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Orthogonal polynomials - Wikipedia
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data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Edit</span> </a></li> </ul> </nav><!-- version 1.0.2 (change every time you update a partial) --> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <p>In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematics">mathematics</a>, an <b>orthogonal polynomial sequence</b> is a family of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Polynomial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Polynomial">polynomials</a> such that any two different polynomials in the sequence are <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthogonality">orthogonal</a> to each other under some <a href="https://en-m-wikipedia-org.translate.goog/wiki/Inner_product?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Inner product">inner product</a>.</p> <p>The most widely used orthogonal polynomials are the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Classical_orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Classical orthogonal polynomials">classical orthogonal polynomials</a>, consisting of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hermite_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hermite polynomials">Hermite polynomials</a>, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Laguerre_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Laguerre polynomials">Laguerre polynomials</a> and the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Jacobi_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jacobi polynomials">Jacobi polynomials</a>. The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gegenbauer_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gegenbauer polynomials">Gegenbauer polynomials</a> form the most important class of Jacobi polynomials; they include the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Chebyshev_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Chebyshev polynomials">Chebyshev polynomials</a>, and the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Legendre_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Legendre polynomials">Legendre polynomials</a> as special cases.</p> <p>The field of orthogonal polynomials developed in the late 19th century from a study of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Continued_fraction?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Continued fraction">continued fractions</a> by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pafnuty_Chebyshev?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pafnuty Chebyshev">P. L. Chebyshev</a> and was pursued by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Andrey_Markov?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Andrey Markov">A. A. Markov</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Thomas_Joannes_Stieltjes?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Thomas Joannes Stieltjes">T. J. Stieltjes</a>. They appear in a wide variety of fields: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Numerical_analysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Numerical analysis">numerical analysis</a> (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Gaussian_quadrature?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gaussian quadrature">quadrature rules</a>), <a href="https://en-m-wikipedia-org.translate.goog/wiki/Probability_theory?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Probability theory">probability theory</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Representation_theory?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Representation theory">representation theory</a> (of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Lie_group?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lie group">Lie groups</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Quantum_group?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quantum group">quantum groups</a>, and related objects), <a href="https://en-m-wikipedia-org.translate.goog/wiki/Enumerative_combinatorics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Enumerative combinatorics">enumerative combinatorics</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Algebraic_combinatorics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Algebraic combinatorics">algebraic combinatorics</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematical_physics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematical physics">mathematical physics</a> (the theory of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Random_matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Random matrix">random matrices</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Integrable_system?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Integrable system">integrable systems</a>, etc.), and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Number_theory?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Number theory">number theory</a>. Some of the mathematicians who have worked on orthogonal polynomials include <a href="https://en-m-wikipedia-org.translate.goog/wiki/G%C3%A1bor_Szeg%C5%91?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gábor Szegő">Gábor Szegő</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sergei_Natanovich_Bernstein?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Sergei Natanovich Bernstein">Sergei Bernstein</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Naum_Akhiezer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Naum Akhiezer">Naum Akhiezer</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Arthur_Erd%C3%A9lyi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Arthur Erdélyi">Arthur Erdélyi</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Yakov_Geronimus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Yakov Geronimus">Yakov Geronimus</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wolfgang_Hahn?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wolfgang Hahn">Wolfgang Hahn</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Theodore_Seio_Chihara?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Theodore Seio Chihara">Theodore Seio Chihara</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mourad_Ismail?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mourad Ismail">Mourad Ismail</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Waleed_Al-Salam?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Waleed Al-Salam">Waleed Al-Salam</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Richard_Askey?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Richard Askey">Richard Askey</a>, and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rehuel_Lobatto?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rehuel Lobatto">Rehuel Lobatto</a>.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Definition_for_1-variable_case_for_a_real_measure"><span class="tocnumber">1</span> <span class="toctext">Definition for 1-variable case for a real measure</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Absolutely_continuous_case"><span class="tocnumber">1.1</span> <span class="toctext">Absolutely continuous case</span></a></li> </ul></li> <li class="toclevel-1 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Examples_of_orthogonal_polynomials"><span class="tocnumber">2</span> <span class="toctext">Examples of orthogonal polynomials</span></a></li> <li class="toclevel-1 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Properties"><span class="tocnumber">3</span> <span class="toctext">Properties</span></a> <ul> <li class="toclevel-2 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Relation_to_moments"><span class="tocnumber">3.1</span> <span class="toctext">Relation to moments</span></a></li> <li class="toclevel-2 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Recurrence_relation"><span class="tocnumber">3.2</span> <span class="toctext">Recurrence relation</span></a></li> <li class="toclevel-2 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Christoffel%E2%80%93Darboux_formula"><span class="tocnumber">3.3</span> <span class="toctext">Christoffel–Darboux formula</span></a></li> <li class="toclevel-2 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Zeros"><span class="tocnumber">3.4</span> <span class="toctext">Zeros</span></a></li> <li class="toclevel-2 tocsection-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Combinatorial_interpretation"><span class="tocnumber">3.5</span> <span class="toctext">Combinatorial interpretation</span></a></li> </ul></li> <li class="toclevel-1 tocsection-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Other_types_of_orthogonal_polynomials"><span class="tocnumber">4</span> <span class="toctext">Other types of orthogonal polynomials</span></a> <ul> <li class="toclevel-2 tocsection-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Multivariate_orthogonal_polynomials"><span class="tocnumber">4.1</span> <span class="toctext">Multivariate orthogonal polynomials</span></a></li> <li class="toclevel-2 tocsection-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Multiple_orthogonal_polynomials"><span class="tocnumber">4.2</span> <span class="toctext">Multiple orthogonal polynomials</span></a></li> <li class="toclevel-2 tocsection-13"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Sobolev_orthogonal_polynomials"><span class="tocnumber">4.3</span> <span class="toctext">Sobolev orthogonal polynomials</span></a></li> <li class="toclevel-2 tocsection-14"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Orthogonal_polynomials_with_matrices"><span class="tocnumber">4.4</span> <span class="toctext">Orthogonal polynomials with matrices</span></a></li> <li class="toclevel-2 tocsection-15"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Quantum_polynomials"><span class="tocnumber">4.5</span> <span class="toctext">Quantum polynomials</span></a></li> </ul></li> <li class="toclevel-1 tocsection-16"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#See_also"><span class="tocnumber">5</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-17"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#References"><span class="tocnumber">6</span> <span class="toctext">References</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Definition_for_1-variable_case_for_a_real_measure">Definition for 1-variable case for a real measure</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Definition for 1-variable case for a real measure" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>Given any non-decreasing function <span class="texhtml"><i>α</i></span> on the real numbers, we can define the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Lebesgue%E2%80%93Stieltjes_integral?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Lebesgue–Stieltjes integral">Lebesgue–Stieltjes integral</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int f(x)\,d\alpha (x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ∫<!-- ∫ --> </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> α<!-- α --> </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int f(x)\,d\alpha (x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70e3fc7c8d85248957ff62d6f5a70111fe0a240" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.228ex; height:5.676ex;" alt="{\displaystyle \int f(x)\,d\alpha (x)}"> </noscript><span class="lazy-image-placeholder" style="width: 13.228ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70e3fc7c8d85248957ff62d6f5a70111fe0a240" data-alt="{\displaystyle \int f(x)\,d\alpha (x)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> of a function <i>f</i>. If this integral is finite for all polynomials <i>f</i>, we can define an inner product on pairs of polynomials <i>f</i> and <i>g</i> by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int f(x)g(x)\,d\alpha (x).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <mi> f </mi> <mo> , </mo> <mi> g </mi> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mo> = </mo> <mo> ∫<!-- ∫ --> </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> α<!-- α --> </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle f,g\rangle =\int f(x)g(x)\,d\alpha (x).} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d44033208c277eebcbbddee709632530ce878f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.466ex; height:5.676ex;" alt="{\displaystyle \langle f,g\rangle =\int f(x)g(x)\,d\alpha (x).}"> </noscript><span class="lazy-image-placeholder" style="width: 26.466ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d44033208c277eebcbbddee709632530ce878f3" data-alt="{\displaystyle \langle f,g\rangle =\int f(x)g(x)\,d\alpha (x).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span></p> <p>This operation is a positive semidefinite <a href="https://en-m-wikipedia-org.translate.goog/wiki/Inner_product_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Inner product space">inner product</a> on the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Vector_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vector space">vector space</a> of all polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthogonality">orthogonality</a> in the usual way, namely that two polynomials are orthogonal if their inner product is zero.</p> <p>Then the sequence <span class="texhtml">(<i>P</i><sub><i>n</i></sub>)<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">∞</sup><br><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>n</i>=0</sub></span></span></span> of orthogonal polynomials is defined by the relations <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> deg </mi> <mo> <!-- --> </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mi> n </mi> <mtext> </mtext> <mo> , </mo> <mspace width="1em"></mspace> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msub> <mo> , </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mo> = </mo> <mn> 0 </mn> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mspace width="1em"></mspace> <mi> m </mi> <mo> ≠<!-- ≠ --> </mo> <mi> n </mi> <mtext> </mtext> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa49eb6d6cc7301bf9d20f2648c89521ff1aca13" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.577ex; height:2.843ex;" alt="{\displaystyle \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.}"> </noscript><span class="lazy-image-placeholder" style="width: 43.577ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa49eb6d6cc7301bf9d20f2648c89521ff1aca13" data-alt="{\displaystyle \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span></p> <p>In other words, the sequence is obtained from the sequence of monomials 1, <i>x</i>, <i>x</i><sup>2</sup>, … by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gram–Schmidt process">Gram–Schmidt process</a> with respect to this inner product.</p> <p>Usually the sequence is required to be <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthonormal?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Orthonormal">orthonormal</a>, namely, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle P_{n},P_{n}\rangle =1,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle P_{n},P_{n}\rangle =1,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21492c780ce745bc879e56e73b7b0783ebdd94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.172ex; height:2.843ex;" alt="{\displaystyle \langle P_{n},P_{n}\rangle =1,}"> </noscript><span class="lazy-image-placeholder" style="width: 13.172ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21492c780ce745bc879e56e73b7b0783ebdd94" data-alt="{\displaystyle \langle P_{n},P_{n}\rangle =1,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> however, other normalisations are sometimes used.</p> <div class="mw-heading mw-heading3"> <h3 id="Absolutely_continuous_case">Absolutely continuous case</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Absolutely continuous case" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Sometimes we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\alpha (x)=W(x)\,dx}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mi> α<!-- α --> </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> W </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d\alpha (x)=W(x)\,dx} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff263f6a7e4c126df386d16dedca9d17a622ec0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.448ex; height:2.843ex;" alt="{\displaystyle d\alpha (x)=W(x)\,dx}"> </noscript><span class="lazy-image-placeholder" style="width: 17.448ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff263f6a7e4c126df386d16dedca9d17a622ec0" data-alt="{\displaystyle d\alpha (x)=W(x)\,dx}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W:[x_{1},x_{2}]\to \mathbb {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> W </mi> <mo> : </mo> <mo stretchy="false"> [ </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ] </mo> <mo stretchy="false"> →<!-- → --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle W:[x_{1},x_{2}]\to \mathbb {R} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73394b2ece307feade24cf330229f9a650ed992e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.76ex; height:2.843ex;" alt="{\displaystyle W:[x_{1},x_{2}]\to \mathbb {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 16.76ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73394b2ece307feade24cf330229f9a650ed992e" data-alt="{\displaystyle W:[x_{1},x_{2}]\to \mathbb {R} }" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> is a non-negative function with support on some interval <span class="texhtml">[<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>]</span> in the real line (where <span class="texhtml"><i>x</i><sub>1</sub> = −∞</span> and <span class="texhtml"><i>x</i><sub>2</sub> = ∞</span> are allowed). Such a <span class="texhtml"><i>W</i></span> is called a <b>weight function</b>.<sup id="cite_ref-1" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Then the inner product is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\,dx.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <mi> f </mi> <mo> , </mo> <mi> g </mi> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mo> = </mo> <msubsup> <mo> ∫<!-- ∫ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> </msubsup> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mi> W </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> x </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\,dx.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3abf88519d7729c66687f102a8fe69ba2b01ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.018ex; height:6.176ex;" alt="{\displaystyle \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\,dx.}"> </noscript><span class="lazy-image-placeholder" style="width: 31.018ex;height: 6.176ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3abf88519d7729c66687f102a8fe69ba2b01ca" data-alt="{\displaystyle \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\,dx.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span> However, there are many examples of orthogonal polynomials where the measure <span class="texhtml"><i>dα</i>(<i>x</i>)</span> has points with non-zero measure where the function <span class="texhtml"><i>α</i></span> is discontinuous, so cannot be given by a weight function <span class="texhtml"><i>W</i></span> as above.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Examples_of_orthogonal_polynomials">Examples of orthogonal polynomials</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Examples of orthogonal polynomials" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>The most commonly used orthogonal polynomials are orthogonal for a measure with support in a real interval. This includes:</p> <ul> <li>The classical orthogonal polynomials (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Jacobi_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jacobi polynomials">Jacobi polynomials</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Laguerre_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Laguerre polynomials">Laguerre polynomials</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hermite_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hermite polynomials">Hermite polynomials</a>, and their special cases <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gegenbauer_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gegenbauer polynomials">Gegenbauer polynomials</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Chebyshev_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Chebyshev polynomials">Chebyshev polynomials</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Legendre_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Legendre polynomials">Legendre polynomials</a>).</li> <li>The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wilson_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wilson polynomials">Wilson polynomials</a>, which generalize the Jacobi polynomials. They include many orthogonal polynomials as special cases, such as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Meixner%E2%80%93Pollaczek_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Meixner–Pollaczek polynomials">Meixner–Pollaczek polynomials</a>, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Continuous_Hahn_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Continuous Hahn polynomials">continuous Hahn polynomials</a>, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Continuous_dual_Hahn_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Continuous dual Hahn polynomials">continuous dual Hahn polynomials</a>, and the classical polynomials, described by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Askey_scheme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Askey scheme">Askey scheme</a></li> <li>The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Askey%E2%80%93Wilson_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Askey–Wilson polynomials">Askey–Wilson polynomials</a> introduce an extra parameter <i>q</i> into the Wilson polynomials.</li> </ul> <p><a href="https://en-m-wikipedia-org.translate.goog/wiki/Discrete_orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Discrete orthogonal polynomials">Discrete orthogonal polynomials</a> are orthogonal with respect to some discrete measure. Sometimes the measure has finite support, in which case the family of orthogonal polynomials is finite, rather than an infinite sequence. The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Racah_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Racah polynomials">Racah polynomials</a> are examples of discrete orthogonal polynomials, and include as special cases the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hahn_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hahn polynomials">Hahn polynomials</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dual_Hahn_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dual Hahn polynomials">dual Hahn polynomials</a>, which in turn include as special cases the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Meixner_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Meixner polynomials">Meixner polynomials</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Krawtchouk_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Krawtchouk polynomials">Krawtchouk polynomials</a>, and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Charlier_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Charlier polynomials">Charlier polynomials</a>.</p> <p>Meixner classified all the orthogonal <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sheffer_sequence?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sheffer sequence">Sheffer sequences</a>: there are only Hermite, Laguerre, Charlier, Meixner, and Meixner–Pollaczek. In some sense Krawtchouk should be on this list too, but they are a finite sequence. These six families correspond to the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Natural_exponential_family?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#The_six_NEF-QVFs" title="Natural exponential family">NEF-QVFs</a> and are <a href="https://en-m-wikipedia-org.translate.goog/wiki/Martingale_(probability_theory)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Martingale (probability theory)">martingale</a> polynomials for certain <a href="https://en-m-wikipedia-org.translate.goog/wiki/L%C3%A9vy_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lévy process">Lévy processes</a>.</p> <p><a href="https://en-m-wikipedia-org.translate.goog/wiki/Sieved_orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sieved orthogonal polynomials">Sieved orthogonal polynomials</a>, such as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sieved_ultraspherical_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sieved ultraspherical polynomials">sieved ultraspherical polynomials</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sieved_Jacobi_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sieved Jacobi polynomials">sieved Jacobi polynomials</a>, and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sieved_Pollaczek_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sieved Pollaczek polynomials">sieved Pollaczek polynomials</a>, have modified recurrence relations.</p> <p>One can also consider orthogonal polynomials for some curve in the complex plane. The most important case (other than real intervals) is when the curve is the unit circle, giving <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials_on_the_unit_circle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthogonal polynomials on the unit circle">orthogonal polynomials on the unit circle</a>, such as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rogers%E2%80%93Szeg%C5%91_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rogers–Szegő polynomials">Rogers–Szegő polynomials</a>.</p> <p>There are some families of orthogonal polynomials that are orthogonal on plane regions such as triangles or disks. They can sometimes be written in terms of Jacobi polynomials. For example, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Zernike_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Zernike polynomials">Zernike polynomials</a> are orthogonal on the unit disk.</p> <p>The advantage of orthogonality between different orders of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hermite_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hermite polynomials">Hermite polynomials</a> is applied to Generalized frequency division multiplexing (GFDM) structure. More than one symbol can be carried in each grid of time-frequency lattice.<sup id="cite_ref-2" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Properties">Properties</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Properties" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>Orthogonal polynomials of one variable defined by a non-negative measure on the real line have the following properties.</p> <div class="mw-heading mw-heading3"> <h3 id="Relation_to_moments">Relation to moments</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Relation to moments" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The orthogonal polynomials <i>P</i><sub><i>n</i></sub> can be expressed in terms of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Moment_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Moment (mathematics)">moments</a></p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{n}=\int x^{n}\,d\alpha (x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mo> ∫<!-- ∫ --> </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> α<!-- α --> </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{n}=\int x^{n}\,d\alpha (x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072cd3d985007fda4caca5df1ee877899409046e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.716ex; height:5.676ex;" alt="{\displaystyle m_{n}=\int x^{n}\,d\alpha (x)}"> </noscript><span class="lazy-image-placeholder" style="width: 17.716ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072cd3d985007fda4caca5df1ee877899409046e" data-alt="{\displaystyle m_{n}=\int x^{n}\,d\alpha (x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>as follows:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(x)=c_{n}\,\det {\begin{bmatrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n-1}&m_{n}&m_{n+1}&\cdots &m_{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{bmatrix}}~,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo> = </mo> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo movablelimits="true" form="prefix"> det </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> <mtd> <mo> ⋯<!-- ⋯ --> </mo> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> <mtd> <mo> ⋯<!-- ⋯ --> </mo> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> ⋮<!-- ⋮ --> </mo> </mtd> <mtd> <mo> ⋮<!-- ⋮ --> </mo> </mtd> <mtd> <mo> ⋮<!-- ⋮ --> </mo> </mtd> <mtd> <mo> ⋱<!-- ⋱ --> </mo> </mtd> <mtd> <mo> ⋮<!-- ⋮ --> </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <mo> ⋯<!-- ⋯ --> </mo> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mi> x </mi> </mtd> <mtd> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> <mtd> <mo> ⋯<!-- ⋯ --> </mo> </mtd> <mtd> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mtext> </mtext> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{n}(x)=c_{n}\,\det {\begin{bmatrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n-1}&m_{n}&m_{n+1}&\cdots &m_{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{bmatrix}}~,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5dcea1a0736046c1dfd6895e7ef1166d775b0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:53.072ex; height:17.509ex;" alt="{\displaystyle P_{n}(x)=c_{n}\,\det {\begin{bmatrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n-1}&m_{n}&m_{n+1}&\cdots &m_{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{bmatrix}}~,}"> </noscript><span class="lazy-image-placeholder" style="width: 53.072ex;height: 17.509ex;vertical-align: -8.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5dcea1a0736046c1dfd6895e7ef1166d775b0b" data-alt="{\displaystyle P_{n}(x)=c_{n}\,\det {\begin{bmatrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n-1}&m_{n}&m_{n+1}&\cdots &m_{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{bmatrix}}~,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>where the constants <i>c</i><sub><i>n</i></sub> are arbitrary (depend on the normalization of <i>P</i><sub><i>n</i></sub>).</p> <p>This comes directly from applying the Gram–Schmidt process to the monomials, imposing each polynomial to be orthogonal with respect to the previous ones. For example, orthogonality with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{0}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.547ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" data-alt="{\displaystyle P_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> prescribes that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{1}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.547ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" data-alt="{\displaystyle P_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> must have the form<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}(x)=c_{1}\left(x-{\frac {\langle P_{0},x\rangle P_{0}}{\langle P_{0},P_{0}\rangle }}\right)=c_{1}(x-m_{1}),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <mi> x </mi> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mrow> <mrow> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> −<!-- − --> </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{1}(x)=c_{1}\left(x-{\frac {\langle P_{0},x\rangle P_{0}}{\langle P_{0},P_{0}\rangle }}\right)=c_{1}(x-m_{1}),} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1fc399c1120196948f919613854fbbde3620309" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.806ex; height:6.509ex;" alt="{\displaystyle P_{1}(x)=c_{1}\left(x-{\frac {\langle P_{0},x\rangle P_{0}}{\langle P_{0},P_{0}\rangle }}\right)=c_{1}(x-m_{1}),}"> </noscript><span class="lazy-image-placeholder" style="width: 43.806ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1fc399c1120196948f919613854fbbde3620309" data-alt="{\displaystyle P_{1}(x)=c_{1}\left(x-{\frac {\langle P_{0},x\rangle P_{0}}{\langle P_{0},P_{0}\rangle }}\right)=c_{1}(x-m_{1}),}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </span></span>which can be seen to be consistent with the previously given expression with the determinant.</p> <div class="mw-heading mw-heading3"> <h3 id="Recurrence_relation">Recurrence relation</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Recurrence relation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The polynomials <i>P</i><sub><i>n</i></sub> satisfy a recurrence relation of the form</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mi> x </mi> <mo> + </mo> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <msub> <mi> P </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> <mo> + </mo> <msub> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b71ab07cff182175eb811817150cbf3fcf56be5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.493ex; height:2.843ex;" alt="{\displaystyle P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 42.493ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b71ab07cff182175eb811817150cbf3fcf56be5" data-alt="{\displaystyle P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>where <i>A<sub>n</sub></i> is not 0. The converse is also true; see <a href="https://en-m-wikipedia-org.translate.goog/wiki/Favard%27s_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Favard's theorem">Favard's theorem</a>.</p> <div class="mw-heading mw-heading3"> <h3 id="Christoffel–Darboux_formula"><span id="Christoffel.E2.80.93Darboux_formula"></span>Christoffel–Darboux formula</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Christoffel–Darboux formula" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </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"> Main article: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Christoffel%E2%80%93Darboux_formula?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Christoffel–Darboux formula">Christoffel–Darboux formula</a> </div> <div class="mw-heading mw-heading3"> <h3 id="Zeros">Zeros</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Zeros" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>If the measure d<i>α</i> is supported on an interval [<i>a</i>, <i>b</i>], all the zeros of <i>P</i><sub><i>n</i></sub> lie in [<i>a</i>, <i>b</i>]. Moreover, the zeros have the following interlacing property: if <i>m</i> < <i>n</i>, there is a zero of <i>P</i><sub><i>n</i></sub> between any two zeros of <i>P</i><sub><i>m</i></sub>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Electrostatic?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Electrostatic">Electrostatic</a> interpretations of the zeros can be given.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:Citation_needed?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2021)">citation needed</span></a></i>]</sup></p> <div class="mw-heading mw-heading3"> <h3 id="Combinatorial_interpretation">Combinatorial interpretation</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Combinatorial interpretation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>From the 1980s, with the work of X. G. Viennot, J. Labelle, Y.-N. Yeh, D. Foata, and others, combinatorial interpretations were found for all the classical orthogonal polynomials. <sup id="cite_ref-3" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Other_types_of_orthogonal_polynomials">Other types of orthogonal polynomials</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Other types of orthogonal polynomials" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <div class="mw-heading mw-heading3"> <h3 id="Multivariate_orthogonal_polynomials">Multivariate orthogonal polynomials</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Multivariate orthogonal polynomials" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Macdonald_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Macdonald polynomials">Macdonald polynomials</a> are orthogonal polynomials in several variables, depending on the choice of an affine root system. They include many other families of multivariable orthogonal polynomials as special cases, including the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Jack_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Jack polynomials">Jack polynomials</a>, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hall%E2%80%93Littlewood_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hall–Littlewood polynomials">Hall–Littlewood polynomials</a>, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Heckman%E2%80%93Opdam_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Heckman–Opdam polynomials">Heckman–Opdam polynomials</a>, and the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Koornwinder_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Koornwinder polynomials">Koornwinder polynomials</a>. The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Askey%E2%80%93Wilson_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Askey–Wilson polynomials">Askey–Wilson polynomials</a> are the special case of Macdonald polynomials for a certain non-reduced root system of rank 1.</p> <div class="mw-heading mw-heading3"> <h3 id="Multiple_orthogonal_polynomials">Multiple orthogonal polynomials</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Multiple orthogonal polynomials" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> Main article: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiple_orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Multiple orthogonal polynomials">Multiple orthogonal polynomials</a> </div> <p>Multiple orthogonal polynomials are polynomials in one variable that are orthogonal with respect to a finite family of measures.</p> <div class="mw-heading mw-heading3"> <h3 id="Sobolev_orthogonal_polynomials">Sobolev orthogonal polynomials</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Sobolev orthogonal polynomials" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> Main article: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sobolev_orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sobolev orthogonal polynomials">Sobolev orthogonal polynomials</a> </div> <p>These are orthogonal polynomials with respect to a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sobolev_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sobolev space">Sobolev</a> inner product, i.e. an inner product with derivatives. Including derivatives has big consequences for the polynomials, in general they no longer share some of the nice features of the classical orthogonal polynomials.</p> <div class="mw-heading mw-heading3"> <h3 id="Orthogonal_polynomials_with_matrices">Orthogonal polynomials with matrices</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Orthogonal polynomials with matrices" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Orthogonal polynomials with matrices have either coefficients that are matrices or the indeterminate is a matrix.</p> <p>There are two popular examples: either the coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{i}\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{a_{i}\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434434a3a4c297856e0eff9f57d2d25053f830b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.354ex; height:2.843ex;" alt="{\displaystyle \{a_{i}\}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.354ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434434a3a4c297856e0eff9f57d2d25053f830b7" data-alt="{\displaystyle \{a_{i}\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> are matrices or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>:</p> <ul> <li>Variante 1: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=A_{n}x^{n}+A_{n-1}x^{n-1}+\cdots +A_{1}x+A_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> P </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <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> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <mo> + </mo> <mo> ⋯<!-- ⋯ --> </mo> <mo> + </mo> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mi> x </mi> <mo> + </mo> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P(x)=A_{n}x^{n}+A_{n-1}x^{n-1}+\cdots +A_{1}x+A_{0}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da93feebcc96657ccc34cde6590bc58c40e9e80d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.213ex; height:3.176ex;" alt="{\displaystyle P(x)=A_{n}x^{n}+A_{n-1}x^{n-1}+\cdots +A_{1}x+A_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 44.213ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da93feebcc96657ccc34cde6590bc58c40e9e80d" data-alt="{\displaystyle P(x)=A_{n}x^{n}+A_{n-1}x^{n-1}+\cdots +A_{1}x+A_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A_{i}\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{A_{i}\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469990a3aa11e4d16c5a304d675b87ce4b3b80aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.868ex; height:2.843ex;" alt="{\displaystyle \{A_{i}\}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.868ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469990a3aa11e4d16c5a304d675b87ce4b3b80aa" data-alt="{\displaystyle \{A_{i}\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times p}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo> ×<!-- × --> </mo> <mi> p </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p\times p} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e6de741a3aa8176f5a487de8e8f602aa75c5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.269ex; height:2.009ex;" alt="{\displaystyle p\times p}"> </noscript><span class="lazy-image-placeholder" style="width: 5.269ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e6de741a3aa8176f5a487de8e8f602aa75c5e4" data-alt="{\displaystyle p\times p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> matrices.</li> <li>Variante 2: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}I_{p}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> P </mi> <mo stretchy="false"> ( </mo> <mi> X </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <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> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> <msup> <mi> X </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> <mo> + </mo> <mo> ⋯<!-- ⋯ --> </mo> <mo> + </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mi> X </mi> <mo> + </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}I_{p}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1ce9ec10b029b0e60a277bb5a7d15b2f6462a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.877ex; height:3.343ex;" alt="{\displaystyle P(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}I_{p}}"> </noscript><span class="lazy-image-placeholder" style="width: 46.877ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1ce9ec10b029b0e60a277bb5a7d15b2f6462a3" data-alt="{\displaystyle P(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}I_{p}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> X </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle X} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> </noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" data-alt="{\displaystyle X}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times p}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo> ×<!-- × --> </mo> <mi> p </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p\times p} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e6de741a3aa8176f5a487de8e8f602aa75c5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.269ex; height:2.009ex;" alt="{\displaystyle p\times p}"> </noscript><span class="lazy-image-placeholder" style="width: 5.269ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e6de741a3aa8176f5a487de8e8f602aa75c5e4" data-alt="{\displaystyle p\times p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-matrix and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{p}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{p}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3226e9ad60e659391806720213f9c5f6123a70f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.082ex; height:2.843ex;" alt="{\displaystyle I_{p}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.082ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3226e9ad60e659391806720213f9c5f6123a70f8" data-alt="{\displaystyle I_{p}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the identity matrix.</li> </ul> <div class="mw-heading mw-heading3"> <h3 id="Quantum_polynomials">Quantum polynomials</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=15&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Quantum polynomials" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Quantum polynomials or q-polynomials are the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Q-analog?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Q-analog">q-analogs</a> of orthogonal polynomials.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=16&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Appell_sequence?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Appell sequence">Appell sequence</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Askey_scheme?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Askey scheme">Askey scheme</a> of hypergeometric orthogonal polynomials</li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Favard%27s_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Favard's theorem">Favard's theorem</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Binomial_type?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Binomial type">Polynomial sequences of binomial type</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Biorthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Biorthogonal polynomials">Biorthogonal polynomials</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Generalized_Fourier_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Generalized Fourier series">Generalized Fourier series</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Secondary_measure?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Secondary measure">Secondary measure</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Sheffer_sequence?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sheffer sequence">Sheffer sequence</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Sturm%E2%80%93Liouville_theory?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sturm–Liouville theory">Sturm–Liouville theory</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Umbral_calculus?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Umbral calculus">Umbral calculus</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Plancherel%E2%80%93Rotach_asymptotics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Plancherel–Rotach asymptotics">Plancherel–Rotach asymptotics</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Orthogonal_polynomials&action=edit&section=17&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <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-1"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://demonstrations.wolfram.com/OrthonormalPolynomialsUnderDifferentInnerProductMeasures/">Demo of orthonormal polynomials obtained for different weight functions</a></span></li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-2">^</a></b></span> <span class="reference-text"><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="CITEREFCatakDurak-Ata2017" class="citation book cs1">Catak, E.; Durak-Ata, L. (2017). "An efficient transceiver design for superimposed waveforms with orthogonal polynomials". <i>2017 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom)</i>. pp. <span class="nowrap">1–</span>5. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1109%252FBlackSeaCom.2017.8277657">10.1109/BlackSeaCom.2017.8277657</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-1-5090-5049-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-1-5090-5049-9"><bdi>978-1-5090-5049-9</bdi></a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:22592277">22592277</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+efficient+transceiver+design+for+superimposed+waveforms+with+orthogonal+polynomials&rft.btitle=2017+IEEE+International+Black+Sea+Conference+on+Communications+and+Networking+%28BlackSeaCom%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E5&rft.date=2017&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A22592277%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FBlackSeaCom.2017.8277657&rft.isbn=978-1-5090-5049-9&rft.aulast=Catak&rft.aufirst=E.&rft.au=Durak-Ata%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+polynomials" class="Z3988"></span></span></li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_polynomials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFViennot2017" class="citation web cs1">Viennot, Xavier (2017). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://viennot.org/abjc4-ch5.html">"The Art of Bijective Combinatorics, Part IV, Combinatorial theory of orthogonal polynomials and continued fractions"</a>. Chennai: IMSc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Art+of+Bijective+Combinatorics%2C+Part+IV%2C+Combinatorial+theory+of+orthogonal+polynomials+and+continued+fractions.&rft.place=Chennai&rft.pub=IMSc&rft.date=2017&rft.aulast=Viennot&rft.aufirst=Xavier&rft_id=https%3A%2F%2Fviennot.org%2Fabjc4-ch5.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+polynomials" class="Z3988"></span></span></li> </ol> </div> </div> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Milton_Abramowitz?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Irene_Stegun?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.math.ubc.ca/~cbm/aands/page_773.htm">"Chapter 22"</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Abramowitz_and_Stegun?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abramowitz and Stegun"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-486-61272-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-486-61272-0"><bdi>978-0-486-61272-0</bdi></a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/LCCN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lccn.loc.gov/64-60036">64-60036</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D0167642">0167642</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/LCCN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.loc.gov/item/65012253">65-12253</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+22&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=Washington+D.C.%3B+New+York&rft.series=Applied+Mathematics+Series&rft.pages=773&rft.edition=Ninth+reprint+with+additional+corrections+of+tenth+original+printing+with+corrections+%28December+1972%29%3B+first&rft.pub=United+States+Department+of+Commerce%2C+National+Bureau+of+Standards%3B+Dover+Publications&rft.date=1983&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0167642%23id-name%3DMR&rft_id=info%3Alccn%2F64-60036&rft.isbn=978-0-486-61272-0&rft_id=http%3A%2F%2Fwww.math.ubc.ca%2F~cbm%2Faands%2Fpage_773.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+polynomials" class="Z3988"></span></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChihara1978" class="citation book cs1">Chihara, Theodore Seio (1978). <i>An Introduction to Orthogonal Polynomials</i>. Gordon and Breach, New York. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-677-04150-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-677-04150-0"><bdi>0-677-04150-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Orthogonal+Polynomials&rft.pub=Gordon+and+Breach%2C+New+York&rft.date=1978&rft.isbn=0-677-04150-0&rft.aulast=Chihara&rft.aufirst=Theodore+Seio&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+polynomials" class="Z3988"></span></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChihara2001" class="citation journal cs1">Chihara, Theodore Seio (2001). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1016%252FS0377-0427%252800%252900632-4">"45 years of orthogonal polynomials: a view from the wings"</a>. 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J.; Foncannon, J. J.; Pekonen, Osmo (2008). "Review of <i>Classical and quantum orthogonal polynomials in one variable</i> by Mourad Ismail". <i><a href="https://en-m-wikipedia-org.translate.goog/wiki/The_Mathematical_Intelligencer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a></i>. <b>30</b>. 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