving the system of equations (and this is the particular trick to our method here) that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(z)={\frac {1-z^{2}}{1-4z^{2}+z^{4}}}={\frac {1}{3-{\sqrt {3}}}}\cdot {\frac {1}{1-\left(2+{\sqrt {3}}\right)z^{2}}}+{\frac {1}{3+{\sqrt {3}}}}\cdot {\frac {1}{1-\left(2-{\sqrt {3}}\right)z^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(z)={\frac {1-z^{2}}{1-4z^{2}+z^{4}}}={\frac {1}{3-{\sqrt {3}}}}\cdot {\frac {1}{1-\left(2+{\sqrt {3}}\right)z^{2}}}+{\frac {1}{3+{\sqrt {3}}}}\cdot {\frac {1}{1-\left(2-{\sqrt {3}}\right)z^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/491e443619a0651765f3ba1fe3b148666e44ede7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:79.931ex; height:7.009ex;" alt="{\displaystyle U(z)={\frac {1-z^{2}}{1-4z^{2}+z^{4}}}={\frac {1}{3-{\sqrt {3}}}}\cdot {\frac {1}{1-\left(2+{\sqrt {3}}\right)z^{2}}}+{\frac {1}{3+{\sqrt {3}}}}\cdot {\frac {1}{1-\left(2-{\sqrt {3}}\right)z^{2}}}.}"> Thus by performing algebraic simplifications to the sequence resulting from the second partial fractions expansions of the generating function in the previous equation, we find that U2n + 1 ≡ 0 and that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{2n}=\left\lceil {\frac {\left(2+{\sqrt {3}}\right)^{n}}{3-{\sqrt {3}}}}\right\rceil \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{2n}=\left\lceil {\frac {\left(2+{\sqrt {3}}\right)^{n}}{3-{\sqrt {3}}}}\right\rceil \,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c25556f3d411ec09e3538938ff5ee5d25e9273ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.404ex; height:7.676ex;" alt="{\displaystyle U_{2n}=\left\lceil {\frac {\left(2+{\sqrt {3}}\right)^{n}}{3-{\sqrt {3}}}}\right\rceil \,,}"> for all integers n ≥ 0. We also note that the same shifted generating function technique applied to the second-order <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence</a> for the <a href="/wiki/Fibonacci_numbers" class="mw-redirect" title="Fibonacci numbers">Fibonacci numbers</a> is the prototypical example of using generating functions to solve recurrence relations in one variable already covered, or at least hinted at, in the subsection on <a href="/wiki/Rational_functions" class="mw-redirect" title="Rational functions">rational functions</a> given above. <div class="mw-heading mw-heading3"><h3 id="Convolution_(Cauchy_products)">Convolution (Cauchy products)</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=45" title="Edit section: Convolution (Cauchy products)">edit</a>]</div> A discrete convolution of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see <a href="/wiki/Cauchy_product" title="Cauchy product">Cauchy product</a>). <ol><li>Consider A(z) and B(z) are ordinary generating functions. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=A(z)B(z)\Leftrightarrow [z^{n}]C(z)=\sum _{k=0}^{n}{a_{k}b_{n-k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(z)=A(z)B(z)\Leftrightarrow [z^{n}]C(z)=\sum _{k=0}^{n}{a_{k}b_{n-k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d329b095b1b3242dd36187b63aa17c64ff3fd4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.455ex; height:7.009ex;" alt="{\displaystyle C(z)=A(z)B(z)\Leftrightarrow [z^{n}]C(z)=\sum _{k=0}^{n}{a_{k}b_{n-k}}}"></li> <li>Consider A(z) and B(z) are exponential generating functions. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=A(z)B(z)\Leftrightarrow \left[{\frac {z^{n}}{n!}}\right]C(z)=\sum _{k=0}^{n}{\binom {n}{k}}a_{k}b_{n-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(z)=A(z)B(z)\Leftrightarrow \left[{\frac {z^{n}}{n!}}\right]C(z)=\sum _{k=0}^{n}{\binom {n}{k}}a_{k}b_{n-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195a38f6bf38db98bf4d6debefb39b3861962dca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.655ex; height:7.009ex;" alt="{\displaystyle C(z)=A(z)B(z)\Leftrightarrow \left[{\frac {z^{n}}{n!}}\right]C(z)=\sum _{k=0}^{n}{\binom {n}{k}}a_{k}b_{n-k}}"></li> <li>Consider the triply convolved sequence resulting from the product of three ordinary generating functions <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=F(z)G(z)H(z)\Leftrightarrow [z^{n}]C(z)=\sum _{j+k+l=n}f_{j}g_{k}h_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>k</mi> <mo>+</mo> <mi>l</mi> <mo>=</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(z)=F(z)G(z)H(z)\Leftrightarrow [z^{n}]C(z)=\sum _{j+k+l=n}f_{j}g_{k}h_{l}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2337ff3c9bf72dfccae661534dabeadbf2e44288" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:50.606ex; height:5.843ex;" alt="{\displaystyle C(z)=F(z)G(z)H(z)\Leftrightarrow [z^{n}]C(z)=\sum _{j+k+l=n}f_{j}g_{k}h_{l}}"></li> <li>Consider the m-fold convolution of a sequence with itself for some positive integer m ≥ 1 (see the example below for an application) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=G(z)^{m}\Leftrightarrow [z^{n}]C(z)=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">⇔</mo> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>⋯</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(z)=G(z)^{m}\Leftrightarrow [z^{n}]C(z)=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec6e5d381123d79161a2e3e189543de7278a1cb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:56.211ex; height:5.843ex;" alt="{\displaystyle C(z)=G(z)^{m}\Leftrightarrow [z^{n}]C(z)=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}}"></li></ol> Multiplication of generating functions, or convolution of their underlying sequences, can correspond to a notion of independent events in certain counting and probability scenarios. For example, if we adopt the notational convention that the <a href="/wiki/Probability_generating_function" class="mw-redirect" title="Probability generating function">probability generating function</a>, or pgf, of a random variable Z is denoted by GZ(z), then we can show that for any two random variables <a href="#cite_note-24">[23]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{X+Y}(z)=G_{X}(z)G_{Y}(z)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{X+Y}(z)=G_{X}(z)G_{Y}(z)\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0b91af38c3ff8fac96d8ca01b94e248c683134" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.588ex; height:2.843ex;" alt="{\displaystyle G_{X+Y}(z)=G_{X}(z)G_{Y}(z)\,,}"> if X and Y are independent. Similarly, the number of ways to pay n ≥ 0 cents in coin denominations of values in the set {1, 5, 10, 25, 50} (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively) is generated by the product <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)={\frac {1}{1-z}}{\frac {1}{1-z^{5}}}{\frac {1}{1-z^{10}}}{\frac {1}{1-z^{25}}}{\frac {1}{1-z^{50}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>50</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(z)={\frac {1}{1-z}}{\frac {1}{1-z^{5}}}{\frac {1}{1-z^{10}}}{\frac {1}{1-z^{25}}}{\frac {1}{1-z^{50}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b9641007a7924a7dd31be40789d7522448ba60" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.736ex; height:5.676ex;" alt="{\displaystyle C(z)={\frac {1}{1-z}}{\frac {1}{1-z^{5}}}{\frac {1}{1-z^{10}}}{\frac {1}{1-z^{25}}}{\frac {1}{1-z^{50}}},}"> and moreover, if we allow the n cents to be paid in coins of any positive integer denomination, we arrive at the generating for the number of such combinations of change being generated by the <a href="/wiki/Partition_function_(mathematics)" title="Partition function (mathematics)">partition function</a> generating function expanded by the infinite <a href="/wiki/Q-Pochhammer_symbol" title="Q-Pochhammer symbol">q-Pochhammer symbol</a> product of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{n=1}^{\infty }\left(1-z^{n}\right)^{-1}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{n=1}^{\infty }\left(1-z^{n}\right)^{-1}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37b3fcdbccab3a7cbab6f2589ea9245d817ac44" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.962ex; height:6.843ex;" alt="{\displaystyle \prod _{n=1}^{\infty }\left(1-z^{n}\right)^{-1}\,.}"> <div class="mw-heading mw-heading4"><h4 id="Example:_Generating_function_for_the_Catalan_numbers">Example: Generating function for the Catalan numbers</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=46" title="Edit section: Example: Generating function for the Catalan numbers">edit</a>]</div> An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the <a href="/wiki/Catalan_numbers" class="mw-redirect" title="Catalan numbers">Catalan numbers</a>, Cn. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x0 · x1 ·⋯· xn so that the order of multiplication is completely specified. For example, C2 = 2 which corresponds to the two expressions x0 · (x1 · x2) and (x0 · x1) · x2. It follows that the sequence satisfies a recurrence relation given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}=\sum _{k=0}^{n-1}C_{k}C_{n-1-k}+\delta _{n,0}=C_{0}C_{n-1}+C_{1}C_{n-2}+\cdots +C_{n-1}C_{0}+\delta _{n,0}\,,\quad n\geq 0\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>≥</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}=\sum _{k=0}^{n-1}C_{k}C_{n-1-k}+\delta _{n,0}=C_{0}C_{n-1}+C_{1}C_{n-2}+\cdots +C_{n-1}C_{0}+\delta _{n,0}\,,\quad n\geq 0\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07ebd1917fc503cde4a8353b6546c571bcb87592" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:80.195ex; height:7.509ex;" alt="{\displaystyle C_{n}=\sum _{k=0}^{n-1}C_{k}C_{n-1-k}+\delta _{n,0}=C_{0}C_{n-1}+C_{1}C_{n-2}+\cdots +C_{n-1}C_{0}+\delta _{n,0}\,,\quad n\geq 0\,,}"> and so has a corresponding convolved generating function, C(z), satisfying <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=z\cdot C(z)^{2}+1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mo>⋅</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(z)=z\cdot C(z)^{2}+1\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f5500143b9b0bde992929153db301d34734858" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.284ex; height:3.176ex;" alt="{\displaystyle C(z)=z\cdot C(z)^{2}+1\,.}"> Since C(0) = 1 ≠ ∞, we then arrive at a formula for this generating function given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)={\frac {1-{\sqrt {1-4z}}}{2z}}=\sum _{n=0}^{\infty }{\frac {1}{n+1}}{\binom {2n}{n}}z^{n}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mi>z</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(z)={\frac {1-{\sqrt {1-4z}}}{2z}}=\sum _{n=0}^{\infty }{\frac {1}{n+1}}{\binom {2n}{n}}z^{n}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ffe3100b7e45e542375517bf7a25eea6df917a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.186ex; height:7.009ex;" alt="{\displaystyle C(z)={\frac {1-{\sqrt {1-4z}}}{2z}}=\sum _{n=0}^{\infty }{\frac {1}{n+1}}{\binom {2n}{n}}z^{n}\,.}"> Note that the first equation implicitly defining C(z) above implies that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)={\frac {1}{1-z\cdot C(z)}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo>⋅</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(z)={\frac {1}{1-z\cdot C(z)}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d7b4bcd26b23d204340ab3232f0e4f5d002b221" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.066ex; height:6.009ex;" alt="{\displaystyle C(z)={\frac {1}{1-z\cdot C(z)}}\,,}"> which then leads to another "simple" (of form) continued fraction expansion of this generating function. <div class="mw-heading mw-heading4"><h4 id="Example:_Spanning_trees_of_fans_and_convolutions_of_convolutions">Example: Spanning trees of fans and convolutions of convolutions</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=47" title="Edit section: Example: Spanning trees of fans and convolutions of convolutions">edit</a>]</div> A fan of order n is defined to be a graph on the vertices {0, 1, ..., n} with 2n − 1 edges connected according to the following rules: Vertex 0 is connected by a single edge to each of the other n vertices, and vertex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is connected by a single edge to the next vertex k + 1 for all 1 ≤ k < n.<a href="#cite_note-25">[24]</a> There is one fan of order one, three fans of order two, eight fans of order three, and so on. A <a href="/wiki/Spanning_tree" title="Spanning tree">spanning tree</a> is a subgraph of a graph which contains all of the original vertices and which contains enough edges to make this subgraph connected, but not so many edges that there is a cycle in the subgraph. We ask how many spanning trees fn of a fan of order n are possible for each n ≥ 1. As an observation, we may approach the question by counting the number of ways to join adjacent sets of vertices. For example, when n = 4, we have that f4 = 4 + 3 · 1 + 2 · 2 + 1 · 3 + 2 · 1 · 1 + 1 · 2 · 1 + 1 · 1 · 2 + 1 · 1 · 1 · 1 = 21, which is a sum over the m-fold convolutions of the sequence gn = n = [zn] <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠z/(1 − z)2⁠ for m ≔ 1, 2, 3, 4. More generally, we may write a formula for this sequence as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}=\sum _{m>0}\sum _{\scriptstyle k_{1}+k_{2}+\cdots +k_{m}=n \atop \scriptstyle k_{1},k_{2},\ldots ,k_{m}>0}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> </mstyle> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mfrac> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>⋯</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}=\sum _{m>0}\sum _{\scriptstyle k_{1}+k_{2}+\cdots +k_{m}=n \atop \scriptstyle k_{1},k_{2},\ldots ,k_{m}>0}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de20fae658d8e3fc868770290d81ce4a726fb01f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:38.049ex; height:7.843ex;" alt="{\displaystyle f_{n}=\sum _{m>0}\sum _{\scriptstyle k_{1}+k_{2}+\cdots +k_{m}=n \atop \scriptstyle k_{1},k_{2},\ldots ,k_{m}>0}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}\,,}"> from which we see that the ordinary generating function for this sequence is given by the next sum of convolutions as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z)=G(z)+G(z)^{2}+G(z)^{3}+\cdots ={\frac {G(z)}{1-G(z)}}={\frac {z}{(1-z)^{2}-z}}={\frac {z}{1-3z+z^{2}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mi>z</mi> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(z)=G(z)+G(z)^{2}+G(z)^{3}+\cdots ={\frac {G(z)}{1-G(z)}}={\frac {z}{(1-z)^{2}-z}}={\frac {z}{1-3z+z^{2}}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2dac0dc0e7513c24beed2dfe58457edb502764" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:79.948ex; height:6.509ex;" alt="{\displaystyle F(z)=G(z)+G(z)^{2}+G(z)^{3}+\cdots ={\frac {G(z)}{1-G(z)}}={\frac {z}{(1-z)^{2}-z}}={\frac {z}{1-3z+z^{2}}}\,,}"> from which we are able to extract an exact formula for the sequence by taking the <a href="/wiki/Partial_fraction_expansion" class="mw-redirect" title="Partial fraction expansion">partial fraction expansion</a> of the last generating function. <div class="mw-heading mw-heading3"><h3 id="Implicit_generating_functions_and_the_Lagrange_inversion_formula">Implicit generating functions and the Lagrange inversion formula</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=48" title="Edit section: Implicit generating functions and the Lagrange inversion formula">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></td><td class="mbox-text"><div class="mbox-text-span">This section needs expansion with: This section needs to be added to the list of techniques with generating functions. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Generating_function&action=edit&section=">adding to it</a>. (April 2017)</div></td></tr></tbody></table> One often encounters generating functions specified by a functional equation, instead of an explicit specification. For example, the generating function T(z) for the number of binary trees on n nodes (leaves included) satisfies <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(z)=z\left(1+T(z)^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(z)=z\left(1+T(z)^{2}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674bd100e61359afedcb806c42a6d4b2f3746ad1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.828ex; height:3.343ex;" alt="{\displaystyle T(z)=z\left(1+T(z)^{2}\right)}"> The <a href="/wiki/Lagrange_inversion_theorem" title="Lagrange inversion theorem">Lagrange inversion theorem</a> is a tool used to explicitly evaluate solutions to such equations. <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> Lagrange inversion formula — Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi (z)\in C[[z]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi (z)\in C[[z]]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15200790093e49d313e8d2b17f3c53f848e6939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.565ex; height:2.843ex;" alt="{\textstyle \phi (z)\in C[[z]]}"> be a formal power series with a non-zero constant term. Then the functional equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(z)=z\phi (T(z))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(z)=z\phi (T(z))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aede79fda056ae25979c2aec9d207760a379fd2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.449ex; height:2.843ex;" alt="{\displaystyle T(z)=z\phi (T(z))}"> admits a unique solution in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T(z)\in C[[z]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle T(z)\in C[[z]]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3dba50d4cc7281989b476fcb32d0a0c51552d8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.816ex; height:2.843ex;" alt="{\textstyle T(z)\in C[[z]]}">, which satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [z^{n}]T(z)=[z^{n-1}]{\frac {1}{n}}(\phi (z))^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [z^{n}]T(z)=[z^{n-1}]{\frac {1}{n}}(\phi (z))^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b79307844cb70ecaa92230ee6b4510ab2887fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.479ex; height:5.176ex;" alt="{\displaystyle [z^{n}]T(z)=[z^{n-1}]{\frac {1}{n}}(\phi (z))^{n}}"> where the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [z^{n}]F(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [z^{n}]F(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53406a968e38369b3deeb6092418b5e0c83d942" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.241ex; height:2.843ex;" alt="{\displaystyle [z^{n}]F(z)}"> returns the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1a8cdd7ee39054e510deeb38ee551cc7616ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.309ex; height:2.343ex;" alt="{\displaystyle z^{n}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf298d777120df944559c5e985b88a824debb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(z)}">. </div> Applying the above theorem to our functional equation yields (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi (z)=1+z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \phi (z)=1+z^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01b92c2d6da6f1064598a44b3cb7203b1aaaaecc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.529ex; height:3.009ex;" alt="{\textstyle \phi (z)=1+z^{2}}">): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [z^{n}]T(z)=[z^{n-1}]{\frac {1}{n}}(1+z^{2})^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [z^{n}]T(z)=[z^{n-1}]{\frac {1}{n}}(1+z^{2})^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb02695add39b5cad4f9edbf5c904605c27d21e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.344ex; height:5.176ex;" alt="{\displaystyle [z^{n}]T(z)=[z^{n-1}]{\frac {1}{n}}(1+z^{2})^{n}}"> Via the binomial theorem expansion, for even <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, the formula returns <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">. This is expected as one can prove that the number of leaves of a binary tree are one more than the number of its internal nodes, so the total sum should always be an odd number. For odd <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, however, we get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [z^{n-1}]{\frac {1}{n}}(1+z^{2})^{n}={\frac {1}{n}}{\dbinom {n}{\frac {n+1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [z^{n-1}]{\frac {1}{n}}(1+z^{2})^{n}={\frac {1}{n}}{\dbinom {n}{\frac {n+1}{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7692ed9c2cad1cc3fa9cecacf4c68f4207cc72a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.782ex; height:6.509ex;" alt="{\displaystyle [z^{n-1}]{\frac {1}{n}}(1+z^{2})^{n}={\frac {1}{n}}{\dbinom {n}{\frac {n+1}{2}}}}"> The expression becomes much neater if we let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> be the number of internal nodes: Now the expression just becomes the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">th Catalan number. <div class="mw-heading mw-heading3"><h3 id="Introducing_a_free_parameter_(snake_oil_method)">Introducing a free parameter (snake oil method)</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=49" title="Edit section: Introducing a free parameter (snake oil method)">edit</a>]</div> Sometimes the sum sn is complicated, and it is not always easy to evaluate. The "Free Parameter" method is another method (called "snake oil" by H. Wilf) to evaluate these sums. Both methods discussed so far have n as limit in the summation. When n does not appear explicitly in the summation, we may consider n as a "free" parameter and treat sn as a coefficient of F(z) = Σ sn zn, change the order of the summations on n and k, and try to compute the inner sum. For example, if we want to compute <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}=\sum _{k=0}^{\infty }{{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}}\,,\quad m,n\in \mathbb {N} _{0}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}=\sum _{k=0}^{\infty }{{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}}\,,\quad m,n\in \mathbb {N} _{0}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a785034ec46d3dd6276ea4191fd96ff6fb6451d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.144ex; height:7.176ex;" alt="{\displaystyle s_{n}=\sum _{k=0}^{\infty }{{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}}\,,\quad m,n\in \mathbb {N} _{0}\,,}"> we can treat n as a "free" parameter, and set <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z)=\sum _{n=0}^{\infty }{\left(\sum _{k=0}^{\infty }{{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}}\right)}z^{n}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(z)=\sum _{n=0}^{\infty }{\left(\sum _{k=0}^{\infty }{{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}}\right)}z^{n}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb24c941958d51db6c89903c32602e20a7a5aac5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.42ex; height:7.509ex;" alt="{\displaystyle F(z)=\sum _{n=0}^{\infty }{\left(\sum _{k=0}^{\infty }{{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}}\right)}z^{n}\,.}"> Interchanging summation ("snake oil") gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z)=\sum _{k=0}^{\infty }{{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}z^{-k}}\sum _{n=0}^{\infty }{{\binom {n+k}{m+2k}}z^{n+k}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(z)=\sum _{k=0}^{\infty }{{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}z^{-k}}\sum _{n=0}^{\infty }{{\binom {n+k}{m+2k}}z^{n+k}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43675a178e899b85c23f21fadc1ec2621b868353" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.719ex; height:7.176ex;" alt="{\displaystyle F(z)=\sum _{k=0}^{\infty }{{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}z^{-k}}\sum _{n=0}^{\infty }{{\binom {n+k}{m+2k}}z^{n+k}}\,.}"> Now the inner sum is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠zm + 2k/(1 − z)m + 2k + 1⁠. Thus <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F(z)&={\frac {z^{m}}{(1-z)^{m+1}}}\sum _{k=0}^{\infty }{{\frac {1}{k+1}}{\binom {2k}{k}}\left({\frac {-z}{(1-z)^{2}}}\right)^{k}}\\[4px]&={\frac {z^{m}}{(1-z)^{m+1}}}\sum _{k=0}^{\infty }{C_{k}\left({\frac {-z}{(1-z)^{2}}}\right)^{k}}&{\text{where }}C_{k}=k{\text{th Catalan number}}\\[4px]&={\frac {z^{m}}{(1-z)^{m+1}}}{\frac {1-{\sqrt {1+{\frac {4z}{(1-z)^{2}}}}}}{\frac {-2z}{(1-z)^{2}}}}\\[4px]&={\frac {-z^{m-1}}{2(1-z)^{m-1}}}\left(1-{\frac {1+z}{1-z}}\right)\\[4px]&={\frac {z^{m}}{(1-z)^{m}}}=z{\frac {z^{m-1}}{(1-z)^{m}}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>z</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>z</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>where </mtext> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>th Catalan number</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>z</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mi>z</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>z</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(z)&={\frac {z^{m}}{(1-z)^{m+1}}}\sum _{k=0}^{\infty }{{\frac {1}{k+1}}{\binom {2k}{k}}\left({\frac {-z}{(1-z)^{2}}}\right)^{k}}\\[4px]&={\frac {z^{m}}{(1-z)^{m+1}}}\sum _{k=0}^{\infty }{C_{k}\left({\frac {-z}{(1-z)^{2}}}\right)^{k}}&{\text{where }}C_{k}=k{\text{th Catalan number}}\\[4px]&={\frac {z^{m}}{(1-z)^{m+1}}}{\frac {1-{\sqrt {1+{\frac {4z}{(1-z)^{2}}}}}}{\frac {-2z}{(1-z)^{2}}}}\\[4px]&={\frac {-z^{m-1}}{2(1-z)^{m-1}}}\left(1-{\frac {1+z}{1-z}}\right)\\[4px]&={\frac {z^{m}}{(1-z)^{m}}}=z{\frac {z^{m-1}}{(1-z)^{m}}}\,.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5255fe5b779e3031fbb9d87bccdeef7c7b2205f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -20.849ex; margin-bottom: -0.323ex; width:86.633ex; height:43.509ex;" alt="{\displaystyle {\begin{aligned}F(z)&={\frac {z^{m}}{(1-z)^{m+1}}}\sum _{k=0}^{\infty }{{\frac {1}{k+1}}{\binom {2k}{k}}\left({\frac {-z}{(1-z)^{2}}}\right)^{k}}\\[4px]&={\frac {z^{m}}{(1-z)^{m+1}}}\sum _{k=0}^{\infty }{C_{k}\left({\frac {-z}{(1-z)^{2}}}\right)^{k}}&{\text{where }}C_{k}=k{\text{th Catalan number}}\\[4px]&={\frac {z^{m}}{(1-z)^{m+1}}}{\frac {1-{\sqrt {1+{\frac {4z}{(1-z)^{2}}}}}}{\frac {-2z}{(1-z)^{2}}}}\\[4px]&={\frac {-z^{m-1}}{2(1-z)^{m-1}}}\left(1-{\frac {1+z}{1-z}}\right)\\[4px]&={\frac {z^{m}}{(1-z)^{m}}}=z{\frac {z^{m-1}}{(1-z)^{m}}}\,.\end{aligned}}}"> Then we obtain <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}={\begin{cases}\displaystyle {\binom {n-1}{m-1}}&{\text{for }}m\geq 1\,,\\{}[n=0]&{\text{for }}m=0\,.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>m</mi> <mo>≥</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}={\begin{cases}\displaystyle {\binom {n-1}{m-1}}&{\text{for }}m\geq 1\,,\\{}[n=0]&{\text{for }}m=0\,.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4243c4a2ad451f0ca9dfa0af5c0b525e602cd979" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:30.714ex; height:8.843ex;" alt="{\displaystyle s_{n}={\begin{cases}\displaystyle {\binom {n-1}{m-1}}&{\text{for }}m\geq 1\,,\\{}[n=0]&{\text{for }}m=0\,.\end{cases}}}"> It is instructive to use the same method again for the sum, but this time take m as the free parameter instead of n. We thus set <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)=\sum _{m=0}^{\infty }\left(\sum _{k=0}^{\infty }{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}\right)z^{m}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(z)=\sum _{m=0}^{\infty }\left(\sum _{k=0}^{\infty }{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}\right)z^{m}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b245a51c698a280fcb5580d023e938028fca69" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.538ex; height:7.509ex;" alt="{\displaystyle G(z)=\sum _{m=0}^{\infty }\left(\sum _{k=0}^{\infty }{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}\right)z^{m}\,.}"> Interchanging summation ("snake oil") gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)=\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}z^{-2k}\sum _{m=0}^{\infty }{\binom {n+k}{m+2k}}z^{m+2k}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>k</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(z)=\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}z^{-2k}\sum _{m=0}^{\infty }{\binom {n+k}{m+2k}}z^{m+2k}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c5f29d302332b35c5440ee897e3bcf3a082dcb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.093ex; height:7.176ex;" alt="{\displaystyle G(z)=\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}z^{-2k}\sum _{m=0}^{\infty }{\binom {n+k}{m+2k}}z^{m+2k}\,.}"> Now the inner sum is (1 + z)n + k. Thus <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}G(z)&=(1+z)^{n}\sum _{k=0}^{\infty }{\frac {1}{k+1}}{\binom {2k}{k}}\left({\frac {-(1+z)}{z^{2}}}\right)^{k}\\[4px]&=(1+z)^{n}\sum _{k=0}^{\infty }C_{k}\,\left({\frac {-(1+z)}{z^{2}}}\right)^{k}&{\text{where }}C_{k}=k{\text{th Catalan number}}\\[4px]&=(1+z)^{n}\,{\frac {1-{\sqrt {1+{\frac {4(1+z)}{z^{2}}}}}}{\frac {-2(1+z)}{z^{2}}}}\\[4px]&=(1+z)^{n}\,{\frac {z^{2}-z{\sqrt {z^{2}+4+4z}}}{-2(1+z)}}\\[4px]&=(1+z)^{n}\,{\frac {z^{2}-z(z+2)}{-2(1+z)}}\\[4px]&=(1+z)^{n}\,{\frac {-2z}{-2(1+z)}}=z(1+z)^{n-1}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>where </mtext> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>th Catalan number</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>4</mn> <mi>z</mi> </msqrt> </mrow> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mi>z</mi> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G(z)&=(1+z)^{n}\sum _{k=0}^{\infty }{\frac {1}{k+1}}{\binom {2k}{k}}\left({\frac {-(1+z)}{z^{2}}}\right)^{k}\\[4px]&=(1+z)^{n}\sum _{k=0}^{\infty }C_{k}\,\left({\frac {-(1+z)}{z^{2}}}\right)^{k}&{\text{where }}C_{k}=k{\text{th Catalan number}}\\[4px]&=(1+z)^{n}\,{\frac {1-{\sqrt {1+{\frac {4(1+z)}{z^{2}}}}}}{\frac {-2(1+z)}{z^{2}}}}\\[4px]&=(1+z)^{n}\,{\frac {z^{2}-z{\sqrt {z^{2}+4+4z}}}{-2(1+z)}}\\[4px]&=(1+z)^{n}\,{\frac {z^{2}-z(z+2)}{-2(1+z)}}\\[4px]&=(1+z)^{n}\,{\frac {-2z}{-2(1+z)}}=z(1+z)^{n-1}\,.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcefeaec5028b5aa43a21eb6ea7efe06a1d0793" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -25.171ex; width:84.08ex; height:51.509ex;" alt="{\displaystyle {\begin{aligned}G(z)&=(1+z)^{n}\sum _{k=0}^{\infty }{\frac {1}{k+1}}{\binom {2k}{k}}\left({\frac {-(1+z)}{z^{2}}}\right)^{k}\\[4px]&=(1+z)^{n}\sum _{k=0}^{\infty }C_{k}\,\left({\frac {-(1+z)}{z^{2}}}\right)^{k}&{\text{where }}C_{k}=k{\text{th Catalan number}}\\[4px]&=(1+z)^{n}\,{\frac {1-{\sqrt {1+{\frac {4(1+z)}{z^{2}}}}}}{\frac {-2(1+z)}{z^{2}}}}\\[4px]&=(1+z)^{n}\,{\frac {z^{2}-z{\sqrt {z^{2}+4+4z}}}{-2(1+z)}}\\[4px]&=(1+z)^{n}\,{\frac {z^{2}-z(z+2)}{-2(1+z)}}\\[4px]&=(1+z)^{n}\,{\frac {-2z}{-2(1+z)}}=z(1+z)^{n-1}\,.\end{aligned}}}"> Thus we obtain <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}=\left[z^{m}\right]z(1+z)^{n-1}=\left[z^{m-1}\right](1+z)^{n-1}={\binom {n-1}{m-1}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>]</mo> </mrow> <mi>z</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>[</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}=\left[z^{m}\right]z(1+z)^{n-1}=\left[z^{m-1}\right](1+z)^{n-1}={\binom {n-1}{m-1}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9929620932cfa7c37627fc88e25a9335fc615384" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.268ex; height:6.176ex;" alt="{\displaystyle s_{n}=\left[z^{m}\right]z(1+z)^{n-1}=\left[z^{m-1}\right](1+z)^{n-1}={\binom {n-1}{m-1}}\,,}"> for m ≥ 1 as before. <div class="mw-heading mw-heading3"><h3 id="Generating_functions_prove_congruences">Generating functions prove congruences</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=50" title="Edit section: Generating functions prove congruences">edit</a>]</div> We say that two generating functions (power series) are congruent modulo m, written A(z) ≡ B(z) (mod m) if their coefficients are congruent modulo m for all n ≥ 0, i.e., an ≡ bn (mod m) for all relevant cases of the integers n (note that we need not assume that m is an integer here—it may very well be polynomial-valued in some indeterminate x, for example). If the "simpler" right-hand-side generating function, B(z), is a rational function of z, then the form of this sequence suggests that the sequence is <a href="/wiki/Periodic_function" title="Periodic function">eventually periodic</a> modulo fixed particular cases of integer-valued m ≥ 2. For example, we can prove that the <a href="/wiki/Euler_numbers" title="Euler numbers">Euler numbers</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle E_{n}\rangle =\langle 1,1,5,61,1385,\ldots \rangle \longmapsto \langle 1,1,2,1,2,1,2,\ldots \rangle {\pmod {3}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>61</mn> <mo>,</mo> <mn>1385</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">⟼</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle E_{n}\rangle =\langle 1,1,5,61,1385,\ldots \rangle \longmapsto \langle 1,1,2,1,2,1,2,\ldots \rangle {\pmod {3}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ac0a4ec94c44449104b175e5c6ab5f189b9eb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.889ex; height:2.843ex;" alt="{\displaystyle \langle E_{n}\rangle =\langle 1,1,5,61,1385,\ldots \rangle \longmapsto \langle 1,1,2,1,2,1,2,\ldots \rangle {\pmod {3}}\,,}"> satisfy the following congruence modulo 3:<a href="#cite_note-26">[25]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }E_{n}z^{n}={\frac {1-z^{2}}{1+z^{2}}}{\pmod {3}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }E_{n}z^{n}={\frac {1-z^{2}}{1+z^{2}}}{\pmod {3}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c535a69fd4a3b15165ce910149b92e1e1f616918" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.947ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }E_{n}z^{n}={\frac {1-z^{2}}{1+z^{2}}}{\pmod {3}}\,.}"> One useful method of obtaining congruences for sequences enumerated by special generating functions modulo any integers (i.e., not only prime powers pk) is given in the section on continued fraction representations of (even non-convergent) ordinary generating functions by J-fractions above. We cite one particular result related to generating series expanded through a representation by continued fraction from Lando's Lectures on Generating Functions as follows: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> Theorem: congruences for series generated by expansions of continued fractions — Suppose that the generating function A(z) is represented by an infinite <a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a> of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(z)={\cfrac {1}{1-c_{1}z-{\cfrac {p_{1}z^{2}}{1-c_{2}z-{\cfrac {p_{2}z^{2}}{1-c_{3}z-{\ddots }}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⋱</mo> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(z)={\cfrac {1}{1-c_{1}z-{\cfrac {p_{1}z^{2}}{1-c_{2}z-{\cfrac {p_{2}z^{2}}{1-c_{3}z-{\ddots }}}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d94698de3ddf0b800d48caa487d29bb49789fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; width:43.204ex; height:15.843ex;" alt="{\displaystyle A(z)={\cfrac {1}{1-c_{1}z-{\cfrac {p_{1}z^{2}}{1-c_{2}z-{\cfrac {p_{2}z^{2}}{1-c_{3}z-{\ddots }}}}}}}}"> and that Ap(z) denotes the pth convergent to this continued fraction expansion defined such that an = [zn] Ap(z) for all 0 ≤ n < 2p. Then: <ol><li>the function Ap(z) is rational for all p ≥ 2 where we assume that one of divisibility criteria of p | p1, p1p2, p1p2p3 is met, that is, p | p1p2⋯pk for some k ≥ 1; and</li> <li>if the integer p divides the product p1p2⋯pk, then we have A(z) ≡ Ak(z) (mod p).</li></ol> </div> Generating functions also have other uses in proving congruences for their coefficients. We cite the next two specific examples deriving special case congruences for the <a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling numbers of the first kind</a> and for the <a href="/wiki/Partition_function_(number_theory)" title="Partition function (number theory)">partition function p(n)</a> which show the versatility of generating functions in tackling problems involving <a href="/wiki/Integer_sequences" class="mw-redirect" title="Integer sequences">integer sequences</a>. <div class="mw-heading mw-heading4"><h4 id="The_Stirling_numbers_modulo_small_integers">The Stirling numbers modulo small integers</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=51" title="Edit section: The Stirling numbers modulo small integers">edit</a>]</div> The <a href="/wiki/Stirling_numbers_of_the_first_kind#Congruences" title="Stirling numbers of the first kind">main article</a> on the Stirling numbers generated by the finite products <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}(x):=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}x^{k}=x(x+1)(x+2)\cdots (x+n-1)\,,\quad n\geq 1\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}(x):=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}x^{k}=x(x+1)(x+2)\cdots (x+n-1)\,,\quad n\geq 1\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d804f4f935ba11b67687566586927a8aa54e237e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.309ex; height:7.009ex;" alt="{\displaystyle S_{n}(x):=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}x^{k}=x(x+1)(x+2)\cdots (x+n-1)\,,\quad n\geq 1\,,}"> provides an overview of the congruences for these numbers derived strictly from properties of their generating function as in Section 4.6 of Wilf's stock reference Generatingfunctionology. We repeat the basic argument and notice that when reduces modulo 2, these finite product generating functions each satisfy <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}(x)=[x(x+1)]\cdot [x(x+1)]\cdots =x^{\left\lceil {\frac {n}{2}}\right\rceil }(x+1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>⋯</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}(x)=[x(x+1)]\cdot [x(x+1)]\cdots =x^{\left\lceil {\frac {n}{2}}\right\rceil }(x+1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10bf6c4b4fe41e59977b00509e965a6c77688de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.015ex; height:4.343ex;" alt="{\displaystyle S_{n}(x)=[x(x+1)]\cdot [x(x+1)]\cdots =x^{\left\lceil {\frac {n}{2}}\right\rceil }(x+1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }\,,}"> which implies that the parity of these <a href="/wiki/Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a> matches that of the binomial coefficient <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}\equiv {\binom {\left\lfloor {\frac {n}{2}}\right\rfloor }{k-\left\lceil {\frac {n}{2}}\right\rceil }}{\pmod {2}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}\equiv {\binom {\left\lfloor {\frac {n}{2}}\right\rfloor }{k-\left\lceil {\frac {n}{2}}\right\rceil }}{\pmod {2}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713807ebb71d50bc27f1143de1bd52f18bf19509" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.07ex; height:7.343ex;" alt="{\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}\equiv {\binom {\left\lfloor {\frac {n}{2}}\right\rfloor }{k-\left\lceil {\frac {n}{2}}\right\rceil }}{\pmod {2}}\,,}"> and consequently shows that [n k] is even whenever k < ⌊ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n/2⁠ ⌋. Similarly, we can reduce the right-hand-side products defining the Stirling number generating functions modulo 3 to obtain slightly more complicated expressions providing that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\begin{bmatrix}n\\m\end{bmatrix}}&\equiv [x^{m}]\left(x^{\left\lceil {\frac {n}{3}}\right\rceil }(x+1)^{\left\lceil {\frac {n-1}{3}}\right\rceil }(x+2)^{\left\lfloor {\frac {n}{3}}\right\rfloor }\right)&&{\pmod {3}}\\&\equiv \sum _{k=0}^{m}{\begin{pmatrix}\left\lceil {\frac {n-1}{3}}\right\rceil \\k\end{pmatrix}}{\begin{pmatrix}\left\lfloor {\frac {n}{3}}\right\rfloor \\m-k-\left\lceil {\frac {n}{3}}\right\rceil \end{pmatrix}}\times 2^{\left\lceil {\frac {n}{3}}\right\rceil +\left\lfloor {\frac {n}{3}}\right\rfloor -(m-k)}&&{\pmod {3}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>3</mn> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>3</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≡</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>3</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>3</mn> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>3</mn> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mo>+</mo> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>3</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{bmatrix}n\\m\end{bmatrix}}&\equiv [x^{m}]\left(x^{\left\lceil {\frac {n}{3}}\right\rceil }(x+1)^{\left\lceil {\frac {n-1}{3}}\right\rceil }(x+2)^{\left\lfloor {\frac {n}{3}}\right\rfloor }\right)&&{\pmod {3}}\\&\equiv \sum _{k=0}^{m}{\begin{pmatrix}\left\lceil {\frac {n-1}{3}}\right\rceil \\k\end{pmatrix}}{\begin{pmatrix}\left\lfloor {\frac {n}{3}}\right\rfloor \\m-k-\left\lceil {\frac {n}{3}}\right\rceil \end{pmatrix}}\times 2^{\left\lceil {\frac {n}{3}}\right\rceil +\left\lfloor {\frac {n}{3}}\right\rfloor -(m-k)}&&{\pmod {3}}\,.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/239af2feb0396383f4f62e5f616f97adfa2e53b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:76.598ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}{\begin{bmatrix}n\\m\end{bmatrix}}&\equiv [x^{m}]\left(x^{\left\lceil {\frac {n}{3}}\right\rceil }(x+1)^{\left\lceil {\frac {n-1}{3}}\right\rceil }(x+2)^{\left\lfloor {\frac {n}{3}}\right\rfloor }\right)&&{\pmod {3}}\\&\equiv \sum _{k=0}^{m}{\begin{pmatrix}\left\lceil {\frac {n-1}{3}}\right\rceil \\k\end{pmatrix}}{\begin{pmatrix}\left\lfloor {\frac {n}{3}}\right\rfloor \\m-k-\left\lceil {\frac {n}{3}}\right\rceil \end{pmatrix}}\times 2^{\left\lceil {\frac {n}{3}}\right\rceil +\left\lfloor {\frac {n}{3}}\right\rfloor -(m-k)}&&{\pmod {3}}\,.\end{aligned}}}"> <div class="mw-heading mw-heading4"><h4 id="Congruences_for_the_partition_function">Congruences for the partition function</h4>[<a href="/w/index.php?title=Generating_function&action=edit&section=52" title="Edit section: Congruences for the partition function">edit</a>]</div> In this example, we pull in some of the machinery of infinite products whose power series expansions generate the expansions of many special functions and enumerate partition functions. In particular, we recall that the <a href="/wiki/Partition_function_(number_theory)" title="Partition function (number theory)">partition function</a> p(n) is generated by the reciprocal infinite <a href="/wiki/Q-Pochhammer_symbol" title="Q-Pochhammer symbol">q-Pochhammer symbol</a> product (or z-Pochhammer product as the case may be) given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }p(n)z^{n}&={\frac {1}{\left(1-z\right)\left(1-z^{2}\right)\left(1-z^{3}\right)\cdots }}\\[4pt]&=1+z+2z^{2}+3z^{3}+5z^{4}+7z^{5}+11z^{6}+\cdots .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo>+</mo> <mn>2</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>11</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }p(n)z^{n}&={\frac {1}{\left(1-z\right)\left(1-z^{2}\right)\left(1-z^{3}\right)\cdots }}\\[4pt]&=1+z+2z^{2}+3z^{3}+5z^{4}+7z^{5}+11z^{6}+\cdots .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee67b6e05ea0ee85c1d1d8a8577a265ffaec6fb5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:57.862ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }p(n)z^{n}&={\frac {1}{\left(1-z\right)\left(1-z^{2}\right)\left(1-z^{3}\right)\cdots }}\\[4pt]&=1+z+2z^{2}+3z^{3}+5z^{4}+7z^{5}+11z^{6}+\cdots .\end{aligned}}}"> This partition function satisfies many known <a href="/wiki/Ramanujan%27s_congruences" title="Ramanujan's congruences">congruence properties</a>, which notably include the following results though there are still many open questions about the forms of related integer congruences for the function:<a href="#cite_note-27">[26]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(5m+4)&\equiv 0{\pmod {5}}\\p(7m+5)&\equiv 0{\pmod {7}}\\p(11m+6)&\equiv 0{\pmod {11}}\\p(25m+24)&\equiv 0{\pmod {5^{2}}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>m</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mi>m</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>11</mn> <mi>m</mi> <mo>+</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>11</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>25</mn> <mi>m</mi> <mo>+</mo> <mn>24</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(5m+4)&\equiv 0{\pmod {5}}\\p(7m+5)&\equiv 0{\pmod {7}}\\p(11m+6)&\equiv 0{\pmod {11}}\\p(25m+24)&\equiv 0{\pmod {5^{2}}}\,.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d57d984971757d8e59c4bf99b02783a9b6dbdd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:30.456ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}p(5m+4)&\equiv 0{\pmod {5}}\\p(7m+5)&\equiv 0{\pmod {7}}\\p(11m+6)&\equiv 0{\pmod {11}}\\p(25m+24)&\equiv 0{\pmod {5^{2}}}\,.\end{aligned}}}"> We show how to use generating functions and manipulations of congruences for formal power series to give a highly elementary proof of the first of these congruences listed above. First, we observe that in the binomial coefficient generating function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(1-z)^{5}}}=\sum _{i=0}^{\infty }{\binom {4+i}{4}}z^{i}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>4</mn> <mo>+</mo> <mi>i</mi> </mrow> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1-z)^{5}}}=\sum _{i=0}^{\infty }{\binom {4+i}{4}}z^{i}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e767082b43dcca7351b7ed93911467cb3c7c1c54" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.782ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{(1-z)^{5}}}=\sum _{i=0}^{\infty }{\binom {4+i}{4}}z^{i}\,,}"> all of the coefficients are divisible by 5 except for those which correspond to the powers 1, z5, z10, ... and moreover in those cases the remainder of the coefficient is 1 modulo 5. Thus, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(1-z)^{5}}}\equiv {\frac {1}{1-z^{5}}}{\pmod {5}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1-z)^{5}}}\equiv {\frac {1}{1-z^{5}}}{\pmod {5}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd2228e23dffa8a6ccf3193f4b01e5a0005d972d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.753ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{(1-z)^{5}}}\equiv {\frac {1}{1-z^{5}}}{\pmod {5}}\,,}"> or equivalently <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1-z^{5}}{(1-z)^{5}}}\equiv 1{\pmod {5}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1-z^{5}}{(1-z)^{5}}}\equiv 1{\pmod {5}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cdc0d955647354cedc9879fd410aefd91acd59c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.932ex; height:6.509ex;" alt="{\displaystyle {\frac {1-z^{5}}{(1-z)^{5}}}\equiv 1{\pmod {5}}\,.}"> It follows that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\left(1-z^{15}\right)\cdots }{\left((1-z)\left(1-z^{2}\right)\left(1-z^{3}\right)\cdots \right)^{5}}}\equiv 1{\pmod {5}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\left(1-z^{15}\right)\cdots }{\left((1-z)\left(1-z^{2}\right)\left(1-z^{3}\right)\cdots \right)^{5}}}\equiv 1{\pmod {5}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6941dfbab037755ccc6a9b42a43202281b42100" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.338ex; height:7.343ex;" alt="{\displaystyle {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\left(1-z^{15}\right)\cdots }{\left((1-z)\left(1-z^{2}\right)\left(1-z^{3}\right)\cdots \right)^{5}}}\equiv 1{\pmod {5}}\,.}"> Using the infinite product expansions of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\cdot {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{\left(1-z\right)\left(1-z^{2}\right)\cdots }}=z\cdot \left((1-z)\left(1-z^{2}\right)\cdots \right)^{4}\times {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{\left(\left(1-z\right)\left(1-z^{2}\right)\cdots \right)^{5}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>z</mi> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\cdot {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{\left(1-z\right)\left(1-z^{2}\right)\cdots }}=z\cdot \left((1-z)\left(1-z^{2}\right)\cdots \right)^{4}\times {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{\left(\left(1-z\right)\left(1-z^{2}\right)\cdots \right)^{5}}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09679c234aaaf38b231afddae31289e747802d1f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.13ex; height:7.343ex;" alt="{\displaystyle z\cdot {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{\left(1-z\right)\left(1-z^{2}\right)\cdots }}=z\cdot \left((1-z)\left(1-z^{2}\right)\cdots \right)^{4}\times {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{\left(\left(1-z\right)\left(1-z^{2}\right)\cdots \right)^{5}}}\,,}"> it can be shown that the coefficient of z5m + 5 in z · ((1 − z)(1 − z2)⋯)4 is divisible by 5 for all m.<a href="#cite_note-28">[27]</a> Finally, since <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }p(n-1)z^{n}&={\frac {z}{(1-z)\left(1-z^{2}\right)\cdots }}\\[6px]&=z\cdot {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{(1-z)\left(1-z^{2}\right)\cdots }}\times \left(1+z^{5}+z^{10}+\cdots \right)\left(1+z^{10}+z^{20}+\cdots \right)\cdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }p(n-1)z^{n}&={\frac {z}{(1-z)\left(1-z^{2}\right)\cdots }}\\[6px]&=z\cdot {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{(1-z)\left(1-z^{2}\right)\cdots }}\times \left(1+z^{5}+z^{10}+\cdots \right)\left(1+z^{10}+z^{20}+\cdots \right)\cdots \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb9132341ba8ba6de85071af9020fafdf221292" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:89.21ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }p(n-1)z^{n}&={\frac {z}{(1-z)\left(1-z^{2}\right)\cdots }}\\[6px]&=z\cdot {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{(1-z)\left(1-z^{2}\right)\cdots }}\times \left(1+z^{5}+z^{10}+\cdots \right)\left(1+z^{10}+z^{20}+\cdots \right)\cdots \end{aligned}}}"> we may equate the coefficients of z5m + 5 in the previous equations to prove our desired congruence result, namely that p(5m + 4) ≡ 0 (mod 5) for all m ≥ 0. <div class="mw-heading mw-heading3"><h3 id="Transformations_of_generating_functions">Transformations of generating functions</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=53" title="Edit section: Transformations of generating functions">edit</a>]</div> There are a number of transformations of generating functions that provide other applications (see the <a href="/wiki/Generating_function_transformation" title="Generating function transformation">main article</a>). A transformation of a sequence's ordinary generating function (OGF) provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas involving a sequence OGF (see <a href="/wiki/Generating_function_transformation#Integral_Transformations" title="Generating function transformation">integral transformations</a>) or weighted sums over the higher-order derivatives of these functions (see <a href="/wiki/Generating_function_transformation#Derivative_Transformations" title="Generating function transformation">derivative transformations</a>). Generating function transformations can come into play when we seek to express a generating function for the sums <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}:=\sum _{m=0}^{n}{\binom {n}{m}}C_{n,m}a_{m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}:=\sum _{m=0}^{n}{\binom {n}{m}}C_{n,m}a_{m},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4019e36c19ef8004758be1534c5dcf5597e3e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.778ex; height:6.843ex;" alt="{\displaystyle s_{n}:=\sum _{m=0}^{n}{\binom {n}{m}}C_{n,m}a_{m},}"> in the form of S(z) = g(z) A(f(z)) involving the original sequence generating function. For example, if the sums are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}:=\sum _{k=0}^{\infty }{\binom {n+k}{m+2k}}a_{k}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}:=\sum _{k=0}^{\infty }{\binom {n+k}{m+2k}}a_{k}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc65030fce606501cc90f679e0cd8783c47a3b66" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.178ex; height:7.009ex;" alt="{\displaystyle s_{n}:=\sum _{k=0}^{\infty }{\binom {n+k}{m+2k}}a_{k}\,}"> then the generating function for the modified sum expressions is given by<a href="#cite_note-29">[28]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(z)={\frac {z^{m}}{(1-z)^{m+1}}}A\left({\frac {z}{(1-z)^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>A</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(z)={\frac {z^{m}}{(1-z)^{m+1}}}A\left({\frac {z}{(1-z)^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9997d81222c5b420ae22422569a4f161c700c0eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.349ex; height:6.343ex;" alt="{\displaystyle S(z)={\frac {z^{m}}{(1-z)^{m+1}}}A\left({\frac {z}{(1-z)^{2}}}\right)}"> (see also the <a href="/wiki/Binomial_transform" title="Binomial transform">binomial transform</a> and the <a href="/wiki/Stirling_transform" title="Stirling transform">Stirling transform</a>). There are also integral formulas for converting between a sequence's OGF, F(z), and its exponential generating function, or EGF, F̂(z), and vice versa given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F(z)&=\int _{0}^{\infty }{\hat {F}}(tz)e^{-t}\,dt\,,\\[4px]{\hat {F}}(z)&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }F\left(ze^{-i\vartheta }\right)e^{e^{i\vartheta }}\,d\vartheta \,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ϑ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϑ</mi> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ϑ</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(z)&=\int _{0}^{\infty }{\hat {F}}(tz)e^{-t}\,dt\,,\\[4px]{\hat {F}}(z)&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }F\left(ze^{-i\vartheta }\right)e^{e^{i\vartheta }}\,d\vartheta \,,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3d2c5436a62f3426646ba7b8d45430a1f6ff3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.654ex; margin-bottom: -0.184ex; width:33.775ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}F(z)&=\int _{0}^{\infty }{\hat {F}}(tz)e^{-t}\,dt\,,\\[4px]{\hat {F}}(z)&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }F\left(ze^{-i\vartheta }\right)e^{e^{i\vartheta }}\,d\vartheta \,,\end{aligned}}}"> provided that these integrals converge for appropriate values of z. <div class="mw-heading mw-heading2"><h2 id="Tables_of_special_generating_functions">Tables of special generating functions</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=54" title="Edit section: Tables of special generating functions">edit</a>]</div> An initial listing of special mathematical series is found <a href="/wiki/List_of_mathematical_series" title="List of mathematical series">here</a>. A number of useful and special sequence generating functions are found in Section 5.4 and 7.4 of Concrete Mathematics and in Section 2.5 of Wilf's Generatingfunctionology. Other special generating functions of note include the entries in the next table, which is by no means complete.<a href="#cite_note-30">[29]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></td><td class="mbox-text"><div class="mbox-text-span">This section needs expansion with: Lists of special and special sequence generating functions. The next table is a start. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Generating_function&action=edit&section=">adding to it</a>. (April 2017)</div></td></tr></tbody></table> <dl><dd><table class="wikitable"> <tbody><tr> <th>Formal power series</th> <th>Generating-function formula</th> <th>Notes </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\binom {m+n}{n}}\left(H_{n+m}-H_{m}\right)z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\binom {m+n}{n}}\left(H_{n+m}-H_{m}\right)z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/460249a49b0156d4ba04a65570fa6864fad439ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.649ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\binom {m+n}{n}}\left(H_{n+m}-H_{m}\right)z^{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(1-z)^{m+1}}}\ln {\frac {1}{1-z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1-z)^{m+1}}}\ln {\frac {1}{1-z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056701e71b4d4b68249106d4443855e91b26f1e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.153ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{(1-z)^{m+1}}}\ln {\frac {1}{1-z}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e63458b04288bbe116a9a8037dfae0b36b2c639a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.15ex; height:2.509ex;" alt="{\displaystyle H_{n}}"> is a first-order <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }B_{n}{\frac {z^{n}}{n!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }B_{n}{\frac {z^{n}}{n!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9902991620add9589e215cdef0b4556c97505c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.869ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }B_{n}{\frac {z^{n}}{n!}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z}{e^{z}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {z}{e^{z}-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/555af2b62daebe76a76ee4bed8838038eab62de4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.924ex; height:4.843ex;" alt="{\displaystyle {\frac {z}{e^{z}-1}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}"> is a <a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli number</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }F_{mn}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }F_{mn}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1bc39d92a5ebcabfc16a6605293b928f5375e0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.207ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }F_{mn}z^{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{m}z}{1-(F_{m-1}+F_{m+1})z+(-1)^{m}z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>z</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>z</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{m}z}{1-(F_{m-1}+F_{m+1})z+(-1)^{m}z^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c42f991771fe4d3d4e02db77eee6453da014ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.557ex; height:6.009ex;" alt="{\displaystyle {\frac {F_{m}z}{1-(F_{m-1}+F_{m+1})z+(-1)^{m}z^{2}}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"> is a <a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci number</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {Z} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {Z} ^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa93f7da8db0f7cbbaaa8ce8ef18cb50d41e129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.942ex; height:2.509ex;" alt="{\displaystyle m\in \mathbb {Z} ^{+}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }\left\{{\begin{matrix}n\\m\end{matrix}}\right\}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }\left\{{\begin{matrix}n\\m\end{matrix}}\right\}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/499e8286cebad1e9c9d71e3c1fceb1e2e2509b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.716ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }\left\{{\begin{matrix}n\\m\end{matrix}}\right\}z^{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z^{-1})^{\overline {-m}}={\frac {z^{m}}{(1-z)(1-2z)\cdots (1-mz)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo>−</mo> <mi>m</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (z^{-1})^{\overline {-m}}={\frac {z^{m}}{(1-z)(1-2z)\cdots (1-mz)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be5fa6e128cd1b24490e71d63cc69e6b5719695b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.615ex; height:6.009ex;" alt="{\displaystyle (z^{-1})^{\overline {-m}}={\frac {z^{m}}{(1-z)(1-2z)\cdots (1-mz)}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\overline {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\overline {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c19902c6e1e5dca4e5265746c4b5e5d65a3c676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.641ex; height:2.843ex;" alt="{\displaystyle x^{\overline {n}}}"> denotes the <a href="/wiki/Rising_factorial" class="mw-redirect" title="Rising factorial">rising factorial</a>, or <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a> and some integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d2d765e4cfd7adfbca9ae0e37e75a2811c0333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle m\geq 0}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }\left[{\begin{matrix}n\\m\end{matrix}}\right]z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }\left[{\begin{matrix}n\\m\end{matrix}}\right]z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ae0024b86ced7034b6a4501cee55be3bd269b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.685ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }\left[{\begin{matrix}n\\m\end{matrix}}\right]z^{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{\overline {m}}=z(z+1)\cdots (z+m-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>=</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{\overline {m}}=z(z+1)\cdots (z+m-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffac8c46e8c9300ca686fba37f3020864d9398c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.224ex; height:3.343ex;" alt="{\displaystyle z^{\overline {m}}=z(z+1)\cdots (z+m-1)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n-1}4^{n}(4^{n}-2)B_{2n}z^{2n}}{(2n)\cdot (2n)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n-1}4^{n}(4^{n}-2)B_{2n}z^{2n}}{(2n)\cdot (2n)!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b94bcd4bb7e733f896ffbe28968e6f681e2e801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.186ex; height:7.009ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n-1}4^{n}(4^{n}-2)B_{2n}z^{2n}}{(2n)\cdot (2n)!}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {\frac {\tan(z)}{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>z</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln {\frac {\tan(z)}{z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339b994d488077aa152a572185fe0d46aacc0c59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.42ex; height:5.676ex;" alt="{\displaystyle \ln {\frac {\tan(z)}{z}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\frac {(1/2)^{\overline {n}}z^{2n}}{(2n+1)\cdot n!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {(1/2)^{\overline {n}}z^{2n}}{(2n+1)\cdot n!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d503c6b4c5e2f1ae98983426c3d5668396010e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.668ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {(1/2)^{\overline {n}}z^{2n}}{(2n+1)\cdot n!}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{-1}\arcsin(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{-1}\arcsin(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84a7fff022e479d3020969ada3a1a650f86cdf50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.67ex; height:3.176ex;" alt="{\displaystyle z^{-1}\arcsin(z)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }H_{n}^{(s)}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }H_{n}^{(s)}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1f682985d26ce5ea119946a42c5c931162aad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.437ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }H_{n}^{(s)}z^{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\operatorname {Li} _{s}(z)}{1-z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Li</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {Li} _{s}(z)}{1-z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f8205efc6fe99de91f2e2027ebca6a11ce0e2b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.837ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {Li} _{s}(z)}{1-z}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Li} _{s}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Li</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} _{s}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e650380250d97d59f35d1572402f4c952f05462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6ex; height:2.843ex;" alt="{\displaystyle \operatorname {Li} _{s}(z)}"> is the <a href="/wiki/Polylogarithm" title="Polylogarithm">polylogarithm</a> function and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}^{(s)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}^{(s)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9596b382ca909028c4c666377e7ae95bfdc5692b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.386ex; height:3.343ex;" alt="{\displaystyle H_{n}^{(s)}}"> is a generalized <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea5968732ea35f67b36093364400e0fd8ca23bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }n^{m}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }n^{m}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608a8220eeb2e5daa17d10fcb79cb048e7c2e4a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.121ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }n^{m}z^{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{0\leq j\leq m}\left\{{\begin{matrix}m\\j\end{matrix}}\right\}{\frac {j!\cdot z^{j}}{(1-z)^{j+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>m</mi> </mrow> </munder> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>j</mi> <mo>!</mo> <mo>⋅</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{0\leq j\leq m}\left\{{\begin{matrix}m\\j\end{matrix}}\right\}{\frac {j!\cdot z^{j}}{(1-z)^{j+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d54594b0404e150e3b06359eebc6d4065b16180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.298ex; height:7.176ex;" alt="{\displaystyle \sum _{0\leq j\leq m}\left\{{\begin{matrix}m\\j\end{matrix}}\right\}{\frac {j!\cdot z^{j}}{(1-z)^{j+1}}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{matrix}n\\m\end{matrix}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{matrix}n\\m\end{matrix}}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec8d5672e9a708baec85f4012902a9c0eaa396b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.278ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{matrix}n\\m\end{matrix}}\right\}}"> is a <a href="/wiki/Stirling_number_of_the_second_kind" class="mw-redirect" title="Stirling number of the second kind">Stirling number of the second kind</a> and where the individual terms in the expansion satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z^{i}}{(1-z)^{i+1}}}=\sum _{k=0}^{i}{\binom {i}{k}}{\frac {(-1)^{k-i}}{(1-z)^{k+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>i</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {z^{i}}{(1-z)^{i+1}}}=\sum _{k=0}^{i}{\binom {i}{k}}{\frac {(-1)^{k-i}}{(1-z)^{k+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a23c8a511fc71588941383c5d4aecab530db81c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.035ex; height:7.343ex;" alt="{\displaystyle {\frac {z^{i}}{(1-z)^{i+1}}}=\sum _{k=0}^{i}{\binom {i}{k}}{\frac {(-1)^{k-i}}{(1-z)^{k+1}}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k<n}{\binom {n-k}{k}}{\frac {n}{n-k}}z^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo><</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k<n}{\binom {n-k}{k}}{\frac {n}{n-k}}z^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8bf104229429a4479b7d0e37a096391946d3d5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.071ex; height:6.843ex;" alt="{\displaystyle \sum _{k<n}{\binom {n-k}{k}}{\frac {n}{n-k}}z^{k}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {1+{\sqrt {1+4z}}}{2}}\right)^{n}+\left({\frac {1-{\sqrt {1+4z}}}{2}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mi>z</mi> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mi>z</mi> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {1+{\sqrt {1+4z}}}{2}}\right)^{n}+\left({\frac {1-{\sqrt {1+4z}}}{2}}\right)^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2cd6a80be8e516d1298ac4cd6d8dce75085ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.176ex; height:6.509ex;" alt="{\displaystyle \left({\frac {1+{\sqrt {1+4z}}}{2}}\right)^{n}+\left({\frac {1-{\sqrt {1+4z}}}{2}}\right)^{n}}"></td> <td> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n_{1},\ldots ,n_{m}\geq 0}\min(n_{1},\ldots ,n_{m})z_{1}^{n_{1}}\cdots z_{m}^{n_{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n_{1},\ldots ,n_{m}\geq 0}\min(n_{1},\ldots ,n_{m})z_{1}^{n_{1}}\cdots z_{m}^{n_{m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f22807cefac95870e2cd2a376fd8eef6b05f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.131ex; height:5.843ex;" alt="{\displaystyle \sum _{n_{1},\ldots ,n_{m}\geq 0}\min(n_{1},\ldots ,n_{m})z_{1}^{n_{1}}\cdots z_{m}^{n_{m}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z_{1}\cdots z_{m}}{(1-z_{1})\cdots (1-z_{m})(1-z_{1}\cdots z_{m})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {z_{1}\cdots z_{m}}{(1-z_{1})\cdots (1-z_{m})(1-z_{1}\cdots z_{m})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e96e44a974e19d9615283d458a93ad128425b14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.051ex; height:5.509ex;" alt="{\displaystyle {\frac {z_{1}\cdots z_{m}}{(1-z_{1})\cdots (1-z_{m})(1-z_{1}\cdots z_{m})}}}"></td> <td>The two-variable case is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(w,z):=\sum _{m,n\geq 0}\min(m,n)w^{m}z^{n}={\frac {wz}{(1-w)(1-z)(1-wz)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>w</mi> <mi>z</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>w</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(w,z):=\sum _{m,n\geq 0}\min(m,n)w^{m}z^{n}={\frac {wz}{(1-w)(1-z)(1-wz)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b961c9aa7d3930988f1a74d115cac4cbc3f8d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:59.834ex; height:6.176ex;" alt="{\displaystyle M(w,z):=\sum _{m,n\geq 0}\min(m,n)w^{m}z^{n}={\frac {wz}{(1-w)(1-z)(1-wz)}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\binom {s}{n}}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>s</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\binom {s}{n}}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b94893b1e51a1051708b62dba9eb531a75b72c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.867ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\binom {s}{n}}z^{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+z)^{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+z)^{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cf46051ebfea509a48b43fe2899437830cd2dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.904ex; height:2.843ex;" alt="{\displaystyle (1+z)^{s}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13da7c2bac9dd6324ac83093fb42f3a9e86fbd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {C} }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\binom {n}{k}}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\binom {n}{k}}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1ae8325fa132fc63f45ba302208bc8a5d1776e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.867ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\binom {n}{k}}z^{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z^{k}}{(1-z)^{k+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {z^{k}}{(1-z)^{k+1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e76587709171c622d8c4f22a34566cac1c8930ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.926ex; height:6.509ex;" alt="{\displaystyle {\frac {z^{k}}{(1-z)^{k+1}}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }\log {(n)}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }\log {(n)}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26bd8cdaf9a0c810a87a9ca2b63b9f480e4f39a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.614ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }\log {(n)}z^{n}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.-{\frac {\partial }{\partial s}}\operatorname {{Li}_{s}(z)} \right|_{s=0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.-{\frac {\partial }{\partial s}}\operatorname {{Li}_{s}(z)} \right|_{s=0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c12f2bd1651494d6861b867c0316335a02b032a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.012ex; height:6.009ex;" alt="{\displaystyle \left.-{\frac {\partial }{\partial s}}\operatorname {{Li}_{s}(z)} \right|_{s=0}}"> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=55" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Moment-generating_function" title="Moment-generating function">Moment-generating function</a></li> <li><a href="/wiki/Probability-generating_function" title="Probability-generating function">Probability-generating function</a></li> <li><a href="/wiki/Generating_function_transformation" title="Generating function transformation">Generating function transformation</a></li> <li><a href="/wiki/Stanley%27s_reciprocity_theorem" title="Stanley's reciprocity theorem">Stanley's reciprocity theorem</a></li> <li><a href="/wiki/Integer_partition" title="Integer partition">Integer partition</a></li> <li><a href="/wiki/Combinatorial_principles" title="Combinatorial principles">Combinatorial principles</a></li> <li><a href="/wiki/Cyclic_sieving" title="Cyclic sieving">Cyclic sieving</a></li> <li><a href="/wiki/Z-transform" title="Z-transform">Z-transform</a></li> <li><a href="/wiki/Umbral_calculus" title="Umbral calculus">Umbral calculus</a></li> <li><a href="/wiki/Coins_in_a_fountain" title="Coins in a fountain">Coins in a fountain</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=56" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-23"><a href="#cite_ref-23">^</a> Incidentally, we also have a corresponding formula when m < 0 given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }g_{n+m}z^{n}={\frac {G(z)-g_{0}-g_{1}z-\cdots -g_{m-1}z^{m-1}}{z^{m}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>−</mo> <mo>⋯</mo> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }g_{n+m}z^{n}={\frac {G(z)-g_{0}-g_{1}z-\cdots -g_{m-1}z^{m-1}}{z^{m}}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/676d3eec4ddb66ab619c332362f2d8648f501cc3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.042ex; height:7.009ex;" alt="{\displaystyle \sum _{n=0}^{\infty }g_{n+m}z^{n}={\frac {G(z)-g_{0}-g_{1}z-\cdots -g_{m-1}z^{m-1}}{z^{m}}}\,.}"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=57" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> This alternative term can already be found in E.N. Gilbert (1956), "Enumeration of Labeled graphs", <a href="/wiki/Canadian_Journal_of_Mathematics" title="Canadian Journal of Mathematics">Canadian Journal of Mathematics</a> 3, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x34z99fCRbsC&dq=%22generating+series%22&pg=PA407">p. 405–411</a>, but its use is rare before the year 2000; since then it appears to be increasing. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKnuth1997" class="citation book cs1"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (1997). "§1.2.9 Generating Functions". Fundamental Algorithms. <a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>. Vol. 1 (3rd ed.). Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-89683-4" title="Special:BookSources/0-201-89683-4"><bdi>0-201-89683-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A71.2.9+Generating+Functions&rft.btitle=Fundamental+Algorithms&rft.series=The+Art+of+Computer+Programming&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=1997&rft.isbn=0-201-89683-4&rft.aulast=Knuth&rft.aufirst=Donald+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFFlajoletSedgewick2009">Flajolet & Sedgewick 2009</a>, p. 95 </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/140418">"Lambert series identity"</a>. Math Overflow. 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+Overflow&rft.atitle=Lambert+series+identity&rft.date=2017&rft_id=https%3A%2F%2Fmathoverflow.net%2Fq%2F140418&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol1976" class="citation cs2"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90163-3" title="Special:BookSources/978-0-387-90163-3"><bdi>978-0-387-90163-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+analytic+number+theory&rft.place=New+York-Heidelberg&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1976&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0335.10001%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0434929%23id-name%3DMR&rft.isbn=978-0-387-90163-3&rft.aulast=Apostol&rft.aufirst=Tom+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> pp.42–43 </li> <li id="cite_note-W56-6"><a href="#cite_ref-W56_6-0">^</a> <a href="#CITEREFWilf1994">Wilf 1994</a>, p. 56 </li> <li id="cite_note-W59-7"><a href="#cite_ref-W59_7-0">^</a> <a href="#CITEREFWilf1994">Wilf 1994</a>, p. 59 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyWrightHeath-BrownSilverman2008" class="citation book cs1">Hardy, G.H.; Wright, E.M.; Heath-Brown, D.R; Silverman, J.H. (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00ghha_922">An Introduction to the Theory of Numbers</a> (6th ed.). Oxford University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00ghha_922/page/n357">339</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780199219858" title="Special:BookSources/9780199219858"><bdi>9780199219858</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.pages=339&rft.edition=6th&rft.pub=Oxford+University+Press&rft.date=2008&rft.isbn=9780199219858&rft.aulast=Hardy&rft.aufirst=G.H.&rft.au=Wright%2C+E.M.&rft.au=Heath-Brown%2C+D.R&rft.au=Silverman%2C+J.H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoth00ghha_922&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1992" class="citation journal cs1">Knuth, D. E. (1992). "Convolution Polynomials". Mathematica J. 2: 67–78. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9207221">math/9207221</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992math......7221K">1992math......7221K</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematica+J.&rft.atitle=Convolution+Polynomials&rft.volume=2&rft.pages=67-78&rft.date=1992&rft_id=info%3Aarxiv%2Fmath%2F9207221&rft_id=info%3Abibcode%2F1992math......7221K&rft.aulast=Knuth&rft.aufirst=D.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivey2007" class="citation journal cs1">Spivey, Michael Z. (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.disc.2007.03.052">"Combinatorial Sums and Finite Differences"</a>. Discrete Math. 307 (24): 3130–3146. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.disc.2007.03.052">10.1016/j.disc.2007.03.052</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2370116">2370116</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Math.&rft.atitle=Combinatorial+Sums+and+Finite+Differences&rft.volume=307&rft.issue=24&rft.pages=3130-3146&rft.date=2007&rft_id=info%3Adoi%2F10.1016%2Fj.disc.2007.03.052&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2370116%23id-name%3DMR&rft.aulast=Spivey&rft.aufirst=Michael+Z.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.disc.2007.03.052&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMathar2012" class="citation arxiv cs1">Mathar, R. J. (2012). "Yet another table of integrals". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1207.5845">1207.5845</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CA">math.CA</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Yet+another+table+of+integrals&rft.date=2012&rft_id=info%3Aarxiv%2F1207.5845&rft.aulast=Mathar&rft.aufirst=R.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> v4 eq. (0.4) </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, Table 265 in §6.1 for finite sum identities involving the Stirling number triangles. </li> <li id="cite_note-GFLECT-13"><a href="#cite_ref-GFLECT_13-0">^</a> <a href="#CITEREFLando2003">Lando 2003</a>, §2.4 </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> Example from <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStanleyFomin1997" class="citation book cs1">Stanley, Richard P.; Fomin, Sergey (1997). "§6.3". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zg5wDqT6T-UC">Enumerative Combinatorics: Volume 2</a>. Cambridge Studies in Advanced Mathematics. Vol. 62. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-78987-5" title="Special:BookSources/978-0-521-78987-5"><bdi>978-0-521-78987-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A76.3&rft.btitle=Enumerative+Combinatorics%3A+Volume+2&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=978-0-521-78987-5&rft.aulast=Stanley&rft.aufirst=Richard+P.&rft.au=Fomin%2C+Sergey&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dzg5wDqT6T-UC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> </li> <li id="cite_note-TAOCPV1-15"><a href="#cite_ref-TAOCPV1_15-0">^</a> <a href="#CITEREFKnuth1997">Knuth 1997</a>, §1.2.9 </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> Solution to <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, p. 569, exercise 7.36 </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFFlajoletSedgewick2009">Flajolet & Sedgewick 2009</a>, §B.4 </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchneider2007" class="citation journal cs1">Schneider, C. (2007). <a rel="nofollow" class="external text" href="http://www.emis.de/journals/SLC/wpapers/s56schneider.html">"Symbolic Summation Assists Combinatorics"</a>. Sém. Lothar. Combin. 56: 1–36.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=S%C3%A9m.+Lothar.+Combin.&rft.atitle=Symbolic+Summation+Assists+Combinatorics&rft.volume=56&rft.pages=1-36&rft.date=2007&rft.aulast=Schneider&rft.aufirst=C.&rft_id=http%3A%2F%2Fwww.emis.de%2Fjournals%2FSLC%2Fwpapers%2Fs56schneider.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> See the usage of these terms in <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, §7.4 on special sequence generating functions. </li> <li id="cite_note-Good_1986-20"><a href="#cite_ref-Good_1986_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGood1986" class="citation journal cs1">Good, I. J. (1986). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176343649">"On applications of symmetric Dirichlet distributions and their mixtures to contingency tables"</a>. <a href="/wiki/Annals_of_Statistics" title="Annals of Statistics">Annals of Statistics</a>. 4 (6): 1159–1189. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176343649">10.1214/aos/1176343649</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Statistics&rft.atitle=On+applications+of+symmetric+Dirichlet+distributions+and+their+mixtures+to+contingency+tables&rft.volume=4&rft.issue=6&rft.pages=1159-1189&rft.date=1986&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176343649&rft.aulast=Good&rft.aufirst=I.+J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176343649&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> For more complete information on the properties of J-fractions see: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlajolet1980" class="citation journal cs1">Flajolet, P. (1980). <a rel="nofollow" class="external text" href="http://algo.inria.fr/flajolet/Publications/Flajolet80b.pdf">"Combinatorial aspects of continued fractions"</a> (PDF). Discrete Mathematics. 32 (2): 125–161. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2880%2990050-3">10.1016/0012-365X(80)90050-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=Combinatorial+aspects+of+continued+fractions&rft.volume=32&rft.issue=2&rft.pages=125-161&rft.date=1980&rft_id=info%3Adoi%2F10.1016%2F0012-365X%2880%2990050-3&rft.aulast=Flajolet&rft.aufirst=P.&rft_id=http%3A%2F%2Falgo.inria.fr%2Fflajolet%2FPublications%2FFlajolet80b.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWall2018" class="citation book cs1">Wall, H.S. (2018) [1948]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=86ReDwAAQBAJ&pg=PR7">Analytic Theory of Continued Fractions</a>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-83044-5" title="Special:BookSources/978-0-486-83044-5"><bdi>978-0-486-83044-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytic+Theory+of+Continued+Fractions&rft.pub=Dover&rft.date=2018&rft.isbn=978-0-486-83044-5&rft.aulast=Wall&rft.aufirst=H.S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D86ReDwAAQBAJ%26pg%3DPR7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li></ul> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> See the following articles: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2016" class="citation arxiv cs1">Schmidt, Maxie D. (2016). "Continued Fractions for Square Series Generating Functions". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1612.02778">1612.02778</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.NT">math.NT</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Continued+Fractions+for+Square+Series+Generating+Functions&rft.date=2016&rft_id=info%3Aarxiv%2F1612.02778&rft.aulast=Schmidt&rft.aufirst=Maxie+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2017" class="citation journal cs1">— (2017). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html">"Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions"</a>. Journal of Integer Sequences. 20. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1610.09691">1610.09691</a>. 17.3.4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Integer+Sequences&rft.atitle=Jacobi-Type+Continued+Fractions+for+the+Ordinary+Generating+Functions+of+Generalized+Factorial+Functions&rft.volume=20&rft.date=2017&rft_id=info%3Aarxiv%2F1610.09691&rft.aulast=Schmidt&rft.aufirst=Maxie+D.&rft_id=https%3A%2F%2Fcs.uwaterloo.ca%2Fjournals%2FJIS%2FVOL20%2FSchmidt%2Fschmidt14.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2017" class="citation arxiv cs1">— (2017). "Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers h ≥ 2". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1702.01374">1702.01374</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CO">math.CO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Jacobi-Type+Continued+Fractions+and+Congruences+for+Binomial+Coefficients+Modulo+Integers+h+%E2%89%A5+2&rft.date=2017&rft_id=info%3Aarxiv%2F1702.01374&rft.aulast=Schmidt&rft.aufirst=Maxie+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li></ul> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, §8.3 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, Example 6 in §7.3 for another method and the complete setup of this problem using generating functions. This more "convoluted" approach is given in Section 7.5 of the same reference. </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFLando2003">Lando 2003</a>, §5 </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFHardyWrightHeath-BrownSilverman2008">Hardy et al. 2008</a>, §19.12 </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyWright" class="citation book cs1">Hardy, G.H.; Wright, E.M. An Introduction to the Theory of Numbers.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.aulast=Hardy&rft.aufirst=G.H.&rft.au=Wright%2C+E.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> p.288, Th.361 </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, p. 535, exercise 5.71 </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> See also the 1031 Generating Functions found in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlouffe1992" class="citation thesis cs1 cs1-prop-foreign-lang-source">Plouffe, Simon (1992). Approximations de séries génératrices et quelques conjectures [Approximations of generating functions and a few conjectures] (Masters) (in French). Université du Québec à Montréal. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0911.4975">0911.4975</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Approximations+de+s%C3%A9ries+g%C3%A9n%C3%A9ratrices+et+quelques+conjectures&rft.inst=Universit%C3%A9+du+Qu%C3%A9bec+%C3%A0+Montr%C3%A9al&rft.date=1992&rft_id=info%3Aarxiv%2F0911.4975&rft.aulast=Plouffe&rft.aufirst=Simon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3>[<a href="/w/index.php?title=Generating_function&action=edit&section=58" title="Edit section: Citations">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAigner2007" class="citation book cs1">Aigner, Martin (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pPEJcu93dzAC">A Course in Enumeration</a>. Graduate Texts in Mathematics. Vol. 238. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-39035-0" title="Special:BookSources/978-3-540-39035-0"><bdi>978-3-540-39035-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Enumeration&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2007&rft.isbn=978-3-540-39035-0&rft.aulast=Aigner&rft.aufirst=Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpPEJcu93dzAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDoubiletRotaStanley1972" class="citation journal cs1">Doubilet, Peter; <a href="/wiki/Gian-Carlo_Rota" title="Gian-Carlo Rota">Rota, Gian-Carlo</a>; <a href="/wiki/Richard_P._Stanley" title="Richard P. Stanley">Stanley, Richard</a> (1972). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bsmsp/1200514223">"On the foundations of combinatorial theory. VI. The idea of generating function"</a>. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. 2: 267–318. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0267.05002">0267.05002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Sixth+Berkeley+Symposium+on+Mathematical+Statistics+and+Probability&rft.atitle=On+the+foundations+of+combinatorial+theory.+VI.+The+idea+of+generating+function&rft.volume=2&rft.pages=267-318&rft.date=1972&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0267.05002%23id-name%3DZbl&rft.aulast=Doubilet&rft.aufirst=Peter&rft.au=Rota%2C+Gian-Carlo&rft.au=Stanley%2C+Richard&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.bsmsp%2F1200514223&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"> Reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRota1975" class="citation book cs1"><a href="/wiki/Gian-Carlo_Rota" title="Gian-Carlo Rota">Rota, Gian-Carlo</a> (1975). "3. The idea of generating function". Finite Operator Calculus. With the collaboration of P. Doubilet, C. Greene, D. Kahaner, <a href="/wiki/Andrew_Odlyzko" title="Andrew Odlyzko">A. Odlyzko</a> and <a href="/wiki/Richard_P._Stanley" title="Richard P. Stanley">R. Stanley</a>. Academic Press. pp. 83–134. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-596650-4" title="Special:BookSources/0-12-596650-4"><bdi>0-12-596650-4</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0328.05007">0328.05007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.+The+idea+of+generating+function&rft.btitle=Finite+Operator+Calculus&rft.pages=83-134&rft.pub=Academic+Press&rft.date=1975&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0328.05007%23id-name%3DZbl&rft.isbn=0-12-596650-4&rft.aulast=Rota&rft.aufirst=Gian-Carlo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlajoletSedgewick2009" class="citation book cs1"><a href="/wiki/Philippe_Flajolet" title="Philippe Flajolet">Flajolet, Philippe</a>; <a href="/wiki/Robert_Sedgewick_(computer_scientist)" title="Robert Sedgewick (computer scientist)">Sedgewick, Robert</a> (2009). <a href="/wiki/Analytic_Combinatorics" class="mw-redirect" title="Analytic Combinatorics">Analytic Combinatorics</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-89806-5" title="Special:BookSources/978-0-521-89806-5"><bdi>978-0-521-89806-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1165.05001">1165.05001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytic+Combinatorics&rft.pub=Cambridge+University+Press&rft.date=2009&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1165.05001%23id-name%3DZbl&rft.isbn=978-0-521-89806-5&rft.aulast=Flajolet&rft.aufirst=Philippe&rft.au=Sedgewick%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGouldenJackson2004" class="citation book cs1">Goulden, Ian P.; <a href="/wiki/David_M._Jackson" title="David M. Jackson">Jackson, David M.</a> (2004). Combinatorial Enumeration. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0486435978" title="Special:BookSources/978-0486435978"><bdi>978-0486435978</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorial+Enumeration&rft.pub=Dover+Publications&rft.date=2004&rft.isbn=978-0486435978&rft.aulast=Goulden&rft.aufirst=Ian+P.&rft.au=Jackson%2C+David+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrahamKnuthPatashnik1994" class="citation book cs1"><a href="/wiki/Ronald_Graham" title="Ronald Graham">Graham, Ronald L.</a>; <a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a>; <a href="/wiki/Oren_Patashnik" title="Oren Patashnik">Patashnik, Oren</a> (1994). "Chapter 7: Generating Functions". <a href="/wiki/Concrete_Mathematics" title="Concrete Mathematics">Concrete Mathematics. A foundation for computer science</a> (2nd ed.). Addison-Wesley. pp. 320–380. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-55802-5" title="Special:BookSources/0-201-55802-5"><bdi>0-201-55802-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0836.00001">0836.00001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+7%3A+Generating+Functions&rft.btitle=Concrete+Mathematics.+A+foundation+for+computer+science&rft.pages=320-380&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1994&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0836.00001%23id-name%3DZbl&rft.isbn=0-201-55802-5&rft.aulast=Graham&rft.aufirst=Ronald+L.&rft.au=Knuth%2C+Donald+E.&rft.au=Patashnik%2C+Oren&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLando2003" class="citation book cs1">Lando, Sergei K. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=A6_4AwAAQBAJ">Lectures on Generating Functions</a>. American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3481-7" title="Special:BookSources/978-0-8218-3481-7"><bdi>978-0-8218-3481-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Generating+Functions&rft.pub=American+Mathematical+Society&rft.date=2003&rft.isbn=978-0-8218-3481-7&rft.aulast=Lando&rft.aufirst=Sergei+K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DA6_4AwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilf1994" class="citation book cs1"><a href="/wiki/Herbert_Wilf" title="Herbert Wilf">Wilf, Herbert S.</a> (1994). <a rel="nofollow" class="external text" href="http://www.math.upenn.edu/%7Ewilf/DownldGF.html">Generatingfunctionology</a> (2nd ed.). Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-751956-4" title="Special:BookSources/0-12-751956-4"><bdi>0-12-751956-4</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0831.05001">0831.05001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generatingfunctionology&rft.edition=2nd&rft.pub=Academic+Press&rft.date=1994&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0831.05001%23id-name%3DZbl&rft.isbn=0-12-751956-4&rft.aulast=Wilf&rft.aufirst=Herbert+S.&rft_id=http%3A%2F%2Fwww.math.upenn.edu%2F%257Ewilf%2FDownldGF.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Generating_function&action=edit&section=59" title="Edit section: External links">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://garsia.math.yorku.ca/~zabrocki/MMM1/MMM1Intro2OGFs.pdf">"Introduction To Ordinary Generating Functions"</a> by Mike Zabrocki, York University, Mathematics and Statistics</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Generating_function">"Generating function"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Generating+function&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DGenerating_function&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGenerating+function" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/ctk/GeneratingFunctions.shtml">Generating Functions, Power Indices and Coin Change</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/GeneratingFunctions/">"Generating Functions"</a> by <a href="/wiki/Ed_Pegg_Jr." title="Ed Pegg Jr.">Ed Pegg Jr.</a>, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>, 2007.</li></ul> <div class="navbox-styles"><style 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