Difference between revisions of "Pochhammer"
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The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by | The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by | ||
| − | $$(a)_n = \ | + | $$(a)_n = \dfrac{\Gamma(a+n)}{\Gamma(a)},$$ |
| − | + | where $\Gamma$ denotes [[gamma]]. | |
| − | \ | ||
| − | \ | ||
=Properties= | =Properties= | ||
[[Sum of reciprocal Pochhammer symbols of a fixed exponent]]<br /> | [[Sum of reciprocal Pochhammer symbols of a fixed exponent]]<br /> | ||
| + | [[Pochhammer symbol with non-negative integer subscript]]<br /> | ||
[[Relationship between Pochhammer and gamma]]<br /> | [[Relationship between Pochhammer and gamma]]<br /> | ||
Revision as of 19:09, 17 June 2017
The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by $$(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
Relationship between Pochhammer and gamma
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 $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$)
- 1964: W.N. Bailey: Generalized Hypergeometric Series ... (next): Section $1.1$
