Difference between revisions of "Pochhammer"
From specialfunctionswiki
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<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: |
Revision as of 12:32, 17 September 2016
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. 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
Theorem: The following formula holds: $$\displaystyle\sum_{k=1}^n \dfrac{1}{(k)_p} = \dfrac{1}{(p-1)\Gamma(p)} - \dfrac{n\Gamma(n)}{(p-1)\Gamma(n+p)}.$$
Proof: █