Difference between revisions of "Pochhammer"

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$$(a)_0 = 1$$
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The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by
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$$(a)_0 = 1;$$
 
$$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$
 
$$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$
 
where $\Gamma$ denotes the [[gamma function]].
 
where $\Gamma$ denotes the [[gamma function]].
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=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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<strong>Proposition:</strong> The following formula holds:
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$$(a)_n = \displaystyle\sum_{k=0}^n (-1)^{n-k}s(n,k)a^k,$$
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where $s(n,k)$ denotes a [[Stirling number of the first kind]].
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<div class="mw-collapsible-content">
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<strong>Proof:</strong> proof goes here █
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</div>
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</div>
  
 
=References=
 
=References=
 
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_256.htm Abramowitz and Stegun]
 
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_256.htm Abramowitz and Stegun]

Revision as of 07:42, 8 February 2015

The Pochhammer symbol is a notation that denotes the "rising factorial" function. It is defined by $$(a)_0 = 1;$$ $$(a)_n=a(a+1)(a+2)\ldots(a+n-1)=\dfrac{\Gamma(a+n)}{\Gamma(a)},$$ where $\Gamma$ denotes the gamma function.

Properties

Proposition: The following formula holds: $$(a)_n = \displaystyle\sum_{k=0}^n (-1)^{n-k}s(n,k)a^k,$$ where $s(n,k)$ denotes a Stirling number of the first kind.

Proof: proof goes here █

References

Abramowitz and Stegun