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  "body": "\n## Definition\n\nLet $p$ be a prime number and let $\\Z_p$ be the ring of [[p-adic integers|V6TJAFZ4MD]].\nFor a positive integer $n$, put\n$$\\Gamma_p(n) = (-1)^n \\prod_{\\substack{0 \u003c i \u003c n \\\\ p \\nmid i}} i.$$\nThe *$p$-adic gamma function* is the unique continuous function $\\Gamma_p \\colon \\Z_p \\to \\Z_p$ that takes these values at the positive integers.\n\n## Remarks\n\nThe positive integers are dense in $\\Z_p$, so such a continuous extension is unique.\nIt exists because $\\Gamma_p(m)$ and $\\Gamma_p(n)$ are $p$-adically close when $m$ and $n$ are; the proof uses a generalisation of Wilson's theorem.\nThe values of $\\Gamma_p$ are units of $\\Z_p$.\n\nExample: for $p = 3$, $\\Gamma_3(1) = -1$, $\\Gamma_3(2) = 1$, $\\Gamma_3(3) = -2$, $\\Gamma_3(4) = 2$ and $\\Gamma_3(5) = -8$.\nEach value follows from the previous one by the rule\n$$\\Gamma_p(x + 1) = \\begin{cases} -x\\, \\Gamma_p(x) \u0026 \\text{if } x \\in \\Z_p^\\times, \\\\ -\\Gamma_p(x) \u0026 \\text{if } x \\in p\\Z_p, \\end{cases}$$\nwhich holds for all $x \\in \\Z_p$; here $\\Gamma_p(0) = 1$.\n\nNon-example: the [[factorial|0XWG0JGDFW]] $n \\mapsto n!$ has no continuous extension to $\\Z_p$.\nIndeed $1 + p^k \\to 1$ in $\\Z_p$ as $k \\to \\infty$, but $(1 + p^k)! = (1 + p^k) \\cdot (p^k)!$ is divisible by $p^k$ and so tends to $0$, while $1! = 1$.\n",
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