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Pochhammer




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Pochhammer[a,n] > Series representations > Generalized power series > Expansions of (-m+Epsilon)n at Epsilon==0/;m-n ∉ N && m ∈ N





http://functions.wolfram.com/06.10.06.0022.01









  


  








Input Form





Pochhammer[-m + \[Epsilon], n] \[Proportional] (-1)^m m! Gamma[-m + n] (\[Epsilon] + \[Epsilon]^2 (-PolyGamma[1 + m] + PolyGamma[-m + n]) + (\[Epsilon]^3/6) (-Pi^2 + 3 PolyGamma[1 + m]^2 - 6 PolyGamma[1 + m] PolyGamma[-m + n] + 3 PolyGamma[-m + n]^2 + 3 PolyGamma[1, 1 + m] + 3 PolyGamma[1, -m + n]) + (1/6) \[Epsilon]^4 (-PolyGamma[1 + m]^3 + 3 PolyGamma[1 + m]^2 PolyGamma[-m + n] + PolyGamma[-m + n]^3 + PolyGamma[1 + m] (Pi^2 - 3 PolyGamma[-m + n]^2 - 3 PolyGamma[1, 1 + m] - 3 PolyGamma[1, -m + n]) + PolyGamma[-m + n] (-Pi^2 + 3 PolyGamma[1, 1 + m] + 3 PolyGamma[1, -m + n]) - PolyGamma[2, 1 + m] + PolyGamma[2, -m + n]) + O[\[Epsilon]^5]) /; !(Element[m - n, Integers] && m - n >= 0) && Element[m, Integers] && m >= 0








Standard Form





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MathML Form







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</mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#1013; </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> &#1013; </mi> <mn> 5 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> &#1013; 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</ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -3 </cn> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <integers /> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02