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  "body": "\n## Definition\n\nLet $m, n, p, q$ be integers with $0 \\le m \\le q$ and $0 \\le n \\le p$.\nLet $a_1, \\dots, a_p$ 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.\nLet $\\Gamma$ be the [[gamma function|0H5SYBSED9]].\nFix $z \\ne 0$ and a branch of $\\log z$, and put $z^{-s} = e^{-s \\log z}$.\nAssume 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 $\\ell \\in \\N$.\nLet $L$ be a path that separates the poles of the factors $\\Gamma(b_j + B_j s)$ from the poles of the factors $\\Gamma(1 - a_k - A_k s)$.\nChoose $L$ so that the integral below converges.\nThe *Fox H-function* is the [[Barnes integral|1PDGBGEHM2]]\n$$H^{m, n}_{p, q}\\left[ z \\,\\middle|\\, \\begin{matrix} (a_1, A_1), \\dots, (a_p, A_p) \\\\ (b_1, B_1), \\dots, (b_q, B_q) \\end{matrix} \\right] = \\frac{1}{2\\pi i} \\int_L \\frac{\\prod_{j = 1}^{m} \\Gamma(b_j + B_j s) \\prod_{k = 1}^{n} \\Gamma(1 - a_k - A_k s)}{\\prod_{j = m + 1}^{q} \\Gamma(1 - b_j - B_j s) \\prod_{k = n + 1}^{p} \\Gamma(a_k + A_k s)}\\, z^{-s}\\, ds.$$\n\n## Remarks\n\nWhen 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 $a_k, b_j$.\nExample: $H^{1, 0}_{0, 1}\\left[ z \\,\\middle|\\, (0, 1) \\right] = e^{-z}$ for $z \u003e 0$.\nCheck: the integrand is $\\Gamma(s) z^{-s}$ on a line $\\operatorname{Re} s = c \u003e 0$; closing to the left, the residues $\\frac{(-1)^\\ell}{\\ell!} z^\\ell$ at $s = -\\ell$ sum to $e^{-z}$.\nNon-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.\nThe 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.\n",
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