Difference between revisions of "Pochhammer"
From specialfunctionswiki
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| − | $$(a)_0 = 1$$ | + | The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by |
| + | $$(a)_0 = 1;$$ | ||
$$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ | $$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ | ||
where $\Gamma$ denotes the [[gamma function]]. | where $\Gamma$ denotes the [[gamma function]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> The following formula holds: | ||
| + | $$(a)_n = \displaystyle\sum_{k=0}^n (-1)^{n-k}s(n,k)a^k,$$ | ||
| + | where $s(n,k)$ denotes a [[Stirling number of the first kind]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> proof goes here █ | ||
| + | </div> | ||
| + | </div> | ||
=References= | =References= | ||
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_256.htm Abramowitz and Stegun] | [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_256.htm Abramowitz and Stegun] | ||
Revision as of 07:42, 8 February 2015
The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by $$(a)_0 = 1;$$ $$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ where $\Gamma$ denotes the gamma function.
Properties
Proposition: The following formula holds: $$(a)_n = \displaystyle\sum_{k=0}^n (-1)^{n-k}s(n,k)a^k,$$ where $s(n,k)$ denotes a Stirling number of the first kind.
Proof: proof goes here █
