Fox H-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 and let $A_1, \dots, A_p$ and $B_1, \dots, B_q$ be positive real numbers. Let $\Gamma$ be the gamma function. Fix $z \ne 0$ and a branch of $\log z$, and put $z^{-s} = e^{-s \log z}$. Assume that no pole $s = -(b_j + \ell)/B_j$ of a factor $\Gamma(b_j + B_j s)$ with $j \le m$ coincides with a pole $s = (1 - a_k + \ell)/A_k$ of a factor $\Gamma(1 - a_k - A_k s)$ with $k \le n$, where $\ell \in \N$. Let $L$ be a path that separates the poles of the factors $\Gamma(b_j + B_j s)$ from the poles of the factors $\Gamma(1 - a_k - A_k s)$. Choose $L$ so that the integral below converges. The Fox H-function is the Barnes integral $$H^{m, n}_{p, q}\left[ z \,\middle|\, \begin{matrix} (a_1, A_1), \dots, (a_p, A_p) \\ (b_1, B_1), \dots, (b_q, B_q) \end{matrix} \right] = \frac{1}{2\pi i} \int_L \frac{\prod_{j = 1}^{m} \Gamma(b_j + B_j s) \prod_{k = 1}^{n} \Gamma(1 - a_k - A_k s)}{\prod_{j = m + 1}^{q} \Gamma(1 - b_j - B_j s) \prod_{k = n + 1}^{p} \Gamma(a_k + A_k s)}\, z^{-s}\, ds.$$