Meijer G-function

Definition

Let $m, n, p, q$ be integers with $0 \le m \le q$ and $0 \le n \le p$. Let $a_1, \dots, a_p$ and $b_1, \dots, b_q$ be complex numbers such that $a_k - b_j$ is not a positive integer for $1 \le k \le n$ and $1 \le j \le m$. Let $\Gamma$ be the gamma function, and let $z \ne 0$ be complex with a fixed branch of $\log z$. The Meijer G-function is the Barnes integral $$G^{m, n}_{p, q}\left( \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \,\middle|\, z \right) = \frac{1}{2\pi i} \int_L \frac{\prod_{j = 1}^{m} \Gamma(b_j - s) \prod_{k = 1}^{n} \Gamma(1 - a_k + s)}{\prod_{j = m + 1}^{q} \Gamma(1 - b_j + s) \prod_{k = n + 1}^{p} \Gamma(a_k - s)}\, z^{s}\, ds.$$ The path $L$ separates the poles $s = b_j + \ell$ of the factors $\Gamma(b_j - s)$ from the poles $s = a_k - 1 - \ell$ of the factors $\Gamma(1 - a_k + s)$, where $\ell \in \N$. Call these the $b$-poles and the $a$-poles. The path $L$ is one of the following:

1. a path from $-i\infty$ to $+i\infty$, where the integral converges;
2. a loop from $+\infty$ to $+\infty$ around all $b$-poles once in the negative direction, when $q \ge 1$ and either $p < q$, or $p = q$ and $\abs{z} < 1$;
3. a loop from $-\infty$ to $-\infty$ around all $a$-poles once in the positive direction, when $p \ge 1$ and either $p > q$, or $p = q$ and $\abs{z} > 1$.