Difference between revisions of "Pochhammer"
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<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: | ||
| − | $$\displaystyle\sum_{k=1}^n \dfrac{1}{(k)_p} = \dfrac{1}{(p-1)\Gamma(p)} - \dfrac{n\Gamma(n)}{(p-1)\Gamma(n+p)} | + | $$\displaystyle\sum_{k=1}^n \dfrac{1}{(k)_p} = \dfrac{1}{(p-1)\Gamma(p)} - \dfrac{n\Gamma(n)}{(p-1)\Gamma(n+p)}.$$ |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 01:20, 10 February 2015
The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by $$(a)_0 = 1;$$ $$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ where $\Gamma$ denotes the gamma function. We are using this symbol to denote the rising factorial (following the notation used by Abramowitz&Stegun and Mathematica) as opposed to denoting the falling factorial (as Wikipedia does).
Properties
Proposition: The following formula holds: $$(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)}.$$
Proof: █
