Difference between revisions of "Pochhammer"
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(Created page with "$$(a)_0 = 1$$ $$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ where $\Gamma$ denotes the [gamma function].") |
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| − | + | 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)},$$ |
| − | where $\Gamma$ denotes the [ | + | where $\Gamma$ denotes [[gamma]]. |
| + | |||
| + | =Properties= | ||
| + | [[Sum of reciprocal Pochhammer symbols of a fixed exponent]]<br /> | ||
| + | [[Pochhammer symbol with non-negative integer subscript]]<br /> | ||
| + | |||
| + | =Notes= | ||
| + | We are using this symbol to denote the [[rising factorial]] (following the notation used by Abramowitz&Stegun and Mathematica) as opposed to denoting the [[falling factorial]] (as Wikipedia does). | ||
| + | |||
| + | =References= | ||
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Hypergeometric pFq|next=Pochhammer symbol with non-negative integer subscript}}: $4.1 (2)$ | ||
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Hypergeometric pFq|next=Pochhammer symbol with non-negative integer subscript}}: $5.1 (3)$ | ||
| + | * {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=findme}}: $18. (1)$ (note: Rainville calls this the "factorial function" and expresses it slightly differently by defining it by the equivalent formula $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$) | ||
| + | * {{BookReference|Generalized Hypergeometric Series|1964|W.N. Bailey|next=Hypergeometric 2F1}}: Section $1.1$ | ||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 23:25, 3 March 2018
The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by $$(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: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $4.1 (2)$
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: 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 $1.1$
