§16.17 Definition

Again assume $a_1,a_2,\mathrm{\dots },a_p$ and $b_1,b_2,\mathrm{\dots },b_q$ are real or complex parameters. Assume also that $m$ and $n$ are integers such that $0\le m\le q$ and $0\le n\le p$, and none of $a_k-b_j$ is a positive integer when $1\le k\le n$ and $1\le j\le m$. Then the Meijer $G$-function is defined via the Mellin--Barnes integral representation:

16.17.1		$$\begin{array}{ll}{G}_{p,q}^{m,n}(z;\mathbf{a};\mathbf{b})& =G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})\\ & =\frac{1}{2\pi \mathrm{i}}{\int }_L\left(\prod _{\mathrm{\ell }=1}^m\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)\prod _{\mathrm{\ell }=1}^n\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)/\left(\prod _{\mathrm{\ell }=m}^{q-1}\mathrm{\Gamma }\left(1-b_{\mathrm{\ell }+1}+s\right)\prod _{\mathrm{\ell }=n}^{p-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }+1}-s\right)\right)\right)z^{s}ds,\end{array}$$
ⓘ Defines: $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;\mathbf{a};\mathbf{b})$: Meijer $G$-function Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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: Annotations for §16.17 and Ch.16

where the integration path $L$ separates the poles of the factors $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ from those of the factors $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$. 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 $-\mathrm{i}\mathrm{\infty }$ to $\mathrm{i}\mathrm{\infty }$. The integral converges if and .

2. (ii)

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

3. (iii)

   $L$ is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$ 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 $p=q\ge 1$.

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

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

16.17.2		$$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})=\sum _{k=1}^m{A}_{p,q,k}^{m,n}(z){}_{p}F_{q-1}(\genfrac{}{}{0pt}{}{1+b_k-a_1,\mathrm{\dots },1+b_k-a_p}{1+b_k-b_1,\mathrm{\dots }*\mathrm{\dots },1+b_k-b_q};{(-1)}^{p-m-n}z),$$
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;\mathbf{a};\mathbf{b})$: Meijer $G$-function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $A_{p,q}^{m,n}(z)$: coefficient Permalink: http://dlmf.nist.gov/16.17.E2 Encodings: TeX, pMML, png See also: Annotations for §16.17 and Ch.16

where $*$ indicates that the entry $1+b_k-b_k$ is omitted. Also,

16.17.3		$$A_{p,q,k}^{m,n}(z)=\prod _{\begin{array}{c}\hfill \mathrm{\ell }=1\hfill \\ \hfill \mathrm{\ell }\ne k\hfill \end{array}}^m\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-b_k\right)\prod _{\mathrm{\ell }=1}^n\mathrm{\Gamma }\left(1+b_k-a_{\mathrm{\ell }}\right)z^{b_k}/\left(\prod _{\mathrm{\ell }=m}^{q-1}\mathrm{\Gamma }\left(1+b_k-b_{\mathrm{\ell }+1}\right)\prod _{\mathrm{\ell }=n}^{p-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }+1}-b_k\right)\right).$$
ⓘ Defines: $A_{p,q}^{m,n}(z)$: coefficient (locally) Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters Permalink: http://dlmf.nist.gov/16.17.E3 Encodings: TeX, pMML, png See also: Annotations for §16.17 and Ch.16