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  "body": "\n## Definition\n\nLet $x$ be an element of a commutative ring with identity, for example a real number, and let $n$ be a natural number.\nThe *falling factorial* of $x$ of order $n$ is\n$$x^{\\underline{n}} = x (x - 1) (x - 2) \\cdots (x - n + 1) = \\prod_{i=0}^{n-1} (x - i).$$\nThe *rising factorial* of $x$ of order $n$ is\n$$x^{\\overline{n}} = x (x + 1) (x + 2) \\cdots (x + n - 1) = \\prod_{i=0}^{n-1} (x + i).$$\nFor $n = 0$ both products are empty, and $x^{\\underline{0}} = x^{\\overline{0}} = 1$.\n\n## Remarks\n\nExample: for $x = 4$ and $n = 2$, $4^{\\underline{2}} = 4 \\cdot 3 = 12$ and $4^{\\overline{2}} = 4 \\cdot 5 = 20$.\nNon-example: the product $4 \\cdot 2 = 8$ is not $4^{\\underline{2}}$, because its factors do not decrease by steps of $1$.\nFor a natural number $m \\ge n$, $m^{\\underline{n}} = m!/(m-n)!$, and $m^{\\underline{n}} = 0$ when $n \u003e m$; see [[falling factorial|0N5M924N3P]] and [[factorial|0WFCGVMRRC]].\nFor a natural number $m \\ge 1$, $m^{\\overline{n}} = (m + n - 1)!/(m - 1)!$, and $1^{\\overline{n}} = n^{\\underline{n}} = n!$.\nThe two are linked by $x^{\\overline{n}} = (x + n - 1)^{\\underline{n}} = (-1)^n (-x)^{\\underline{n}}$.\nFor real $x \u003e 0$, $x^{\\overline{n}} = \\Gamma(x + n)/\\Gamma(x)$, where $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nOther notations: $(x)_n$ is used for the falling factorial in combinatorics and for the rising factorial (the Pochhammer symbol) in special functions; the rising factorial is also written $x^{(n)}$.\nThe [[q-Pochhammer symbol|Y1X8ZZQ7VM]] is a $q$-analogue of the rising factorial.\n",
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