Difference between revisions of "Pochhammer"
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The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by | The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by | ||
$$(a)_0 = 1;$$ | $$(a)_0 = 1;$$ | ||
| − | $$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ | + | $$(a)_n \equiv a^{\overline{n}}=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ |
where $\Gamma$ denotes the [[gamma function]]. We are using this symbol to denote the [[rising factorial]] (following the notation used by Abramowitz&Stegun and Mathematica) as opposed to denoting the [[falling factorial]] (as Wikipedia does). | where $\Gamma$ denotes the [[gamma function]]. We are using this symbol to denote the [[rising factorial]] (following the notation used by Abramowitz&Stegun and Mathematica) as opposed to denoting the [[falling factorial]] (as Wikipedia does). | ||
=Properties= | =Properties= | ||
| − | < | + | [[Sum of reciprocal Pochhammer symbols of a fixed exponent]]<br /> |
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=References= | =References= | ||
Revision as of 12:36, 17 September 2016
The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by $$(a)_0 = 1;$$ $$(a)_n \equiv a^{\overline{n}}=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ where $\Gamma$ denotes the gamma function. We are using this symbol to denote the rising factorial (following the notation used by Abramowitz&Stegun and Mathematica) as opposed to denoting the falling factorial (as Wikipedia does).
Properties
Sum of reciprocal Pochhammer symbols of a fixed exponent
