Difference between revisions of "Pochhammer"
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where $s(n,k)$ denotes a [[Stirling number of the first kind]]. | where $s(n,k)$ denotes a [[Stirling number of the first kind]]. | ||
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| − | <strong>Proof:</strong> | + | <strong>Proof:</strong> █ |
| + | </div> | ||
| + | </div> | ||
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| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\displaystyle\sum_{k=1}^n \dfrac{1}{(k)_p} = \dfrac{1}{(p-1)\Gamma(p)} - \dfrac{n\Gamma(n)}{(p-1)\Gamma(n+p)},$$ | ||
| + | where $s(n,k)$ denotes a [[Stirling number of the first kind]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 07:47, 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: █
Theorem: The following formula holds: $$\displaystyle\sum_{k=1}^n \dfrac{1}{(k)_p} = \dfrac{1}{(p-1)\Gamma(p)} - \dfrac{n\Gamma(n)}{(p-1)\Gamma(n+p)},$$ where $s(n,k)$ denotes a Stirling number of the first kind.
Proof: █
