Difference between revisions of "Pochhammer"
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$$(a)_0 = 1;$$ | $$(a)_0 = 1;$$ | ||
$$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ | $$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ | ||
| − | where $\Gamma$ denotes the [[gamma function]]. | + | where $\Gamma$ denotes the [[gamma function]]. 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= | ||
Revision as of 22:40, 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. 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: $$\displaystyle\sum_{k=1}^n \dfrac{1}{(k)_p} = \dfrac{1}{(p-1)\Gamma(p)} - \dfrac{n\Gamma(n)}{(p-1)\Gamma(n+p)},$$ where $s(n,k)$ denotes a Stirling number of the first kind.
Proof: █
