Pochhammer symbol: Series representations (formula 06.10.06.0007)

Pochhammer

$Mathematica Notation$

Gamma, Beta, Erf

Pochhammer[a,n]

Series representations

Asymptotic series expansions

Expansions at n==infinity

http://functions.wolfram.com/06.10.06.0007.01

Input Form

Pochhammer[a, n] \[Proportional] ((Sqrt[2 Pi] n^(a + n - 1/2))/ (E^n Gamma[a])) (1 + (1 - 6 a + 6 a^2)/(12 n) + (1 - 36 a + 120 a^2 - 120 a^3 + 36 a^4)/(288 n^2) + (1/(51840 n^3)) (-139 - 450 a + 8190 a^2 - 20160 a^3 + 18900 a^4 - 7560 a^5 + 1080 a^6) + (1/(2488320 n^4)) (-571 + 23352 a + 34464 a^2 - 465840 a^3 + 945000 a^4 - 828576 a^5 + 362880 a^6 - 77760 a^7 + 6480 a^8) + O[1/n^5]) /; (Abs[n] -> Infinity) && Abs[Arg[a + n]] < Pi

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "n"], Pochhammer] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mfrac> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> n </mi> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mn> 36 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 120 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 120 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 36 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 288 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mn> 1080 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7560 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 18900 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 20160 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8190 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 450 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 139 </mn> </mrow> <mrow> <mn> 51840 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2488320 </mn> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6480 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 77760 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 362880 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 828576 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 945000 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 465840 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 34464 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 23352 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 571 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mn> 5 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> n </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mi> π </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Pochhammer </ci> <ci> a </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <apply> <power /> <ci> n </ci> <apply> <plus /> <ci> a </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 120 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 120 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 36 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 288 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1080 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7560 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 18900 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20160 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8190 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 450 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -139 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 51840 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2488320 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 6480 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 77760 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 362880 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 828576 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 945000 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 465840 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 34464 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 23352 </cn> <ci> a </ci> </apply> <cn type='integer'> -571 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> n </ci> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> n </ci> </apply> <infinity /> </apply> <apply> <lt /> <apply> <abs /> <apply> <arg /> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> </apply> </apply> <pi /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29