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  "body": "\n## Definition\n\nLet $a, b, c \\in \\C$ with $c \\notin \\set{0, -1, -2, \\ldots}$.\nFor $x \\in \\C$ and an integer $n \\ge 0$, let $(x)_n = x(x+1)\\cdots(x+n-1)$ be the [[Pochhammer symbol|YZHM3HSM0F]], with $(x)_0 = 1$.\nThe *hypergeometric series* is the [[power series|0MZB06ST2B]]\n$${}_2F_1(a, b; c; z) = \\sum_{n=0}^{\\infty} \\frac{(a)_n (b)_n}{(c)_n} \\, \\frac{z^n}{n!}.$$\n\n## Remarks\n\nIf $a$ or $b$ is a non-positive integer $-m$, then $(a)_n$ or $(b)_n$ vanishes for $n \u003e m$ and the series is a polynomial of degree at most $m$.\nOtherwise the ratio test gives [[radius of convergence|0N288P2TB5]] $1$.\nOn the circle $\\abs{z} = 1$ the series converges absolutely when $\\operatorname{Re}(c - a - b) \u003e 0$.\n\nExample: for $a = b = 1$, $c = 2$, the coefficient is $\\frac{n! \\, n!}{(n+1)! \\, n!} = \\frac{1}{n+1}$, so ${}_2F_1(1, 1; 2; z) = \\sum_n \\frac{z^n}{n+1} = -\\frac{\\log(1 - z)}{z}$ for $0 \u003c \\abs{z} \u003c 1$.\nAlso ${}_2F_1(a, b; b; z) = \\sum_n \\frac{(a)_n}{n!} z^n = (1 - z)^{-a}$ by the binomial series.\n\nNon-example: $c = 0$ is excluded, because $(0)_n = 0$ for every $n \\ge 1$ and the coefficients are undefined.\nFor $c = -m$ a negative integer the series is still meaningful when $a = -k$ with $0 \\le k \\le m$ and the sum is cut at $n = k$, but this needs a separate convention.\n\nThe notation $F(a, b; c; z)$ is also common.\n",
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    "name": "Hypergeometric series",
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  "msc": [
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