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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 such that $a_k - b_j$ is not a positive integer for $1 \\le k \\le n$ and $1 \\le j \\le m$.\nLet $\\Gamma$ be the [[gamma function|0H5SYBSED9]], and let $z \\ne 0$ be complex with a fixed branch of $\\log z$.\nThe *Meijer G-function* is the [[Barnes integral|1PDGBGEHM2]]\n$$G^{m, n}_{p, q}\\left( \\begin{matrix} a_1, \\dots, a_p \\\\ b_1, \\dots, b_q \\end{matrix} \\,\\middle|\\, z \\right) = \\frac{1}{2\\pi i} \\int_L \\frac{\\prod_{j = 1}^{m} \\Gamma(b_j - s) \\prod_{k = 1}^{n} \\Gamma(1 - a_k + s)}{\\prod_{j = m + 1}^{q} \\Gamma(1 - b_j + s) \\prod_{k = n + 1}^{p} \\Gamma(a_k - s)}\\, z^{s}\\, ds.$$\nThe 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 $\\ell \\in \\N$.\nCall these the $b$-poles and the $a$-poles.\nThe path $L$ is one of the following:\n\n1. a path from $-i\\infty$ to $+i\\infty$, where the integral converges;\n2. a loop from $+\\infty$ to $+\\infty$ around all $b$-poles once in the negative direction, when $q \\ge 1$ and either $p \u003c q$, or $p = q$ and $\\abs{z} \u003c 1$;\n3. a loop from $-\\infty$ to $-\\infty$ around all $a$-poles once in the positive direction, when $p \\ge 1$ and either $p \u003e q$, or $p = q$ and $\\abs{z} \u003e 1$.\n\n## Remarks\n\nThe condition on $a_k - b_j$ makes the two families of poles disjoint, so a separating path exists.\nWhen more than one choice of $L$ is possible, they give the same value.\nExample: $G^{1, 0}_{0, 1}(- ; 0 \\mid z) = e^{-z}$, where $-$ marks the empty list of parameters $a_k$.\nCheck: the integrand is $\\Gamma(-s) z^s$, and with the loop of type 2 the residues at $s = \\ell$ give $\\sum_{\\ell \\ge 0} \\frac{(-1)^\\ell}{\\ell!} z^\\ell$.\nNon-example: $m = 2$ with $q = 1$ is not allowed, since $m \\le q$ fails; there are not enough parameters $b_j$ for two factors $\\Gamma(b_j - s)$.\nEvery [[generalized hypergeometric function|VDBAW9PN88]] is a G-function: if no $a_k$ is a nonpositive integer, no $b_j$ is a nonpositive integer, $p \\le q + 1$, and $\\abs{z} \u003c 1$ when $p = q + 1$, then\n$${}_pF_q(a_1, \\dots, a_p; b_1, \\dots, b_q; z) = \\frac{\\prod_{j} \\Gamma(b_j)}{\\prod_{k} \\Gamma(a_k)}\\, G^{1, p}_{p, q + 1}\\left( \\begin{matrix} 1 - a_1, \\dots, 1 - a_p \\\\ 0, 1 - b_1, \\dots, 1 - b_q \\end{matrix} \\,\\middle|\\, -z \\right).$$\n",
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