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  "body": "\n## Definition\n\nLet $p, q \\ge 0$ be integers.\nLet $a_1, \\ldots, a_p, b_1, \\ldots, b_q \\in \\C$ and let $A_1, \\ldots, A_p, B_1, \\ldots, B_q \u003e 0$ be real.\nAssume that $a_i + A_i n$ is never a pole of the [[gamma function|0H5SYBSED9]] $\\Gamma$, for $n = 0, 1, 2, \\ldots$, and use the convention $1/\\Gamma(-m) = 0$ for $m = 0, 1, 2, \\ldots$.\nThe *Fox–Wright function* is the [[power series|0MZB06ST2B]]\n$${}_p\\Psi_q\\left[\\begin{matrix} (a_1, A_1), \\ldots, (a_p, A_p) \\\\ (b_1, B_1), \\ldots, (b_q, B_q) \\end{matrix}; z\\right] = \\sum_{n=0}^{\\infty} \\frac{\\Gamma(a_1 + A_1 n) \\cdots \\Gamma(a_p + A_p n)}{\\Gamma(b_1 + B_1 n) \\cdots \\Gamma(b_q + B_q n)} \\, \\frac{z^n}{n!}.$$\n\n## Remarks\n\nLet $\\Delta = 1 + \\sum_j B_j - \\sum_i A_i$.\nIf $\\Delta \u003e 0$ the series converges for every $z \\in \\C$.\nIf $\\Delta = 0$ it converges for $\\abs{z} \u003c \\prod_i A_i^{-A_i} \\prod_j B_j^{B_j}$.\n\nExample: with $p = q = 1$, $(a_1, A_1) = (1, 1)$ and $(b_1, B_1) = (\\beta, \\alpha)$, the coefficient is $\\frac{\\Gamma(1 + n)}{\\Gamma(\\beta + \\alpha n)} \\frac{1}{n!} = \\frac{1}{\\Gamma(\\alpha n + \\beta)}$, so the series is the [[Mittag-Leffler function|12G9M694YG]] $E_{\\alpha, \\beta}(z)$; here $\\Delta = \\alpha \u003e 0$.\n\nWhen all $A_i$ and $B_j$ equal $1$,\n$${}_p\\Psi_q = \\frac{\\Gamma(a_1) \\cdots \\Gamma(a_p)}{\\Gamma(b_1) \\cdots \\Gamma(b_q)} \\, {}_pF_q(a_1, \\ldots, a_p; b_1, \\ldots, b_q; z),$$\na multiple of the [[generalized hypergeometric function|VDBAW9PN88]].\n\nNon-example: with $p = 1$, $q = 0$ and $A_1 = 2$, one has $\\Delta = -1 \u003c 0$, and the series $\\sum \\Gamma(1 + 2n) z^n / n!$ diverges for every $z \\neq 0$.\n",
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