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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/;n ∈ Z && n<=m && m ∈ N





http://functions.wolfram.com/06.10.06.0025.01









  


  








Input Form





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








Standard Form





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MathML Form







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</mo> <msup> <mi> &#1013; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> &#1013; </mi> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8804; </mo> <mi> m </mi> </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 /> <ci> &#1013; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <ci> n </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <ci> &#1013; 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</ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <leq /> <ci> n </ci> <ci> m </ci> </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