Difference between revisions of "Pochhammer"
(→Properties) |
|||
| Line 24: | Line 24: | ||
=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] | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Revision as of 18:29, 24 May 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
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: █
