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variants of this functions
HypergeometricPFQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQ[{a1,a2},{b1,b2},z] > Series representations > Asymptotic series expansions > Case of double poles





http://functions.wolfram.com/07.25.06.0010.01









  


  










Input Form





HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 1] + n}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 1] + n])) z^\[Chi] E^z Sum[Subscript[c, k]/z^k, {k, 0, Infinity}] + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ (Gamma[Subscript[a, 1] + n] Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]])) Sum[((Pochhammer[Subscript[a, 1], n + k]/(k! (k + n)!)) Pochhammer[1 - Subscript[b, 1] + Subscript[a, 1], n + k] Pochhammer[1 - Subscript[b, 2] + Subscript[a, 1], n + k] (PolyGamma[1 + k] + PolyGamma[1 + k + n] - PolyGamma[k + n + Subscript[a, 1]] - PolyGamma[ -k - n - Subscript[a, 1] + Subscript[b, 1]] - PolyGamma[-k - n - Subscript[a, 1] + Subscript[b, 2]]))/(-z)^k, {k, 0, Infinity}])/(z^n (-z)^Subscript[a, 1]) + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ (n! Gamma[Subscript[a, 1]] Gamma[Subscript[b, 1] - Subscript[a, 1] - n] Gamma[Subscript[b, 2] - Subscript[a, 1] - n])) Log[-z] HypergeometricPFQ[{Subscript[a, 1] + n, 1 - Subscript[b, 1] + Subscript[a, 1] + n, 1 - Subscript[b, 2] + Subscript[a, 1] + n}, {n + 1}, -(1/z)])/(z^n (-z)^Subscript[a, 1]) + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ Gamma[Subscript[a, 1] + n]) Sum[(Pochhammer[Subscript[a, 1], k] Gamma[n - k])/ (Gamma[Subscript[b, 1] - Subscript[a, 1] - k] Gamma[Subscript[b, 2] - Subscript[a, 1] - k] k!)/z^k, {k, 0, n - 1}])/(-z)^Subscript[a, 1] /; (Abs[z] -> Infinity) && Element[n, Integers] && n >= 0 && \[Chi] == 2 Subscript[a, 1] + n - Subscript[b, 1] - Subscript[b, 2] && Subscript[c, 0] == 1 && Subscript[c, 1] == (Subscript[A, 2] - 1) \[Chi] + \[GothicCapitalB] - \[GothicCapitalA] && Subscript[c, k] == (1/k) ((1 - Subscript[B, 2] + 4 Subscript[a, 1] + 2 Subscript[a, 1]^2 + 2 n + 2 Subscript[a, 1] n + n^2 - Subscript[A, 2] Subscript[B, 2] + \[GothicCapitalA] + \[GothicCapitalB] + (2 Subscript[B, 2] - 3 (Subscript[A, 2] + 1)) k + 2 k^2) Subscript[c, k - 1] - (k - Subscript[A, 2] + Subscript[b, 1] - 1) (k - Subscript[A, 2] + Subscript[b, 2] - 1) (k - \[Chi] - 2) Subscript[c, k - 2]) && Subscript[A, 2] == 2 Subscript[a, 1] + n && Subscript[B, 2] == Subscript[b, 1] + Subscript[b, 2] && \[GothicCapitalA] == Subscript[a, 1] (Subscript[a, 1] + n) && \[GothicCapitalB] == Subscript[b, 1] Subscript[b, 2]










Standard Form





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MathML Form







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RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;n&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;n&quot;]]], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mi> z </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#967; </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#967; </mi> <mo> &#10869; </mo> <mrow> <mi> n </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> c </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mrow> <mi> &#120069; </mi> <mo> - </mo> <mi> &#120068; </mi> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> A </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#967; </mi> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> k </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> B </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> A </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msubsup> <mi> a </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mi> &#120068; </mi> <mo> + </mo> <mi> &#120069; </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mrow> <msub> <mi> A </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> B </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <msub> <mi> B </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> c </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <msub> <mi> A </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <msub> <mi> A </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#967; </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> c </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> A </mi> <mn> 2 </mn> </msub> <mo> &#10869; </mo> <mrow> <mi> n </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> B </mi> <mn> 2 </mn> </msub> <mo> &#10869; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#120068; </mi> <mo> &#10869; </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#120069; </mi> <mo> &#10869; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> log </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <power /> <apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <ci> Subscript 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Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.