§16.17 Definition

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 $m$ and $n$ 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 $G$-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: $\mathop{{G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\/}\nolimits\!\left(\NVar{z% };{\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or $\mathop{{G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\/}\nolimits\!\left(\NVar{z% };\NVar{\mathbf{a}};\NVar{\mathbf{b}}\right)$: Meijer $G$-function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $d\NVar{x}$: differential of $x$, $\int$: integral, $p$: nonnegative integer, $q$: nonnegative integer, $z$: 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 See also: info for 16.17

where the integration path $L$ 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 $L$, illustrated in Figure 16.17.1 in the case $m=1$, $n=2$:

1. (i)

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

2. (ii)

$L$ 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 $z$ ($\neq 0$) if $p, and for $0<|z|<1$ if $p=q\geq 1$.

3. (iii)

$L$ 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 $z$ if $p>q$, and for $|z|>1$ if $p=q\geq 1$.

When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer $G$-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 $\mathop{{G^{m,n}_{p,q}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}}\right)=\sum_{k=1}^{m}A_{p,q,k}^{m,n}(z)\mathop{{{}_{p}F_{q-1}}\/}% \nolimits\!\left({1+b_{k}-a_{1},\dots,1+b_{k}-a_{p}\atop 1+b_{k}-b_{1},\ldots*% \dots,1+b_{k}-b_{q}};(-1)^{p-m-n}z\right),$

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

 16.17.3 $A_{p,q,k}^{m,n}(z)=\ifrac{\prod\limits_{\begin{subarray}{c}\ell=1\\ \ell\neq k\end{subarray}}^{m}\mathop{\Gamma\/}\nolimits\!\left(b_{\ell}-b_{k}% \right)\prod\limits_{\ell=1}^{n}\mathop{\Gamma\/}\nolimits\!\left(1+b_{k}-a_{% \ell}\right)z^{b_{k}}}{\left(\prod\limits_{\ell=m}^{q-1}\mathop{\Gamma\/}% \nolimits\!\left(1+b_{k}-b_{\ell+1}\right)\prod\limits_{\ell=n}^{p-1}\mathop{% \Gamma\/}\nolimits\!\left(a_{\ell+1}-b_{k}\right)\right)}.$ Defines: $A_{p,q}^{m,n}(z)$: coefficient (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_{1},\ldots,a_{p}$: real or complex parameters and $b,b_{1},\ldots,b_{q}$: real or complex parameters Permalink: http://dlmf.nist.gov/16.17.E3 Encodings: TeX, pMML, png See also: info for 16.17