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  
$$\left\{ \begin{array}{ll}
+
$$(a)_n = \dfrac{\Gamma(a+n)}{\Gamma(a)},$$
(a)_0 &= 1 \\
+
where $\Gamma$ denotes [[gamma]].
(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=
 
=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 />
  
 
=Notes=  
 
=Notes=  
Line 12: Line 11:
  
 
=References=
 
=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|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]]

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