16 Generalized Hypergeometric Functions and Meijer

-Function 16.16 Transformations of Variables 16.18 Special Cases

§16.17 Definition

Notes:: See Luke (1969a, Chapter 5).
Keywords:: Meijer -function
Permalink:: http://dlmf.nist.gov/16.17

Again assume $a_{1},a_{2},\dots,a_{p}$ and $b_{1},b_{2},\dots,b_{q}$ are real or complex parameters. Assume also that and are integers such that $0\leq m\leq q$ and $0\leq n\leq p$ , and none of $a_{k}-b_{j}$ is a positive integer when $1\leq k\leq n$ and $1\leq j\leq m$ . Then the Meijer -function is defined via the Mellin–Barnes integral representation:

16.17.1 $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;\mathbf{a};\mathbf{b}\right)=% \mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1}% ,\dots,b_{q}}\right)=\frac{1}{2\pi i}\int_{{L}}\left({\textstyle\ifrac{\prod% \limits_{{\ell=1}}^{m}\mathop{\Gamma\/}\nolimits\!\left(b_{{\ell}}-s\right)% \prod\limits_{{\ell=1}}^{n}\mathop{\Gamma\/}\nolimits\!\left(1-a_{{\ell}}+s% \right)}{\left(\prod\limits_{{\ell=m}}^{{q-1}}\mathop{\Gamma\/}\nolimits\!% \left(1-b_{{\ell+1}}+s\right)\prod\limits_{{\ell=n}}^{{p-1}}\mathop{\Gamma\/}% \nolimits\!\left(a_{{\ell+1}}-s\right)\right)}}\right)z^{s}ds,$

Defines:: $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1}% ,\dots,b_{q}}\right)$ : Meijer -function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: Figure 16.17.1, Figure 16.17.1, §16.19
Permalink:: http://dlmf.nist.gov/16.17.E1
Encodings:: TeX, pMML, png

where the integration path separates the poles of the factors $\mathop{\Gamma\/}\nolimits\!\left(b_{{\ell}}-s\right)$ from those of the factors $\mathop{\Gamma\/}\nolimits\!\left(1-a_{{\ell}}+s\right)$ . There are three possible choices for , illustrated in Figure 16.17.1 in the case , :

(i)

goes from $-i\infty$ to $i\infty$ . The integral converges if and $|\mathop{\mathrm{ph}\/}\nolimits z|<(m+n-\frac{1}{2}(p+q))\pi$ .
(ii)

is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the $\mathop{\Gamma\/}\nolimits\!\left(b_{\ell}-s\right)$ once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all ( $\neq 0$ ) if , and for if $p=q\geq 1$ .
(iii)

is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\mathop{\Gamma\/}\nolimits\!\left(1-a_{\ell}+s\right)$ once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all if , and for if $p=q\geq 1$ .


Case (i)	Case (ii)	Case (iii)

Figure 16.17.1: s-plane. Path

for the integral representation (16.17.1) of the Meijer

-function.

Referenced by:: §16.17
Permalink:: http://dlmf.nist.gov/16.17.F1
Encodings:: pdf, pdf, pdf, png, png, png

When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer -function.

Assume $p\leq q$ , no two of the bottom parameters $b_{j}$ , $j=1,\dots,m$ , differ by an integer, and $a_{j}-b_{k}$ is not a positive integer when $j=1,2,\dots,n$ and $k=1,2,\dots,m$ . Then

16.17.2 $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1}% ,\dots,b_{q}}\right)=\sum_{{k=1}}^{m}A_{{p,q,k}}^{{m,n}}(z)\mathop{{{}_{{p}}F_% {{q-1}}}\/}\nolimits\!\left({1+b_{k}-a_{1},\dots,1+b_{k}-a_{p}\atop 1+b_{k}-b_% {1},\ldots*\dots,1+b_{k}-b_{q}};(-1)^{{p-m-n}}z\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1}% ,\dots,b_{q}}\right)$ : Meijer -function, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $A_{{p,q}}^{{m,n}}(z)$ : coefficient
Permalink:: http://dlmf.nist.gov/16.17.E2
Encodings:: TeX, pMML, png

where indicates that the entry $1+b_{k}-b_{k}$ is omitted. Also,

16.17.3 $A_{{p,q,k}}^{{m,n}}(z)=\ifrac{\prod\limits_{{\substack{\ell=1\\ \ell\neq k}}}^{m}\mathop{\Gamma\/}\nolimits\!\left(b_{\ell}-b_{k}\right)\prod% \limits_{{\ell=1}}^{n}\mathop{\Gamma\/}\nolimits\!\left(1+b_{k}-a_{{\ell}}% \right)z^{{b_{k}}}}{\left(\prod\limits_{{\ell=m}}^{{q-1}}\mathop{\Gamma\/}% \nolimits\!\left(1+b_{k}-b_{{\ell+1}}\right)\prod\limits_{{\ell=n}}^{{p-1}}% \mathop{\Gamma\/}\nolimits\!\left(a_{{\ell+1}}-b_{k}\right)\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $A_{{p,q}}^{{m,n}}(z)$ : coefficient
Permalink:: http://dlmf.nist.gov/16.17.E3
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 16.16 Transformations of Variables 16.18 Special Cases