Difference between revisions of "Pochhammer"
From specialfunctionswiki
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=References= | =References= | ||
| + | * {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=findme}}: $18. (1)$ (note: Rainville calls this the "factorial function" and expresses it slightly differently by defining it by the equivalent formula $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$) | ||
[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]] | [[Category:SpecialFunction]] | ||
Revision as of 14:03, 25 January 2017
The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by $$\left\{ \begin{array}{ll} (a)_0 &= 1 \\ (a)_n \equiv a^{\overline{n}} &= \displaystyle\prod_{k=0}^{n-1} a+k=a(a+1)(a+2)\ldots(a+n-1). \end{array} \right.$$
Properties
Sum of reciprocal Pochhammer symbols of a fixed exponent
Notes
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).
References
- 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $18. (1)$ (note: Rainville calls this the "factorial function" and expresses it slightly differently by defining it by the equivalent formula $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$)
