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  "body": "\n## Definition\n\nLet $a$ be an element of a commutative ring, for example a complex number, and let $n$ be a natural number.\nThe *Pochhammer symbol* (or *rising factorial*) of $a$ of order $n$ is\n$$(a)_n = a (a + 1) (a + 2) \\cdots (a + n - 1) = \\prod_{i=0}^{n-1} (a + i).$$\nFor $n = 0$ the product is empty, and $(a)_0 = 1$.\n\n## Remarks\n\nExample: $(1)_n = 1 \\cdot 2 \\cdots n = n!$, the [[factorial|0XWG0JGDFW]]; and $(2)_3 = 2 \\cdot 3 \\cdot 4 = 24$.\nFor a negative integer $a = -m$ and $n \u003e m$, one factor is $0$, so $(-m)_n = 0$; this is why hypergeometric series with such a parameter terminate.\nOther notations are $a^{\\overline{n}}$ and $a^{(n)}$.\nThe rising factorial of $a$ is $(-1)^n$ times the [[falling factorial|0SJNCJCCQG]] of $-a$: $(a)_n = (-1)^n (-a)(-a-1)\\cdots(-a-n+1)$.\nIf $a \\in \\C$ is not in $\\set{0, -1, -2, \\ldots}$, then $(a)_n = \\Gamma(a + n)/\\Gamma(a)$, where $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nIn Mathlib the rising factorial as a polynomial in $a$ is `ascPochhammer`.\n\n## Explanations\n\n### L1 human\n\nThe Pochhammer symbol is a factorial that may start at any number and goes up in steps of $1$.\nFor example $(5)_3 = 5 \\cdot 6 \\cdot 7 = 210$.\n",
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