# hypergeometric functions

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##### 1: 19.15 Advantages of Symmetry
Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s $F_{D}$ (Carlson (1961b)). …
##### 2: 17.1 Special Notation
###### §17.1 Special Notation
 $k,j,m,n,r,s$ nonnegative integers. …
The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$, and the $q$-analogs of the Appell functions $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)$, $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)$, $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)$, and $\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)$. Another function notation used is the “idem” function: …
##### 3: 13.2 Definitions and Basic Properties
13.2.2 $M\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{{\left(b% \right)_{s}}s!}z^{s}=1+\frac{a}{b}z+\frac{a(a+1)}{b(b+1)2!}z^{2}+\cdots,$
13.2.3 ${\mathbf{M}}\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{% \Gamma\left(b+s\right)s!}z^{s},$
13.2.4 $M\left(a,b,z\right)=\Gamma\left(b\right){\mathbf{M}}\left(a,b,z\right).$
13.2.6 $U\left(a,b,z\right)\sim z^{-a},$ $z\to\infty$, $|\operatorname{ph}z|\leq\frac{3}{2}\pi-\delta$,
##### 4: 13.14 Definitions and Basic Properties
13.14.2 $M_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}M\left(\tfrac% {1}{2}+\mu-\kappa,1+2\mu,z\right),$
13.14.3 $W_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu}U\left(\tfrac% {1}{2}+\mu-\kappa,1+2\mu,z\right),$
Except when $z=0$, each branch of the functions $\ifrac{M_{\kappa,\mu}\left(z\right)}{\Gamma\left(2\mu+1\right)}$ and $W_{\kappa,\mu}\left(z\right)$ is entire in $\kappa$ and $\mu$. …
13.14.26 $\mathscr{W}\left\{M_{\kappa,\mu}\left(z\right),W_{\kappa,\mu}\left(z\right)% \right\}=-\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa% \right)},$
13.14.28 $\mathscr{W}\left\{M_{\kappa,-\mu}\left(z\right),W_{\kappa,\mu}\left(z\right)% \right\}=-\frac{\Gamma\left(1-2\mu\right)}{\Gamma\left(\frac{1}{2}-\mu-\kappa% \right)},$
##### 5: 16.4 Argument Unity
Denote, formally, the bilateral hypergeometric function
16.4.16 ${{}_{p}H_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\sum_{k% =-\infty}^{\infty}\frac{{\left(a_{1}\right)_{k}}\dots{\left(a_{p}\right)_{k}}}% {{\left(b_{1}\right)_{k}}\dots{\left(b_{q}\right)_{k}}}z^{k}.$
##### 6: 35.6 Confluent Hypergeometric Functions of Matrix Argument
###### §35.6 Confluent HypergeometricFunctions of Matrix Argument
35.6.2 $\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)|\mathbf{X}|% ^{a-\frac{1}{2}(m+1)}\*{|\mathbf{I}+\mathbf{X}|}^{b-a-\frac{1}{2}(m+1)}\mathrm% {d}\mathbf{X},$ $\Re(a)>\frac{1}{2}(m-1)$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
##### 8: 15.2 Definitions and Analytical Properties
###### §15.2(i) Gauss Series
15.2.1 $F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(b% \right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}% {c(c+1)2!}z^{2}+\cdots=\frac{\Gamma\left(c\right)}{\Gamma\left(a\right)\Gamma% \left(b\right)}\sum_{s=0}^{\infty}\frac{\Gamma\left(a+s\right)\Gamma\left(b+s% \right)}{\Gamma\left(c+s\right)s!}z^{s},$
15.2.2 $\mathbf{F}\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{% \left(b\right)_{s}}}{\Gamma\left(c+s\right)s!}z^{s},$ $|z|<1$,
##### 9: 35.8 Generalized Hypergeometric Functions of Matrix Argument
###### §35.8(i) Definition
35.8.1 ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right% )=\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{|\kappa|=k}\frac{{\left[a_{1}\right]_{% \kappa}}\cdots{\left[a_{p}\right]_{\kappa}}}{{\left[b_{1}\right]_{\kappa}}% \cdots{\left[b_{q}\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right).$
##### 10: 16.2 Definition and Analytic Properties
###### §16.2(i) Generalized Hypergeometric Series
16.2.5 ${{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b};z\right)=\ifrac{{{}_{p}F_{% q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)}{\left(\Gamma% \left(b_{1}\right)\cdots\Gamma\left(b_{q}\right)\right)}=\sum_{k=0}^{\infty}% \frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{\Gamma\left(b_{1% }+k\right)\cdots\Gamma\left(b_{q}+k\right)}\frac{z^{k}}{k!};$