Difference between revisions of "Pochhammer"
From specialfunctionswiki
| Line 12: | Line 12: | ||
=References= | =References= | ||
* {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=Hypergeometric pFq|next=Pochhammer symbol with non-negative integer subscript}}: $4.1 (2)$ | * {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=Hypergeometric pFq|next=Pochhammer symbol with non-negative integer subscript}}: $4.1 (2)$ | ||
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=Hypergeometric pFq|next=Pochhammer symbol with non-negative integer subscript}}: $5.1 (3)$ | ||
* {{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 $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$) | * {{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 $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$) | ||
* {{BookReference|Generalized Hypergeometric Series|1964|W.N. Bailey|next=Hypergeometric 2F1}}: Section $1.1$ | * {{BookReference|Generalized Hypergeometric Series|1964|W.N. Bailey|next=Hypergeometric 2F1}}: Section $1.1$ | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Revision as of 19:20, 21 June 2017
The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by $$(a)_n = \dfrac{\Gamma(a+n)}{\Gamma(a)},$$ where $\Gamma$ denotes gamma.
Properties
Sum of reciprocal Pochhammer symbols of a fixed exponent
Pochhammer symbol with non-negative integer subscript
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
- 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $4.1 (2)$
- 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $5.1 (3)$
- 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 $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$)
- 1964: W.N. Bailey: Generalized Hypergeometric Series ... (next): Section $1.1$
