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<!DOCTYPE html><html lang="en"> <head> <meta http-equiv="content-type" content="text/html; charset=UTF-8"> <title>DLMF: Notations</title> <link rel="shortcut icon" href=".././style/DLMF-16.png" type="image/png"> <script type="text/javascript"></script> <script type="text/javascript" src=".././style/jquery.js"></script> <script type="text/javascript" src=".././style/jquery.leaveNotice.js"></script> <script type="text/javascript" src=".././style/DLMF.js"></script> <script async="" type="text/javascript" id="_fed_an_ua_tag" src="https://dap.digitalgov.gov/Universal-Federated-Analytics-Min.js?agency=DOC&subagency=NIST&pua=UA-37115410-44&yt=true&exts=ppsx,pps,f90,sch,rtf,wrl,txz,m1v,xlsm,msi,xsd,f,tif,eps,mpg,xml,pl,xlt,c"></script><script async="" src="https://www.googletagmanager.com/gtag/js?id=G-BZB64W4M0L"></script><script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-BZB64W4M0L'); </script> <link 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rel="chapter" href=".././14" title="Chapter 14 Legendre and Related Functions"> <link rel="chapter" href=".././15" title="Chapter 15 Hypergeometric Function"> <link rel="chapter" href=".././16" title="Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function"> <link rel="chapter" href=".././17" title="Chapter 17 𝑞-Hypergeometric and Related Functions"> <link rel="chapter" href=".././18" title="Chapter 18 Orthogonal Polynomials"> <link rel="chapter" href=".././19" title="Chapter 19 Elliptic Integrals"> <link rel="chapter" href=".././20" title="Chapter 20 Theta Functions"> <link rel="chapter" href=".././21" title="Chapter 21 Multidimensional Theta Functions"> <link rel="chapter" href=".././22" title="Chapter 22 Jacobian Elliptic Functions"> <link rel="chapter" href=".././23" title="Chapter 23 Weierstrass Elliptic and Modular Functions"> <link rel="chapter" href=".././24" title="Chapter 24 Bernoulli and Euler Polynomials"> <link rel="chapter" href=".././25" title="Chapter 25 Zeta and Related Functions"> <link rel="chapter" href=".././26" title="Chapter 26 Combinatorial Analysis"> <link rel="chapter" href=".././27" title="Chapter 27 Functions of Number Theory"> <link rel="chapter" href=".././28" title="Chapter 28 Mathieu Functions and Hill’s Equation"> <link rel="chapter" href=".././29" title="Chapter 29 Lamé Functions"> <link rel="chapter" href=".././30" title="Chapter 30 Spheroidal Wave Functions"> <link rel="chapter" href=".././31" title="Chapter 31 Heun Functions"> <link rel="chapter" href=".././32" title="Chapter 32 Painlevé Transcendents"> <link rel="chapter" href=".././33" title="Chapter 33 Coulomb Functions"> <link rel="chapter" href=".././34" title="Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols"> <link rel="chapter" href=".././35" title="Chapter 35 Functions of Matrix Argument"> <link rel="chapter" href=".././36" title="Chapter 36 Integrals with Coalescing Saddles"> <link rel="bibliography" href=".././bib/" title="Bibliography"> <link rel="index" href=".././idx/" title="Index"> <link rel="document" href=".././lof/" title="List of Figures"> <link rel="document" href=".././lot/" title="List of Tables"> <link rel="document" href=".././software/" title="Software Index"> <link rel="document" href=".././errata/" title="Errata"> <link rel="document" href=".././help/" title="Need Help?"> <link rel="document" href=".././search/" title="Advanced Search"> <link rel="document" href=".././about/" title="About the Project"> <link rel="glossary" href=".././not/A" title="Notations A ‣ Notations"> <link rel="glossary" href=".././not/B" title="Notations B ‣ Notations"> <link rel="glossary" href=".././not/C" title="Notations C ‣ Notations"> <link rel="glossary" href=".././not/D" title="Notations D ‣ Notations"> <link rel="glossary" href=".././not/E" title="Notations E ‣ Notations"> <link rel="glossary" href=".././not/F" title="Notations F ‣ Notations"> <link rel="glossary" href=".././not/G" title="Notations G ‣ Notations"> <link rel="glossary" href=".././not/H" title="Notations H ‣ Notations"> <link rel="glossary" href=".././not/I" title="Notations I ‣ Notations"> <link rel="glossary" href=".././not/J" title="Notations J ‣ Notations"> <link rel="glossary" href=".././not/K" title="Notations K ‣ Notations"> <link rel="glossary" href=".././not/L" title="Notations L ‣ Notations"> <link rel="glossary" href=".././not/M" title="Notations M ‣ Notations"> <link rel="glossary" href=".././not/N" title="Notations N ‣ Notations"> <link rel="glossary" href=".././not/O" title="Notations O ‣ Notations"> <link rel="glossary" href=".././not/P" title="Notations P ‣ Notations"> <link rel="glossary" href=".././not/Q" title="Notations Q ‣ Notations"> <link rel="glossary" href=".././not/R" title="Notations R ‣ Notations"> <link rel="glossary" href=".././not/S" title="Notations S ‣ Notations"> <link rel="glossary" href=".././not/T" title="Notations T ‣ Notations"> <link rel="glossary" href=".././not/U" title="Notations U ‣ Notations"> <link rel="glossary" href=".././not/V" title="Notations V ‣ Notations"> <link rel="glossary" href=".././not/W" title="Notations W ‣ Notations"> <link rel="glossary" href=".././not/X" title="Notations X ‣ Notations"> <link rel="glossary" href=".././not/Y" title="Notations Y ‣ Notations"> <link rel="glossary" href=".././not/Z" title="Notations Z ‣ Notations"> </head> <body class="color_default textfont_default titlefont_default fontsize_default navbar_default"> <div class="ltx_page_navbar"> <div class="ltx_page_navlogo"><a href=".././" title="Digital Library of Mathematical Functions"><span>DLMF</span></a></div> <div class="ltx_page_navitems"> <form method="get" action=".././search/search"> <ul> <li><a href=".././idx/">Index</a></li> <li><a href=".././not/">Notations</a></li> <li><small><input type="text" name="q" value="" size="6" class="ltx_page_navitem_search"><button type="submit">Search</button></small></li> <li><a href=".././help/" class="ltx_help">Help?</a></li> <li><a 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class="ltx_ref">C</a>♦<a href=".././not/D" title="Notations D ‣ Notations" class="ltx_ref">D</a>♦<a href=".././not/E" title="Notations E ‣ Notations" class="ltx_ref">E</a>♦<a href=".././not/F" title="Notations F ‣ Notations" class="ltx_ref">F</a>♦<a href=".././not/G" title="Notations G ‣ Notations" class="ltx_ref">G</a>♦<a href=".././not/H" title="Notations H ‣ Notations" class="ltx_ref">H</a>♦<a href=".././not/I" title="Notations I ‣ Notations" class="ltx_ref">I</a>♦<a href=".././not/J" title="Notations J ‣ Notations" class="ltx_ref">J</a>♦<a href=".././not/K" title="Notations K ‣ Notations" class="ltx_ref">K</a>♦<a href=".././not/L" title="Notations L ‣ Notations" class="ltx_ref">L</a>♦<a href=".././not/M" title="Notations M ‣ Notations" class="ltx_ref">M</a>♦<a href=".././not/N" title="Notations N ‣ Notations" class="ltx_ref">N</a>♦<a href=".././not/O" title="Notations O ‣ Notations" class="ltx_ref">O</a>♦<a href=".././not/P" title="Notations P ‣ Notations" class="ltx_ref">P</a>♦<a href=".././not/Q" title="Notations Q ‣ Notations" class="ltx_ref">Q</a>♦<a href=".././not/R" title="Notations R ‣ Notations" class="ltx_ref">R</a>♦<a href=".././not/S" title="Notations S ‣ Notations" class="ltx_ref">S</a>♦<a href=".././not/T" title="Notations T ‣ Notations" class="ltx_ref">T</a>♦<a href=".././not/U" title="Notations U ‣ Notations" class="ltx_ref">U</a>♦<a href=".././not/V" title="Notations V ‣ Notations" class="ltx_ref">V</a>♦<a href=".././not/W" title="Notations W ‣ Notations" class="ltx_ref">W</a>♦<a href=".././not/X" title="Notations X ‣ Notations" class="ltx_ref">X</a>♦<a href=".././not/Y" title="Notations Y ‣ Notations" class="ltx_ref">Y</a>♦<a href=".././not/Z" title="Notations Z ‣ Notations" class="ltx_ref">Z</a>♦</div></nav> <div class="ltx_page_columns"> <div class="ltx_page_column1"> <dl class="ltx_glossarylist ltx_list_notation"> <dt id="AA.n69" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m99.png" altimg-height="23px" altimg-valign="-2px" altimg-width="45px" alttext="{\NVar{\mathbf{A}}}^{-1}" display="inline"><msup><mi class="ltx_nvar">𝐀</mi><mrow><mo href=".././1.2#Px12" title="matrix inverse">−</mo><mn href=".././1.2#Px12" title="matrix inverse">1</mn></mrow></msup></math></dt> <dd>matrix inverse; <a href=".././1.2#Px12" title="The Inverse ‣ §1.2(vi) Square Matrices ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§1.2(vi)</span></a> </dd> <dt id="AA.n1" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m61.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\mathbf{A}" display="inline"><mi>𝐀</mi></math></dt> <dd>m by n matrix; <a href=".././1.2#Px4.p1" title="General 𝑚×𝑛 Matrices ‣ §1.2(v) Matrices, Vectors, Scalar Products, and Norms ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§1.2(v)</span></a> </dd> <dt id="AA.n62" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m2.png" altimg-height="18px" altimg-valign="-2px" altimg-width="10px" alttext="!" display="inline"><mo href=".././front/introduction#common.t1.r15" title="factorial (as in 𝑛!)">!</mo></math></dt> <dd>factorial (as in <math class="ltx_Math" altimg="m90.png" altimg-height="18px" altimg-valign="-2px" altimg-width="23px" alttext="n!" display="inline"><mrow><mi>n</mi><mo href=".././front/introduction#common.t1.r15" title="factorial (as in 𝑛!)">!</mo></mrow></math>); <a href=".././front/introduction#common.t1.r15" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n77" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m3.png" altimg-height="24px" altimg-valign="-8px" altimg-width="19px" alttext="!_{\NVar{q}}" display="inline"><msub><mo href=".././5.18#E2" title="𝑞-factorial (as in 𝑛!_𝑞)">!</mo><mi class="ltx_nvar">q</mi></msub></math></dt> <dd> <math class="ltx_Math" altimg="m94.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi>q</mi></math>-factorial (as in <math class="ltx_Math" altimg="m91.png" altimg-height="24px" altimg-valign="-8px" altimg-width="32px" alttext="n!_{q}" display="inline"><mrow><mi>n</mi><msub><mo href=".././5.18#E2" title="𝑞-factorial (as in 𝑛!_𝑞)">!</mo><mi>q</mi></msub></mrow></math>); <a href=".././5.18#E2" title="In §5.18(i) 𝑞-Factorials ‣ §5.18 𝑞-Gamma and 𝑞-Beta Functions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(5.18.2)</span></a> </dd> <dt id="AA.n61" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m1.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="!!" display="inline"><mo href=".././front/introduction#common.t1.r16" title="double factorial">!!</mo></math></dt> <dd>double factorial; <a href=".././front/introduction#common.t1.r16" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n2" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m32.png" altimg-height="13px" altimg-valign="-2px" altimg-width="10px" alttext="\cdot" display="inline"><mo>⋅</mo></math></dt> <dd> <math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="44px" alttext="\mathbf{a}\cdot\mathbf{b}" display="inline"><mrow><mi>𝐚</mi><mo lspace="0.222em" rspace="0.222em">⋅</mo><mi>𝐛</mi></mrow></math>: vector dot (or scalar) product; <a href=".././1.6#E2" title="In Dot Product (or Scalar Product) ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.6.2)</span></a> </dd> <dt id="AA.n3" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m10.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="*" display="inline"><mo>∗</mo></math></dt> <dd> <math class="ltx_Math" altimg="m87.png" altimg-height="22px" altimg-valign="-6px" altimg-width="47px" alttext="f*g" display="inline"><mrow><mi>f</mi><mo lspace="0.222em" rspace="0.222em">∗</mo><mi>g</mi></mrow></math>: convolution product; <a href=".././2.6#E34" title="In §2.6(iii) Fractional Integrals ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(2.6.34)</span></a> </dd> <dt id="AA.n4" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-4px" altimg-width="20px" alttext="\times" display="inline"><mo>×</mo></math></dt> <dd> <math class="ltx_Math" altimg="m64.png" altimg-height="20px" altimg-valign="-4px" altimg-width="55px" alttext="\mathbf{a}\times\mathbf{b}" display="inline"><mrow><mi>𝐚</mi><mo lspace="0.222em" rspace="0.222em">×</mo><mi>𝐛</mi></mrow></math>: vector cross product; <a href=".././1.6#E9" title="In Cross Product (or Vector Product) ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.6.9)</span></a> </dd> <dt id="AA.n52" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m78.png" altimg-height="18px" altimg-valign="-4px" altimg-width="20px" alttext="\times" display="inline"><mo href=".././23.1" title="𝐺×𝐻: Cartesian product of groups">×</mo></math></dt> <dd> <math class="ltx_Math" altimg="m12.png" altimg-height="20px" altimg-valign="-4px" altimg-width="65px" alttext="G\times H" display="inline"><mrow><mi>G</mi><mo href=".././23.1" lspace="0.222em" rspace="0.222em" title="𝐺×𝐻: Cartesian product of groups">×</mo><mi>H</mi></mrow></math>: Cartesian product of groups; <a href=".././23.1" title="§23.1 Special Notation ‣ Notation ‣ Chapter 23 Weierstrass Elliptic and Modular Functions" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§23.1</span></a> </dd> <dt id="AA.n81" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m11.png" altimg-height="25px" altimg-valign="-7px" altimg-width="15px" alttext="/" display="inline"><mo href=".././21.1#t1.r18" title="set of elements modulo an element">/</mo></math></dt> <dd> <math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="S_{1}/S_{2}" display="inline"><mrow><msub><mi>S</mi><mn>1</mn></msub><mo href=".././21.1#t1.r18" title="set of elements modulo an element">/</mo><msub><mi>S</mi><mn>2</mn></msub></mrow></math>: set of all elements of <math class="ltx_Math" altimg="m15.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="S_{1}" display="inline"><msub><mi>S</mi><mn>1</mn></msub></math> modulo elements of <math class="ltx_Math" altimg="m16.png" altimg-height="21px" altimg-valign="-5px" altimg-width="26px" alttext="S_{2}" display="inline"><msub><mi>S</mi><mn>2</mn></msub></math>; <a href=".././21.1#t1.r18" title="§21.1 Special Notation ‣ Notation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§21.1</span></a> </dd> <dt id="AA.n80" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m76.png" altimg-height="25px" altimg-valign="-7px" altimg-width="15px" alttext="\setminus" display="inline"><mo href=".././front/introduction#common.t2.r19" title="set subtraction">∖</mo></math></dt> <dd>set subtraction; <a href=".././front/introduction#common.t2.r19" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n5" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m24.png" altimg-height="11px" altimg-valign="-2px" altimg-width="38px" alttext="\Longrightarrow" display="inline"><mo stretchy="false">⟹</mo></math></dt> <dd>implies; <a href=".././front/introduction#common.t1.r13" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n6" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m23.png" altimg-height="11px" altimg-valign="-2px" altimg-width="42px" alttext="\Longleftrightarrow" display="inline"><mo stretchy="false">⟺</mo></math></dt> <dd>is equivalent to; <a href=".././front/introduction#common.t1.r14" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n50" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m77.png" altimg-height="11px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href=".././2.1#E1" title="asymptotic equality">∼</mo></math></dt> <dd>asymptotic equality; <a href=".././2.1#E1" title="In §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(2.1.1)</span></a> </dd> <dt id="AA.n49" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m77.png" altimg-height="11px" altimg-valign="-2px" altimg-width="20px" alttext="\sim" display="inline"><mo href=".././2.1#iii.p1" title="Poincaré asymptotic expansion">∼</mo></math></dt> <dd>Poincaré asymptotic expansion; <a href=".././2.1#iii.p1" title="§2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§2.1(iii)</span></a> </dd> <dt id="AA.n7" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\nabla" display="inline"><mo>∇</mo></math></dt> <dd>backward difference operator; <a href=".././3.10#Px8.p2" title="Steed’s Algorithm ‣ §3.10(iii) Numerical Evaluation of Continued Fractions ‣ §3.10 Continued Fractions ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§3.10(iii)</span></a> </dd> <dt id="AA.n8" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m70.png" altimg-height="18px" altimg-valign="-2px" altimg-width="22px" alttext="\nabla" display="inline"><mo>∇</mo></math></dt> <dd>del operator; <a href=".././1.6#E19" title="In Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.6.19)</span></a> </dd> <dt id="AA.n9" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m72.png" altimg-height="21px" altimg-valign="-2px" altimg-width="31px" alttext="\nabla^{2}" display="inline"><msup><mo>∇</mo><mn>2</mn></msup></math></dt> <dd>Laplacian for spherical coordinates; <a href=".././1.5#Px6" title="Spherical Coordinates ‣ §1.5(ii) Coordinate Systems ‣ §1.5 Calculus of Two or More Variables ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§1.5(ii)</span></a> </dd> <dt id="AA.n10" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m68.png" altimg-height="22px" altimg-valign="-6px" altimg-width="34px" alttext="\nabla\NVar{f}" display="inline"><mrow><mo rspace="0.167em">∇</mo><mi class="ltx_nvar">f</mi></mrow></math></dt> <dd>gradient of differentiable scalar function <math class="ltx_Math" altimg="m88.png" altimg-height="22px" altimg-valign="-6px" altimg-width="17px" alttext="f" display="inline"><mi>f</mi></math>; <a href=".././1.6#E20" title="In Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.6.20)</span></a> </dd> <dt id="AA.n11" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m69.png" altimg-height="18px" altimg-valign="-2px" altimg-width="52px" alttext="\nabla\cdot\NVar{\mathbf{F}}" display="inline"><mrow><mo>∇</mo><mo lspace="0em" rspace="0.222em">⋅</mo><mi class="ltx_nvar">𝐅</mi></mrow></math></dt> <dd>divergence of vector-valued function <math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi>𝐅</mi></math>; <a href=".././1.6#E21" title="In Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.6.21)</span></a> </dd> <dt id="AA.n12" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m71.png" altimg-height="20px" altimg-valign="-4px" altimg-width="62px" alttext="\nabla\times\NVar{\mathbf{F}}" display="inline"><mrow><mo>∇</mo><mo lspace="0em" rspace="0.222em">×</mo><mi class="ltx_nvar">𝐅</mi></mrow></math></dt> <dd>curl of vector-valued function <math class="ltx_Math" altimg="m62.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbf{F}" display="inline"><mi>𝐅</mi></math>; <a href=".././1.6#E22" title="In Del Operator ‣ §1.6(iii) Vector-Valued Functions ‣ §1.6 Vectors and Vector-Valued Functions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.6.22)</span></a> </dd> <dt id="AA.n67" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m41.png" altimg-height="27px" altimg-valign="-8px" altimg-width="18px" alttext="\int" display="inline"><mo href=".././1.4#iv" title="integral">∫</mo></math></dt> <dd>integral; <a href=".././1.4#iv" title="§1.4(iv) Indefinite Integrals ‣ §1.4 Calculus of One Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§1.4(iv)</span></a> </dd> <dt id="AA.n14" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m43.png" altimg-height="34px" altimg-valign="-9px" altimg-width="52px" alttext="\int_{\NVar{a}}^{(\NVar{b}+)}" display="inline"><msubsup><mo href=".././1.4#iv" title="integral">∫</mo><mi class="ltx_nvar">a</mi><mrow><mo stretchy="false">(</mo><mrow><mi class="ltx_nvar">b</mi><mo rspace="0em">+</mo></mrow><mo stretchy="false">)</mo></mrow></msubsup></math></dt> <dd>loop integral in <math class="ltx_Math" altimg="m60.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="\mathbb{C}" display="inline"><mi href=".././front/introduction#common.t1.r1" title="complex plane">ℂ</mi></math>: path begins at <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math>, encircles <math class="ltx_Math" altimg="m86.png" altimg-height="18px" altimg-valign="-2px" altimg-width="13px" alttext="b" display="inline"><mi>b</mi></math> once in the positive sense, and returns to <math class="ltx_Math" altimg="m85.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="a" display="inline"><mi>a</mi></math>.; <a href=".././5.9#Px1.p1" title="Hankel’s Loop Integral ‣ §5.9(i) Gamma Function ‣ §5.9 Integral Representations ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§5.9(i)</span></a> </dd> <dt id="AA.n13" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m42.png" altimg-height="34px" altimg-valign="-9px" altimg-width="131px" alttext="\int_{P}^{(1+,0+,1-,0-)}" display="inline"><msubsup><mo href=".././1.4#iv" title="integral">∫</mo><mi>P</mi><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo rspace="0em">+</mo></mrow><mo>,</mo><mrow><mn>0</mn><mo rspace="0em">+</mo></mrow><mo>,</mo><mrow><mn>1</mn><mo rspace="0em">−</mo></mrow><mo>,</mo><mrow><mn>0</mn><mo rspace="0em">−</mo></mrow><mo stretchy="false">)</mo></mrow></msubsup></math></dt> <dd>Pochhammer’s loop integral; <a href=".././5.12#Px2.p1" title="Pochhammer’s Integral ‣ §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§5.12</span></a> </dd> <dt id="AA.n15" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m40.png" altimg-height="27px" altimg-valign="-8px" altimg-width="82px" alttext="\int\NVar{\cdots}\,{\mathrm{d}}_{\NVar{q}}\NVar{x}" display="inline"><mrow><mo href=".././1.4#iv" title="integral">∫</mo><mrow><mi class="ltx_nvar" mathvariant="normal">⋯</mi><mo lspace="0em">⁢</mo><mrow><msub><mo href=".././17.2#v.p1" rspace="0em" title="𝑞-differential">d</mo><mi class="ltx_nvar">q</mi></msub><mi class="ltx_nvar">x</mi></mrow></mrow></mrow></math></dt> <dd> <math class="ltx_Math" altimg="m94.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi>q</mi></math>-integral; <a href=".././17.2#v.p1" title="§17.2(v) Integrals ‣ §17.2 Calculus ‣ Properties ‣ Chapter 17 𝑞-Hypergeometric and Related Functions" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§17.2(v)</span></a> </dd> <dt id="AA.n74" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m74.png" altimg-height="33px" altimg-valign="-9px" altimg-width="24px" alttext="\pvint_{\NVar{a}}^{\NVar{b}}" display="inline"><msubsup><mo href=".././1.4#E24" title="Cauchy principal value">⨍</mo><mi class="ltx_nvar">a</mi><mi class="ltx_nvar">b</mi></msubsup></math></dt> <dd>Cauchy principal value; <a href=".././1.4#E24" title="In Cauchy Principal Values ‣ §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.4.24)</span></a> </dd> <dt id="AA.n17" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m27.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\NVar{f}(\NVar{c}-)" display="inline"><mrow><mi class="ltx_nvar">f</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mrow><mi class="ltx_nvar">c</mi><mo rspace="0em">−</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></math></dt> <dd>limit on left (or from below); <a href=".././1.4#E3" title="In §1.4(ii) Continuity ‣ §1.4 Calculus of One Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.4.3)</span></a> </dd> <dt id="AA.n16" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m26.png" altimg-height="25px" altimg-valign="-7px" altimg-width="58px" alttext="\NVar{f}(\NVar{c}+)" display="inline"><mrow><mi class="ltx_nvar">f</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mrow><mi class="ltx_nvar">c</mi><mo rspace="0em">+</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></math></dt> <dd>limit on right (or from above); <a href=".././1.4#E1" title="In §1.4(ii) Continuity ‣ §1.4 Calculus of One Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.4.1)</span></a> </dd> <dt id="AA.n57" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m73.png" altimg-height="17px" altimg-valign="-2px" altimg-width="15px" alttext="\overline{\NVar{z}}" display="inline"><mover accent="true"><mi class="ltx_nvar">z</mi><mo href=".././1.9#E11" title="complex conjugate">¯</mo></mover></math></dt> <dd>complex conjugate; <a href=".././1.9#E11" title="In Complex Conjugate ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.9.11)</span></a> </dd> <dt id="AA.n18" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m102.png" altimg-height="18px" altimg-valign="-2px" altimg-width="27px" alttext="{\NVar{x}}^{\underline{\NVar{n}}}" display="inline"><msup><mi class="ltx_nvar">x</mi><munder accentunder="true"><mi class="ltx_nvar">n</mi><mo>¯</mo></munder></msup></math></dt> <dd>falling factorial; <a href=".././26.1#Px1.p1" title="Alternative Notations ‣ §26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§26.1</span></a> </dd> <dt id="AA.n19" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m101.png" altimg-height="22px" altimg-valign="-2px" altimg-width="27px" alttext="{\NVar{x}}^{\overline{\NVar{n}}}" display="inline"><msup><mi class="ltx_nvar">x</mi><mover accent="true"><mi class="ltx_nvar">n</mi><mo>¯</mo></mover></msup></math></dt> <dd>rising factorial; <a href=".././26.1#Px1.p1" title="Alternative Notations ‣ §26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§26.1</span></a> </dd> <dt id="AA.n21" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m59.png" altimg-height="25px" altimg-valign="-7px" altimg-width="26px" alttext="\left|\NVar{z}\right|" display="inline"><mrow><mo>|</mo><mi class="ltx_nvar">z</mi><mo>|</mo></mrow></math></dt> <dd>modulus (or absolute value); <a href=".././1.9#E7" title="In Modulus and Phase ‣ §1.9(i) Complex Numbers ‣ §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.9.7)</span></a> </dd> <dt id="AA.n22" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m81.png" altimg-height="25px" altimg-valign="-7px" altimg-width="37px" alttext="\|\NVar{\mathbf{a}}\|" display="inline"><mrow><mo stretchy="false">‖</mo><mi class="ltx_nvar">𝐚</mi><mo stretchy="false">‖</mo></mrow></math></dt> <dd>magnitude of vector; <a href=".././1.6#E3" title="In Magnitude and Angle of Vector 𝐚 ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.6.3)</span></a> </dd> <dt id="AA.n23" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m82.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="\|\NVar{\mathbf{x}}\|_{2}" display="inline"><msub><mrow><mo stretchy="false">‖</mo><mi class="ltx_nvar">𝐱</mi><mo stretchy="false">‖</mo></mrow><mn>2</mn></msub></math></dt> <dd>Euclidean norm of a vector; <a href=".././3.2#iii.p1" title="§3.2(iii) Condition of Linear Systems ‣ §3.2 Linear Algebra ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§3.2(iii)</span></a> </dd> <dt id="AA.n24" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m80.png" altimg-height="26px" altimg-valign="-8px" altimg-width="53px" alttext="\|\NVar{\mathbf{A}}\|_{p}" display="inline"><msub><mrow><mo stretchy="false">‖</mo><mi class="ltx_nvar">𝐀</mi><mo stretchy="false">‖</mo></mrow><mi>p</mi></msub></math></dt> <dd> <math class="ltx_Math" altimg="m93.png" altimg-height="17px" altimg-valign="-6px" altimg-width="15px" alttext="p" display="inline"><mi>p</mi></math>-norm of a matrix; <a href=".././3.2#iii.p2" title="§3.2(iii) Condition of Linear Systems ‣ §3.2 Linear Algebra ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§3.2(iii)</span></a> </dd> <dt id="AA.n25" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m84.png" altimg-height="26px" altimg-valign="-8px" altimg-width="47px" alttext="\|\NVar{\mathbf{x}}\|_{p}" display="inline"><msub><mrow><mo stretchy="false">‖</mo><mi class="ltx_nvar">𝐱</mi><mo stretchy="false">‖</mo></mrow><mi>p</mi></msub></math></dt> <dd> <math class="ltx_Math" altimg="m93.png" altimg-height="17px" altimg-valign="-6px" altimg-width="15px" alttext="p" display="inline"><mi>p</mi></math>-norm of a vector; <a href=".././3.2#iii.p1" title="§3.2(iii) Condition of Linear Systems ‣ §3.2 Linear Algebra ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§3.2(iii)</span></a> </dd> <dt id="AA.n26" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m83.png" altimg-height="25px" altimg-valign="-7px" altimg-width="55px" alttext="\|\NVar{\mathbf{x}}\|_{\infty}" display="inline"><msub><mrow><mo stretchy="false">‖</mo><mi class="ltx_nvar">𝐱</mi><mo stretchy="false">‖</mo></mrow><mi mathvariant="normal">∞</mi></msub></math></dt> <dd>infinity (or maximum) norm of a vector; <a href=".././3.2#iii.p1" title="§3.2(iii) Condition of Linear Systems ‣ §3.2 Linear Algebra ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§3.2(iii)</span></a> </dd> <dt id="AA.n20" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m75.png" altimg-height="42px" altimg-valign="-10px" altimg-width="113px" alttext="\scriptstyle\NVar{b_{0}}+\cfrac{\NVar{a_{1}}}{\NVar{b_{1}}+\cfrac{\NVar{a_{2}}% }{\NVar{b_{2}}+}}\cdots" display="inline"><mrow><msub><mi class="ltx_nvar" mathsize="0.700em">b</mi><mn class="ltx_nvar" mathsize="0.710em">0</mn></msub><mo mathsize="0.700em">+</mo><mrow><mstyle scriptlevel="+1"><mrow><mfrac><msub><mi class="ltx_nvar">a</mi><mn class="ltx_nvar">1</mn></msub><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">1</mn></msub><mo>+</mo></mrow></mfrac><mfrac><msub><mi class="ltx_nvar">a</mi><mn class="ltx_nvar">2</mn></msub><mrow><msub><mi class="ltx_nvar">b</mi><mn class="ltx_nvar">2</mn></msub><mo>+</mo></mrow></mfrac></mrow></mstyle><mo>⁢</mo><mi mathsize="0.700em" mathvariant="normal">⋯</mi></mrow></mrow></math></dt> <dd>continued fraction; <a href=".././1.12#i" title="§1.12(i) Notation ‣ §1.12 Continued Fractions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§1.12(i)</span></a> </dd> <dt id="AA.n53" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m54.png" altimg-height="25px" altimg-valign="-7px" altimg-width="35px" alttext="\left\lceil\NVar{x}\right\rceil" display="inline"><mrow><mo href=".././front/introduction#common.t1.r18" title="ceiling of 𝑥">⌈</mo><mi class="ltx_nvar">x</mi><mo href=".././front/introduction#common.t1.r18" title="ceiling of 𝑥">⌉</mo></mrow></math></dt> <dd>ceiling of <math class="ltx_Math" altimg="m96.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math>; <a href=".././front/introduction#common.t1.r18" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n63" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m55.png" altimg-height="25px" altimg-valign="-7px" altimg-width="35px" alttext="\left\lfloor\NVar{x}\right\rfloor" display="inline"><mrow><mo href=".././front/introduction#common.t1.r17" title="floor of 𝑥">⌊</mo><mi class="ltx_nvar">x</mi><mo href=".././front/introduction#common.t1.r17" title="floor of 𝑥">⌋</mo></mrow></math></dt> <dd>floor of <math class="ltx_Math" altimg="m96.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="x" display="inline"><mi>x</mi></math>; <a href=".././front/introduction#common.t1.r17" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n60" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m49.png" altimg-height="25px" altimg-valign="-7px" altimg-width="146px" alttext="\left[\NVar{z_{0},z_{1},\dots,z_{n}}\right]\NVar{f}" display="inline"><mrow><mrow><mo href=".././3.3#E34" title="divided difference">[</mo><mrow><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">0</mn></msub><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar">z</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar">z</mi><mi class="ltx_nvar">n</mi></msub></mrow><mo href=".././3.3#E34" title="divided difference">]</mo></mrow><mo>⁡</mo><mi class="ltx_nvar">f</mi></mrow></math></dt> <dd>divided difference; <a href=".././3.3#E34" title="In §3.3(iii) Divided Differences ‣ §3.3 Interpolation ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(3.3.34)</span></a> </dd> </dl> </div> <div class="ltx_page_column2"> <dl class="ltx_glossarylist ltx_list_notation"> <dt id="AA.n82" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m105.png" altimg-height="26px" altimg-valign="-8px" altimg-width="38px" alttext="{\left[\NVar{a}\right]_{\NVar{\kappa}}}" display="inline"><msub><mrow><mo href=".././35.4#E1" title="partitional shifted factorial">[</mo><mi class="ltx_nvar">a</mi><mo href=".././35.4#E1" title="partitional shifted factorial">]</mo></mrow><mi class="ltx_nvar">κ</mi></msub></math></dt> <dd>partitional shifted factorial; <a href=".././35.4#E1" title="In §35.4(i) Definitions ‣ §35.4 Partitions and Zonal Polynomials ‣ Properties ‣ Chapter 35 Functions of Matrix Argument" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(35.4.1)</span></a> </dd> <dt id="AA.n27" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m28.png" altimg-height="28px" altimg-valign="-7px" altimg-width="64px" alttext="\NVar{f}^{[\NVar{n}]}(\NVar{z})" display="inline"><mrow><msup><mi class="ltx_nvar">f</mi><mrow><mo stretchy="false">[</mo><mi class="ltx_nvar">n</mi><mo stretchy="false">]</mo></mrow></msup><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi class="ltx_nvar">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></dt> <dd> <math class="ltx_Math" altimg="m92.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="n" display="inline"><mi>n</mi></math>th <math class="ltx_Math" altimg="m94.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi>q</mi></math>-derivative; <a href=".././17.2#iv" title="§17.2(iv) Derivatives ‣ §17.2 Calculus ‣ Properties ‣ Chapter 17 𝑞-Hypergeometric and Related Functions" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§17.2(iv)</span></a> </dd> <dt id="AA.n54" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m18.png" altimg-height="25px" altimg-valign="-7px" altimg-width="45px" alttext="[\NVar{a},\NVar{b}]" display="inline"><mrow><mo href=".././front/introduction#common.t1.r30" stretchy="false" title="closed interval">[</mo><mi class="ltx_nvar">a</mi><mo href=".././front/introduction#common.t1.r30" title="closed interval">,</mo><mi class="ltx_nvar">b</mi><mo href=".././front/introduction#common.t1.r30" stretchy="false" title="closed interval">]</mo></mrow></math></dt> <dd>closed interval; <a href=".././front/introduction#common.t1.r30" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n55" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m17.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="[\NVar{a},\NVar{b})" display="inline"><mrow><mo href=".././front/introduction#common.t2.r1" stretchy="false" title="half-closed interval">[</mo><mi class="ltx_nvar">a</mi><mo href=".././front/introduction#common.t2.r1" title="half-closed interval">,</mo><mi class="ltx_nvar">b</mi><mo href=".././front/introduction#common.t2.r1" stretchy="false" title="half-closed interval">)</mo></mrow></math></dt> <dd>half-closed interval; <a href=".././front/introduction#common.t2.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n28" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m19.png" altimg-height="25px" altimg-valign="-7px" altimg-width="179px" alttext="[\NVar{a},\NVar{z}]!=\Gamma\left(a+1,z\right)" display="inline"><mrow><mrow><mrow><mo href=".././front/introduction#common.t1.r30" stretchy="false" title="closed interval">[</mo><mi class="ltx_nvar">a</mi><mo href=".././front/introduction#common.t1.r30" title="closed interval">,</mo><mi class="ltx_nvar">z</mi><mo href=".././front/introduction#common.t1.r30" stretchy="false" title="closed interval">]</mo></mrow><mo href=".././front/introduction#common.t1.r15" title="factorial (as in 𝑛!)">!</mo></mrow><mo>=</mo><mrow><mi href=".././8.2#E2" mathvariant="normal" title="incomplete gamma function">Γ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math></dt> <dd>notation used by <cite class="ltx_cite ltx_citemacro_citet">Dingle (<a href=".././bib/D#bib670" title="Asymptotic Expansions: Their Derivation and Interpretation" class="ltx_ref">1973</a>)</cite>; <a href=".././8.1#p4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§8.1</span></a> <br class="ltx_break"><span class="ltx_text" style="font-size:70%;">(with <a href=".././8.2#E2" title="(8.2.2) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-7px" altimg-width="68px" alttext="\Gamma\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href=".././8.2#E2" mathvariant="normal" title="incomplete gamma function">Γ</mi><mo>⁡</mo><mrow><mo>(</mo><mi class="ltx_nvar">a</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: incomplete gamma function</a>)</span> </dd> <dt id="AA.n44" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m97.png" altimg-height="26px" altimg-valign="-8px" altimg-width="57px" alttext="{[\NVar{p}/\NVar{q}]_{\NVar{f}}}" display="inline"><msub><mrow><mo href=".././3.11#iv.p1" stretchy="false" title="Padé approximant">[</mo><mrow><mi class="ltx_nvar">p</mi><mo href=".././3.11#iv.p1" title="Padé approximant">/</mo><mi class="ltx_nvar">q</mi></mrow><mo href=".././3.11#iv.p1" stretchy="false" title="Padé approximant">]</mo></mrow><mi class="ltx_nvar">f</mi></msub></math></dt> <dd>Padé approximant; <a href=".././3.11#iv.p1" title="§3.11(iv) Padé Approximations ‣ §3.11 Approximation Techniques ‣ Areas ‣ Chapter 3 Numerical Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§3.11(iv)</span></a> </dd> <dt id="AA.n29" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m48.png" altimg-height="29px" altimg-valign="-9px" altimg-width="203px" alttext="\left[\NVar{n}\atop\NVar{k}\right]=(-1)^{n-k}s\left(n,k\right)" display="inline"><mrow><mrow><mo>[</mo><mfrac linethickness="0pt"><mi class="ltx_nvar">n</mi><mi class="ltx_nvar">k</mi></mfrac><mo>]</mo></mrow><mo>=</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>⁢</mo><mrow><mi href=".././26.8#i.p1" title="Stirling number of the first kind">s</mi><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow></mrow></math></dt> <dd>notation used by <cite class="ltx_cite ltx_citemacro_citet">Knuth (<a href=".././bib/K#bib1296" title="Two notes on notation" class="ltx_ref">1992</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Graham<span class="ltx_text ltx_bib_etal"> et al.</span> (<a href=".././bib/G#bib974" title="Concrete Mathematics: A Foundation for Computer Science" class="ltx_ref">1994</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Rosen<span class="ltx_text ltx_bib_etal"> et al.</span> (<a href=".././bib/R#bib1970" title="Handbook of Discrete and Combinatorial Mathematics" class="ltx_ref">2000</a>)</cite>; <a href=".././26.1#Px1.p2" title="Alternative Notations ‣ §26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§26.1</span></a> <br class="ltx_break"><span class="ltx_text" style="font-size:70%;">(with <a href=".././26.8#i.p1" title="§26.8(i) Definitions ‣ §26.8 Set Partitions: Stirling Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m95.png" altimg-height="25px" altimg-valign="-7px" altimg-width="67px" alttext="s\left(\NVar{n},\NVar{k}\right)" display="inline"><mrow><mi href=".././26.8#i.p1" title="Stirling number of the first kind">s</mi><mo>⁡</mo><mrow><mo>(</mo><mi class="ltx_nvar">n</mi><mo>,</mo><mi class="ltx_nvar">k</mi><mo>)</mo></mrow></mrow></math>: Stirling number of the first kind</a>)</span> </dd> <dt id="AA.n78" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m37.png" altimg-height="36px" altimg-valign="-16px" altimg-width="137px" alttext="\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}" display="inline"><msub><mrow><mo href=".././26.16#p3" title="𝑞-multinomial coefficient">[</mo><mfrac linethickness="0.0pt"><mrow><msub><mi class="ltx_nvar">a</mi><mn class="ltx_nvar">1</mn></msub><mo>+</mo><msub><mi class="ltx_nvar">a</mi><mn class="ltx_nvar">2</mn></msub><mo>+</mo><mi mathvariant="normal">⋯</mi><mo>+</mo><msub><mi class="ltx_nvar">a</mi><mi class="ltx_nvar">n</mi></msub></mrow><mrow><msub><mi class="ltx_nvar">a</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><msub><mi class="ltx_nvar">a</mi><mn class="ltx_nvar">2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">a</mi><mi class="ltx_nvar">n</mi></msub></mrow></mfrac><mo href=".././26.16#p3" title="𝑞-multinomial coefficient">]</mo></mrow><mi class="ltx_nvar">q</mi></msub></math></dt> <dd> <math class="ltx_Math" altimg="m94.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi>q</mi></math>-multinomial coefficient; <a href=".././26.16#p3" title="§26.16 Multiset Permutations ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§26.16</span></a> </dd> <dt id="AA.n76" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m39.png" altimg-height="33px" altimg-valign="-13px" altimg-width="45px" alttext="\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}" display="inline"><msub><mrow><mo href=".././17.2#E27" title="𝑞-binomial coefficient (or Gaussian polynomial)">[</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar">n</mi><mi class="ltx_nvar">m</mi></mfrac><mo href=".././17.2#E27" title="𝑞-binomial coefficient (or Gaussian polynomial)">]</mo></mrow><mi class="ltx_nvar">q</mi></msub></math></dt> <dd> <math class="ltx_Math" altimg="m94.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi>q</mi></math>-binomial coefficient (or Gaussian polynomial); <a href=".././17.2#E27" title="In §17.2(ii) Binomial Coefficients ‣ §17.2 Calculus ‣ Properties ‣ Chapter 17 𝑞-Hypergeometric and Related Functions" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(17.2.27)</span></a> </dd> <dt id="AA.n58" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m103.png" altimg-height="25px" altimg-valign="-7px" altimg-width="34px" alttext="{\left(\NVar{S}\right)}" display="inline"><mrow><mo href=".././26.2#Px2" title="cycle">(</mo><mi class="ltx_nvar">S</mi><mo href=".././26.2#Px2" title="cycle">)</mo></mrow></math></dt> <dd>cycle; <a href=".././26.2#Px2" title="Cycle ‣ §26.2 Basic Definitions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§26.2</span></a> </dd> <dt id="AA.n30" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="143px" alttext="(\NVar{z-1})!=\Gamma\left(z\right)" display="inline"><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi class="ltx_nvar">z</mi><mo class="ltx_nvar">−</mo><mn class="ltx_nvar">1</mn></mrow><mo stretchy="false">)</mo></mrow><mo href=".././front/introduction#common.t1.r15" title="factorial (as in 𝑛!)">!</mo></mrow><mo>=</mo><mrow><mi href=".././5.2#E1" mathvariant="normal" title="gamma function">Γ</mi><mo>⁡</mo><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math></dt> <dd>alternative notation; <a href=".././5.1#p4" title="§5.1 Special Notation ‣ Notation ‣ Chapter 5 Gamma Function" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§5.1</span></a> <br class="ltx_break"><span class="ltx_text" style="font-size:70%;">(with <a href=".././5.2#E1" title="(5.2.1) ‣ Euler’s Integral ‣ §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><math class="ltx_Math" altimg="m22.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="\Gamma\left(\NVar{z}\right)" display="inline"><mrow><mi href=".././5.2#E1" mathvariant="normal" title="gamma function">Γ</mi><mo>⁡</mo><mrow><mo>(</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: gamma function</a>)</span> </dd> <dt id="AA.n45" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m104.png" altimg-height="26px" altimg-valign="-8px" altimg-width="43px" alttext="{\left(\NVar{a}\right)_{\NVar{n}}}" display="inline"><msub><mrow><mo href=".././5.2#iii" title="Pochhammer’s symbol (or shifted factorial)">(</mo><mi class="ltx_nvar">a</mi><mo href=".././5.2#iii" title="Pochhammer’s symbol (or shifted factorial)">)</mo></mrow><mi class="ltx_nvar">n</mi></msub></math></dt> <dd>Pochhammer’s symbol (or shifted factorial); <a href=".././5.2#iii" title="§5.2(iii) Pochhammer’s Symbol ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§5.2(iii)</span></a> </dd> <dt id="AA.n73" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="50px" alttext="(\NVar{a},\NVar{b})" display="inline"><mrow><mo href=".././front/introduction#common.t1.r29" stretchy="false" title="open interval">(</mo><mi class="ltx_nvar">a</mi><mo href=".././front/introduction#common.t1.r29" title="open interval">,</mo><mi class="ltx_nvar">b</mi><mo href=".././front/introduction#common.t1.r29" stretchy="false" title="open interval">)</mo></mrow></math></dt> <dd>open interval; <a href=".././front/introduction#common.t1.r29" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n72" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="47px" alttext="(\NVar{a},\NVar{b}]" display="inline"><mrow><mo href=".././front/introduction#common.t2.r1" stretchy="false" title="half-closed interval">(</mo><mi class="ltx_nvar">a</mi><mo href=".././front/introduction#common.t2.r1" title="half-closed interval">,</mo><mi class="ltx_nvar">b</mi><mo href=".././front/introduction#common.t2.r1" stretchy="false" title="half-closed interval">]</mo></mrow></math></dt> <dd>half-closed interval; <a href=".././front/introduction#common.t2.r1" title="Common Notations and Definitions ‣ Mathematical Introduction" class="ltx_ref"><span class="ltx_text ltx_ref_title">Common Notations and Definitions</span></a> </dd> <dt id="AA.n31" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="183px" alttext="(\NVar{a},\NVar{z})!=\gamma\left(a+1,z\right)" display="inline"><mrow><mrow><mrow><mo href=".././front/introduction#common.t1.r29" stretchy="false" title="open interval">(</mo><mi class="ltx_nvar">a</mi><mo href=".././front/introduction#common.t1.r29" title="open interval">,</mo><mi class="ltx_nvar">z</mi><mo href=".././front/introduction#common.t1.r29" stretchy="false" title="open interval">)</mo></mrow><mo href=".././front/introduction#common.t1.r15" title="factorial (as in 𝑛!)">!</mo></mrow><mo>=</mo><mrow><mi href=".././8.2#E1" title="incomplete gamma function">γ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow></math></dt> <dd>notation used by <cite class="ltx_cite ltx_citemacro_citet">Dingle (<a href=".././bib/D#bib670" title="Asymptotic Expansions: Their Derivation and Interpretation" class="ltx_ref">1973</a>)</cite>; <a href=".././8.1#p4" title="§8.1 Special Notation ‣ Notation ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§8.1</span></a> <br class="ltx_break"><span class="ltx_text" style="font-size:70%;">(with <a href=".././8.2#E1" title="(8.2.1) ‣ §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions" class="ltx_ref"><math class="ltx_Math" altimg="m33.png" altimg-height="25px" altimg-valign="-7px" altimg-width="67px" alttext="\gamma\left(\NVar{a},\NVar{z}\right)" display="inline"><mrow><mi href=".././8.2#E1" title="incomplete gamma function">γ</mi><mo>⁡</mo><mrow><mo>(</mo><mi class="ltx_nvar">a</mi><mo>,</mo><mi class="ltx_nvar">z</mi><mo>)</mo></mrow></mrow></math>: incomplete gamma function</a>)</span> </dd> <dt id="AA.n64" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m47.png" altimg-height="25px" altimg-valign="-7px" altimg-width="60px" alttext="\left(\NVar{m},\NVar{n}\right)" display="inline"><mrow><mo href=".././27.1#t1.r3" title="greatest common divisor (gcd)">(</mo><mrow><mi class="ltx_nvar">m</mi><mo>,</mo><mi class="ltx_nvar">n</mi></mrow><mo href=".././27.1#t1.r3" title="greatest common divisor (gcd)">)</mo></mrow></math></dt> <dd>greatest common divisor (gcd); <a href=".././27.1#t1.r3" title="§27.1 Special Notation ‣ Notation ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§27.1</span></a> </dd> <dt id="AA.n38" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="55px" alttext="(\NVar{n}|\NVar{P})" display="inline"><mrow><mo href=".././27.9#p3" stretchy="false" title="Jacobi symbol">(</mo><mrow><mi class="ltx_nvar">n</mi><mo fence="false" href=".././27.9#p3" title="Jacobi symbol">|</mo><mi class="ltx_nvar">P</mi></mrow><mo href=".././27.9#p3" stretchy="false" title="Jacobi symbol">)</mo></mrow></math></dt> <dd>Jacobi symbol; <a href=".././27.9#p3" title="§27.9 Quadratic Characters ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§27.9</span></a> </dd> <dt id="AA.n43" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m8.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="(\NVar{n}|\NVar{p})" display="inline"><mrow><mo href=".././27.9" stretchy="false" title="Legendre symbol">(</mo><mrow><mi class="ltx_nvar">n</mi><mo fence="false" href=".././27.9" title="Legendre symbol">|</mo><mi class="ltx_nvar">p</mi></mrow><mo href=".././27.9" stretchy="false" title="Legendre symbol">)</mo></mrow></math></dt> <dd>Legendre symbol; <a href=".././27.9" title="§27.9 Quadratic Characters ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§27.9</span></a> </dd> <dt id="AA.n75" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m45.png" altimg-height="26px" altimg-valign="-8px" altimg-width="62px" alttext="\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}" display="inline"><msub><mrow><mo href=".././17.2#i.p1" title="𝑞-Pochhammer symbol (or 𝑞-shifted factorial)">(</mo><mi class="ltx_nvar">a</mi><mo href=".././17.2#i.p1" title="𝑞-Pochhammer symbol (or 𝑞-shifted factorial)">;</mo><mi class="ltx_nvar">q</mi><mo href=".././17.2#i.p1" title="𝑞-Pochhammer symbol (or 𝑞-shifted factorial)">)</mo></mrow><mi class="ltx_nvar">n</mi></msub></math></dt> <dd> <math class="ltx_Math" altimg="m94.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi>q</mi></math>-Pochhammer symbol (or <math class="ltx_Math" altimg="m94.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi>q</mi></math>-shifted factorial); <a href=".././17.2#i.p1" title="§17.2(i) 𝑞-Calculus ‣ §17.2 Calculus ‣ Properties ‣ Chapter 17 𝑞-Hypergeometric and Related Functions" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§17.2(i)</span></a> </dd> <dt id="AA.n79" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m44.png" altimg-height="26px" altimg-valign="-8px" altimg-width="167px" alttext="\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}" display="inline"><msub><mrow><mo href=".././17.2#i.p1" title="multiple 𝑞-Pochhammer symbol">(</mo><mrow><msub><mi class="ltx_nvar">a</mi><mn class="ltx_nvar">1</mn></msub><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar">a</mi><mn class="ltx_nvar">2</mn></msub><mo class="ltx_nvar">,</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo class="ltx_nvar">,</mo><msub><mi class="ltx_nvar">a</mi><mi class="ltx_nvar">r</mi></msub></mrow><mo href=".././17.2#i.p1" title="multiple 𝑞-Pochhammer symbol">;</mo><mi class="ltx_nvar">q</mi><mo href=".././17.2#i.p1" title="multiple 𝑞-Pochhammer symbol">)</mo></mrow><mi class="ltx_nvar">n</mi></msub></math></dt> <dd>multiple <math class="ltx_Math" altimg="m94.png" altimg-height="17px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi>q</mi></math>-Pochhammer symbol; <a href=".././17.2#i.p1" title="§17.2(i) 𝑞-Calculus ‣ §17.2 Calculus ‣ Properties ‣ Chapter 17 𝑞-Hypergeometric and Related Functions" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§17.2(i)</span></a> </dd> <dt id="AA.n32" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m46.png" altimg-height="25px" altimg-valign="-7px" altimg-width="234px" alttext="\left(\NVar{j_{1}}\;\NVar{m_{1}}\;\NVar{j_{2}}\;\NVar{m_{2}}|\NVar{j_{1}}\;% \NVar{j_{2}}\;\NVar{j_{3}}\,\,\NVar{m_{3}}\right)" display="inline"><mrow><mo>(</mo><mrow><mrow><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">1</mn></msub><mo>⁢</mo><msub><mi class="ltx_nvar">m</mi><mn class="ltx_nvar">1</mn></msub><mo>⁢</mo><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">2</mn></msub><mo>⁢</mo><msub><mi class="ltx_nvar">m</mi><mn class="ltx_nvar">2</mn></msub></mrow><mo fence="false">|</mo><mrow><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">1</mn></msub><mo>⁢</mo><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">2</mn></msub><mo>⁢</mo><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">3</mn></msub><mo>⁢</mo><msub><mi class="ltx_nvar">m</mi><mn class="ltx_nvar">3</mn></msub></mrow></mrow><mo>)</mo></mrow></math></dt> <dd>Clebsch–Gordan coefficient; <a href=".././34.1#p4" title="§34.1 Special Notation ‣ Notation ‣ Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§34.1</span></a> </dd> <dt id="AA.n48" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m98.png" altimg-height="20px" altimg-valign="-2px" altimg-width="32px" alttext="{\NVar{\mathbf{A}}}^{*}" display="inline"><msup><mi class="ltx_nvar">𝐀</mi><mo href=".././1.3#Px10" title="adjoint of matrix">∗</mo></msup></math></dt> <dd>adjoint of matrix; <a href=".././1.3#Px10" title="Self-Adjoint Operators on 𝐄_𝑛 ‣ §1.3(iv) Matrices as Linear Operators ‣ §1.3 Determinants, Linear Operators, and Spectral Expansions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§1.3(iv)</span></a> </dd> <dt id="AA.n37" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m100.png" altimg-height="23px" altimg-valign="-2px" altimg-width="36px" alttext="{\NVar{\mathbf{A}}}^{{\rm H}}" display="inline"><msup><mi class="ltx_nvar">𝐀</mi><mi href=".././1.2#E30" mathvariant="normal" title="Hermitian conjugate of matrix">H</mi></msup></math></dt> <dd>Hermitian conjugate of matrix; <a href=".././1.2#E30" title="In General 𝑚×𝑛 Matrices ‣ §1.2(v) Matrices, Vectors, Scalar Products, and Norms ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.2.30)</span></a> </dd> <dt id="AA.n85" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m25.png" altimg-height="22px" altimg-valign="-2px" altimg-width="35px" alttext="\NVar{\mathbf{A}}^{\mathrm{T}}" display="inline"><msup><mi class="ltx_nvar">𝐀</mi><mi href=".././1.2#E28" mathvariant="normal" title="transpose of matrix">T</mi></msup></math></dt> <dd>transpose of matrix; <a href=".././1.2#E28" title="In General 𝑚×𝑛 Matrices ‣ §1.2(v) Matrices, Vectors, Scalar Products, and Norms ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.2.28)</span></a> </dd> <dt id="AA.n51" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m34.png" altimg-height="29px" altimg-valign="-9px" altimg-width="38px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}" display="inline"><mrow><mo href=".././1.2#i" title="binomial coefficient">(</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar">m</mi><mi class="ltx_nvar">n</mi></mfrac><mo href=".././1.2#i" title="binomial coefficient">)</mo></mrow></math></dt> <dd>binomial coefficient; <a href=".././1.2#i" title="§1.2(i) Binomial Coefficients ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§1.2(i)</span></a> </dd> <dt id="AA.n70" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m35.png" altimg-height="32px" altimg-valign="-12px" altimg-width="132px" alttext="\genfrac{(}{)}{0.0pt}{}{\NVar{n_{1}}+\NVar{n_{2}}+\dots+\NVar{n_{k}}}{\NVar{n_% {1}},\NVar{n_{2}},\ldots,\NVar{n_{k}}}" display="inline"><mrow><mo href=".././26.4#i.p1" title="multinomial coefficient">(</mo><mfrac linethickness="0.0pt"><mrow><msub><mi class="ltx_nvar">n</mi><mn class="ltx_nvar">1</mn></msub><mo>+</mo><msub><mi class="ltx_nvar">n</mi><mn class="ltx_nvar">2</mn></msub><mo>+</mo><mi mathvariant="normal">⋯</mi><mo>+</mo><msub><mi class="ltx_nvar">n</mi><mi class="ltx_nvar">k</mi></msub></mrow><mrow><msub><mi class="ltx_nvar">n</mi><mn class="ltx_nvar">1</mn></msub><mo>,</mo><msub><mi class="ltx_nvar">n</mi><mn class="ltx_nvar">2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi class="ltx_nvar">n</mi><mi class="ltx_nvar">k</mi></msub></mrow></mfrac><mo href=".././26.4#i.p1" title="multinomial coefficient">)</mo></mrow></math></dt> <dd>multinomial coefficient; <a href=".././26.4#i.p1" title="§26.4(i) Definitions ‣ §26.4 Lattice Paths: Multinomial Coefficients and Set Partitions ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§26.4(i)</span></a> </dd> <dt id="AA.n84" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m31.png" altimg-height="54px" altimg-valign="-22px" altimg-width="159px" alttext="\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}" display="inline"><mrow><mo href=".././34.2#E4" title="3⁢𝑗 symbol">(</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">1</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">2</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">3</mn></msub></mtd></mtr><mtr><mtd><msub><mi class="ltx_nvar">m</mi><mn class="ltx_nvar">1</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">m</mi><mn class="ltx_nvar">2</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">m</mi><mn class="ltx_nvar">3</mn></msub></mtd></mtr></mtable><mo href=".././34.2#E4" title="3⁢𝑗 symbol">)</mo></mrow></math></dt> <dd> <math class="ltx_Math" altimg="m65.png" altimg-height="21px" altimg-valign="-6px" altimg-width="24px" alttext="\mathit{3j}" display="inline"><mrow><mn class="ltx_mathvariant_italic" href=".././34.2#E4" mathvariant="italic" title="3⁢𝑗 symbol">3</mn><mo href=".././34.2#E4" title="3⁢𝑗 symbol">⁢</mo><mi href=".././34.2#E4" title="3⁢𝑗 symbol">j</mi></mrow></math> symbol; <a href=".././34.2#E4" title="In §34.2 Definition: 3⁢𝑗 Symbol ‣ Properties ‣ Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(34.2.4)</span></a> </dd> <dt id="AA.n56" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="60px" alttext="[{\NVar{\mathbf{A}}},{\NVar{\mathbf{B}}}]" display="inline"><mrow><mo href=".././1.2#E66" stretchy="false" title="commutator">[</mo><mi class="ltx_nvar">𝐀</mi><mo href=".././1.2#E66" title="commutator">,</mo><mi class="ltx_nvar">𝐁</mi><mo href=".././1.2#E66" stretchy="false" title="commutator">]</mo></mrow></math></dt> <dd>commutator; <a href=".././1.2#E66" title="In The Commutator ‣ §1.2(vi) Square Matrices ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.2.66)</span></a> </dd> <dt id="AA.n47" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m38.png" altimg-height="29px" altimg-valign="-9px" altimg-width="32px" alttext="\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href=".././26.13#E3" title="Stirling cycle number of the first kind">[</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar">n</mi><mi class="ltx_nvar">k</mi></mfrac><mo href=".././26.13#E3" title="Stirling cycle number of the first kind">]</mo></mrow></math></dt> <dd>Stirling cycle number of the first kind; <a href=".././26.13#E3" title="In §26.13 Permutations: Cycle Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(26.13.3)</span></a> </dd> <dt id="AA.n65" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m53.png" altimg-height="25px" altimg-valign="-7px" altimg-width="53px" alttext="\left\langle\NVar{f},\NVar{g}\right\rangle" display="inline"><mrow><mo href=".././1.18#E12" title="inner product over functions">⟨</mo><mi class="ltx_nvar">f</mi><mo href=".././1.18#E12" title="inner product over functions">,</mo><mi class="ltx_nvar">g</mi><mo href=".././1.18#E12" title="inner product over functions">⟩</mo></mrow></math></dt> <dd>inner product over functions; <a href=".././1.18#E12" title="In §1.18(ii) 𝐿² spaces on intervals in ℝ ‣ §1.18 Linear Second Order Differential Operators and Eigenfunction Expansions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.18.12)</span></a> </dd> <dt id="AA.n66" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m51.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle" display="inline"><mrow><mo href=".././1.2#E40" title="inner product over vectors">⟨</mo><mi class="ltx_nvar">𝐮</mi><mo href=".././1.2#E40" title="inner product over vectors">,</mo><mi class="ltx_nvar">𝐯</mi><mo href=".././1.2#E40" title="inner product over vectors">⟩</mo></mrow></math></dt> <dd>inner product over vectors; <a href=".././1.2#E40" title="In Row and Column Vectors ‣ §1.2(v) Matrices, Vectors, Scalar Products, and Norms ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.2.40)</span></a> </dd> <dt id="AA.n86" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m58.png" altimg-height="25px" altimg-valign="-7px" altimg-width="38px" alttext="\left\|{\NVar{\mathbf{v}}}\right\|" display="inline"><mrow><mo href=".././1.2#E46" title="vector norm (𝑙²)">‖</mo><mi class="ltx_nvar">𝐯</mi><mo href=".././1.2#E46" title="vector norm (𝑙²)">‖</mo></mrow></math></dt> <dd>vector norm (<math class="ltx_Math" altimg="m89.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="l" display="inline"><mi>l</mi></math>²); <a href=".././1.2#E46" title="In Vector Norms ‣ §1.2(v) Matrices, Vectors, Scalar Products, and Norms ‣ §1.2 Elementary Algebra ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.2.46)</span></a> </dd> <dt id="AA.n33" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m79.png" altimg-height="25px" altimg-valign="-7px" altimg-width="49px" alttext="\{\NVar{\ldots}\}" display="inline"><mrow><mo stretchy="false">{</mo><mi class="ltx_nvar" mathvariant="normal">…</mi><mo stretchy="false">}</mo></mrow></math></dt> <dd>sequence, asymptotic sequence (or scale), or enumerable set; <a href=".././2.1#v.p1" title="§2.1(v) Generalized Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§2.1(v)</span></a> </dd> <dt id="AA.n46" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m57.png" altimg-height="25px" altimg-valign="-7px" altimg-width="55px" alttext="\left\{\NVar{z},\NVar{\zeta}\right\}" display="inline"><mrow><mo href=".././1.13#E20" title="Schwarzian derivative">{</mo><mi class="ltx_nvar">z</mi><mo href=".././1.13#E20" title="Schwarzian derivative">,</mo><mi class="ltx_nvar">ζ</mi><mo href=".././1.13#E20" title="Schwarzian derivative">}</mo></mrow></math></dt> <dd>Schwarzian derivative; <a href=".././1.13#E20" title="In Liouville Transformation ‣ §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(1.13.20)</span></a> </dd> <dt id="AA.n34" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m56.png" altimg-height="29px" altimg-valign="-9px" altimg-width="138px" alttext="\left\{\NVar{n}\atop\NVar{k}\right\}=S\left(n,k\right)" display="inline"><mrow><mrow><mo>{</mo><mfrac linethickness="0pt"><mi class="ltx_nvar">n</mi><mi class="ltx_nvar">k</mi></mfrac><mo>}</mo></mrow><mo>=</mo><mrow><mi href=".././26.8#i.p3" title="Stirling number of the second kind">S</mi><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow></math></dt> <dd>notation used by <cite class="ltx_cite ltx_citemacro_citet">Knuth (<a href=".././bib/K#bib1296" title="Two notes on notation" class="ltx_ref">1992</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Graham<span class="ltx_text ltx_bib_etal"> et al.</span> (<a href=".././bib/G#bib974" title="Concrete Mathematics: A Foundation for Computer Science" class="ltx_ref">1994</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Rosen<span class="ltx_text ltx_bib_etal"> et al.</span> (<a href=".././bib/R#bib1970" title="Handbook of Discrete and Combinatorial Mathematics" class="ltx_ref">2000</a>)</cite>; <a href=".././26.1#Px1.p3" title="Alternative Notations ‣ §26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§26.1</span></a> <br class="ltx_break"><span class="ltx_text" style="font-size:70%;">(with <a href=".././26.8#i.p3" title="§26.8(i) Definitions ‣ §26.8 Set Partitions: Stirling Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="71px" alttext="S\left(\NVar{n},\NVar{k}\right)" display="inline"><mrow><mi href=".././26.8#i.p3" title="Stirling number of the second kind">S</mi><mo>⁡</mo><mrow><mo>(</mo><mi class="ltx_nvar">n</mi><mo>,</mo><mi class="ltx_nvar">k</mi><mo>)</mo></mrow></mrow></math>: Stirling number of the second kind</a>)</span> </dd> <dt id="AA.n83" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m30.png" altimg-height="54px" altimg-valign="-22px" altimg-width="131px" alttext="\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}" display="inline"><mrow><mo href=".././34.4#E1" title="6⁢𝑗 symbol">{</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">1</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">2</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">3</mn></msub></mtd></mtr><mtr><mtd><msub><mi class="ltx_nvar">l</mi><mn class="ltx_nvar">1</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">l</mi><mn class="ltx_nvar">2</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">l</mi><mn class="ltx_nvar">3</mn></msub></mtd></mtr></mtable><mo href=".././34.4#E1" title="6⁢𝑗 symbol">}</mo></mrow></math></dt> <dd> <math class="ltx_Math" altimg="m66.png" altimg-height="21px" altimg-valign="-6px" altimg-width="24px" alttext="\mathit{6j}" display="inline"><mrow><mn class="ltx_mathvariant_italic" href=".././34.4#E1" mathvariant="italic" title="6⁢𝑗 symbol">6</mn><mo href=".././34.4#E1" title="6⁢𝑗 symbol">⁢</mo><mi href=".././34.4#E1" title="6⁢𝑗 symbol">j</mi></mrow></math> symbol; <a href=".././34.4#E1" title="In §34.4 Definition: 6⁢𝑗 Symbol ‣ Properties ‣ Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(34.4.1)</span></a> </dd> <dt id="AA.n71" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m29.png" altimg-height="79px" altimg-valign="-34px" altimg-width="161px" alttext="\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}" display="inline"><mrow><mo href=".././34.6#E1" title="9⁢𝑗 symbol">{</mo><mtable columnspacing="5pt" rowspacing="0pt"><mtr><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">11</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">12</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">13</mn></msub></mtd></mtr><mtr><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">21</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">22</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">23</mn></msub></mtd></mtr><mtr><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">31</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">32</mn></msub></mtd><mtd><msub><mi class="ltx_nvar">j</mi><mn class="ltx_nvar">33</mn></msub></mtd></mtr></mtable><mo href=".././34.6#E1" title="9⁢𝑗 symbol">}</mo></mrow></math></dt> <dd> <math class="ltx_Math" altimg="m67.png" altimg-height="21px" altimg-valign="-6px" altimg-width="24px" alttext="\mathit{9j}" display="inline"><mrow><mn class="ltx_mathvariant_italic" href=".././34.6#E1" mathvariant="italic" title="9⁢𝑗 symbol">9</mn><mo href=".././34.6#E1" title="9⁢𝑗 symbol">⁢</mo><mi href=".././34.6#E1" title="9⁢𝑗 symbol">j</mi></mrow></math> symbol; <a href=".././34.6#E1" title="In §34.6 Definition: 9⁢𝑗 Symbol ‣ Properties ‣ Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(34.6.1)</span></a> </dd> <dt id="AA.n35" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m52.png" altimg-height="25px" altimg-valign="-7px" altimg-width="54px" alttext="\left\langle\NVar{f},\NVar{\phi}\right\rangle" display="inline"><mrow><mo href=".././1.16#i.p5" title="action of distribution on test function">⟨</mo><mi class="ltx_nvar">f</mi><mo href=".././1.16#i.p5" title="action of distribution on test function">,</mo><mi class="ltx_nvar">ϕ</mi><mo href=".././1.16#i.p5" title="action of distribution on test function">⟩</mo></mrow></math></dt> <dd>tempered distribution; <a href=".././2.6#E11" title="In §2.6(ii) Stieltjes Transform ‣ §2.6 Distributional Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations" class="ltx_ref"><span class="ltx_text ltx_ref_tag">(2.6.11)</span></a> </dd> <dt id="AA.n59" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m50.png" altimg-height="25px" altimg-valign="-7px" altimg-width="56px" alttext="\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle" display="inline"><mrow><mo href=".././1.16#i.p5" title="action of distribution on test function">⟨</mo><mi class="ltx_nvar" mathvariant="normal">Λ</mi><mo href=".././1.16#i.p5" title="action of distribution on test function">,</mo><mi class="ltx_nvar">ϕ</mi><mo href=".././1.16#i.p5" title="action of distribution on test function">⟩</mo></mrow></math></dt> <dd>action of distribution on test function; <a href=".././1.16#i.p5" title="§1.16(i) Test Functions ‣ §1.16 Distributions ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§1.16(i)</span></a> </dd> <dt id="AA.n36" class="ltx_glossaryentry ltx_list_notation"><math class="ltx_Math" altimg="m36.png" altimg-height="29px" altimg-valign="-9px" altimg-width="34px" alttext="\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}" display="inline"><mrow><mo href=".././26.14#i.p3" title="Eulerian number">⟨</mo><mfrac linethickness="0.0pt"><mi class="ltx_nvar">n</mi><mi class="ltx_nvar">k</mi></mfrac><mo href=".././26.14#i.p3" title="Eulerian number">⟩</mo></mrow></math></dt> <dd>Eulerian number; <a href=".././26.14#i.p3" title="§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§26.14(i)</span></a> </dd> </dl> </div> </div> </section> </div> <div class="ltx_page_footer"> <div class="ltx_siblings"> <a href=".././idx/Z" title="In Index" class="ltx_ref" rel="prev"><span class="ltx_text ltx_ref_title">Index Z</span></a><a href=".././not/A" title="In Notations" class="ltx_ref" rel="next"><span class="ltx_text ltx_ref_title">Notations A</span></a> </div> <div class="ltx_footer_links ltx_centering"> <a href=".././about/notices">© 2010–2025 NIST</a> / <a 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