Difference between revisions of "Pochhammer"

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The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by  
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The Pochhammer symbol $(a)_n$ is a notation that denotes the "rising factorial" function. It is defined by  
$$(a)_0 = 1;$$
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$$(a)_n = \dfrac{\Gamma(a+n)}{\Gamma(a)},$$
$$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$
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where $\Gamma$ denotes [[gamma]].
where $\Gamma$ denotes the [[gamma function]]. 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).  
 
  
 
=Properties=
 
=Properties=
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[[Sum of reciprocal Pochhammer symbols of a fixed exponent]]<br />
<strong>Proposition:</strong> The following formula holds:
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[[Pochhammer symbol with non-negative integer subscript]]<br />
$$(a)_n = \displaystyle\sum_{k=0}^n (-1)^{n-k}s(n,k)a^k,$$
 
where $s(n,k)$ denotes a [[Stirling number of the first kind]].
 
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<strong>Proof:</strong>  █
 
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=Notes=
<strong>Theorem:</strong> The following formula holds:
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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).  
$$\displaystyle\sum_{k=1}^n \dfrac{1}{(k)_p} = \dfrac{1}{(p-1)\Gamma(p)} - \dfrac{n\Gamma(n)}{(p-1)\Gamma(n+p)}.$$
 
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<strong>Proof:</strong>  █
 
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=References=
 
=References=
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_256.htm Abramowitz and Stegun]
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* {{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}}: $4.1 (2)$
 
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* {{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}}: $5.1 (3)$
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* {{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 $(\alpha)_n=\displaystyle\prod_{k=1}^n (\alpha+k-1)$)
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* {{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