---
tag: Y403TS95KG
kind: definition
name: Fox H-function
msc: [33C60]
defines: EAZMKENA4N
uses: [1PDGBGEHM2, 19JNNN5KV7, 0H5SYBSED9]
sources:
  - url: https://en.wikipedia.org/wiki/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|0H5SYBSED9]].
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|1PDGBGEHM2]]
$$H^{m, n}_{p, q}\left[ z \,\middle|\, \begin{matrix} (a_1, A_1), \dots, (a_p, A_p) \\ (b_1, B_1), \dots, (b_q, B_q) \end{matrix} \right] = \frac{1}{2\pi i} \int_L \frac{\prod_{j = 1}^{m} \Gamma(b_j + B_j s) \prod_{k = 1}^{n} \Gamma(1 - a_k - A_k s)}{\prod_{j = m + 1}^{q} \Gamma(1 - b_j - B_j s) \prod_{k = n + 1}^{p} \Gamma(a_k + A_k s)}\, z^{-s}\, ds.$$

## Remarks

When all $A_k$ and $B_j$ equal $1$, the substitution $s \mapsto -s$ turns the integral into that of the [[Meijer G-function|19JNNN5KV7]], so $H^{m, n}_{p, q}$ equals $G^{m, n}_{p, q}$ with the same parameters $a_k, b_j$.
Example: $H^{1, 0}_{0, 1}\left[ z \,\middle|\, (0, 1) \right] = e^{-z}$ for $z > 0$.
Check: the integrand is $\Gamma(s) z^{-s}$ on a line $\operatorname{Re} s = c > 0$; closing to the left, the residues $\frac{(-1)^\ell}{\ell!} z^\ell$ at $s = -\ell$ sum to $e^{-z}$.
Non-example: a scale $B_1 = 0$ is not allowed; then $\Gamma(b_1 + B_1 s)$ is a constant and the integrand has no poles to collect, so the scales must be positive.
The Fox–Wright function and the Mittag-Leffler function $E_\alpha(z) = \sum_{k \ge 0} \frac{z^k}{\Gamma(\alpha k + 1)}$ are special H-functions.