§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:

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

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