Difference between revisions of "Falling factorial"
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(Created page with "The falling factorial $x^{\underline{k}}$ for nonnegative integer $k$ is given by $$x^{\underline{k}}=x(x-1)\ldots (x-k+1).$$") |
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| − | The falling factorial $x^{\underline{k}}$ for | + | The falling factorial $x^{\underline{k}}$ for positive [[integer]] $k$ is given by |
$$x^{\underline{k}}=x(x-1)\ldots (x-k+1).$$ | $$x^{\underline{k}}=x(x-1)\ldots (x-k+1).$$ | ||
| + | If $k$ is not an integer, we use the following formula to interpret $x^{\underline{k}}$: | ||
| + | $$x^{\underline{k}} = \dfrac{\Gamma(x+1)}{\Gamma(x-k+1)},$$ | ||
| + | where $\Gamma$ denotes the [[gamma function]]. | ||
| + | |||
| + | =Properties= | ||
| + | |||
| + | =References= | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 12:27, 11 August 2016
The falling factorial $x^{\underline{k}}$ for positive integer $k$ is given by $$x^{\underline{k}}=x(x-1)\ldots (x-k+1).$$ If $k$ is not an integer, we use the following formula to interpret $x^{\underline{k}}$: $$x^{\underline{k}} = \dfrac{\Gamma(x+1)}{\Gamma(x-k+1)},$$ where $\Gamma$ denotes the gamma function.
