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  "body": "\n## Definition\n\nLet $p, q \\ge 0$ be integers, let $a_1, \\ldots, a_p \\in \\C$, and let $b_1, \\ldots, b_q \\in \\C \\setminus \\set{0, -1, -2, \\ldots}$.\nLet $(x)_n$ be the [[Pochhammer symbol|YZHM3HSM0F]].\nThe *generalized hypergeometric series* is the [[power series|0MZB06ST2B]]\n$${}_pF_q(a_1, \\ldots, a_p; b_1, \\ldots, b_q; z) = \\sum_{n=0}^{\\infty} \\frac{(a_1)_n \\cdots (a_p)_n}{(b_1)_n \\cdots (b_q)_n} \\, \\frac{z^n}{n!}.$$\nThe *generalized hypergeometric function* is its sum where it converges, together with its analytic continuation.\n\n## Remarks\n\nAssume that no $a_i$ is a non-positive integer, so that the series does not terminate.\nBy the ratio test, its [[radius of convergence|0N288P2TB5]] is:\n\n1. infinite when $p \\le q$, so the function is entire;\n2. equal to $1$ when $p = q + 1$, and then the function continues analytically to $\\C \\setminus [1, \\infty)$;\n3. equal to $0$ when $p \u003e q + 1$.\n\nExamples: ${}_0F_0(;;z) = e^z$ and ${}_1F_0(a;;z) = (1 - z)^{-a}$ for $\\abs{z} \u003c 1$.\nAlso $J_\\nu(x) = \\frac{(x/2)^\\nu}{\\Gamma(\\nu + 1)} \\, {}_0F_1(;\\nu + 1; -x^2/4)$ for the Bessel function of the first kind.\n\nNon-example: ${}_2F_0(1, 1;; z) = \\sum_n n! \\, z^n$ is a formal series with radius of convergence $0$; it is only an asymptotic series and defines no function near $0$.\n\nA lower parameter equal to an upper one cancels: ${}_{p+1}F_{q+1}(a, c; b, c; z) = {}_pF_q(a; b; z)$, with $a$ and $b$ standing for the lists of parameters.\n",
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