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variants of this functions
GegenbauerC






Mathematica Notation

Traditional Notation









Hypergeometric Functions > GegenbauerC[nu,lambda,z] > Series representations > Generalized power series > Expansions at z==infinity > For the function itself > Expansions in 1/(1-z)





http://functions.wolfram.com/07.14.06.0079.01









  


  










Input Form





GegenbauerC[\[Nu], \[Lambda], z] == Subscript[F, Infinity][z, \[Nu], \[Lambda]] /; (Subscript[F, m][z, \[Nu], \[Lambda]] == ((2^\[Nu] Gamma[\[Nu] + \[Lambda]])/(Gamma[\[Nu] + 1] Gamma[\[Lambda]])) (z - 1)^\[Nu] Sum[((Pochhammer[-\[Nu], k] Pochhammer[ 1/2 - \[Nu] - \[Lambda], k])/(Pochhammer[1 - 2 \[Nu] - 2 \[Lambda], k] k!)) (2/(1 - z))^k, {k, 0, m}] - ((2^(-\[Nu] - 2 \[Lambda]) Sin[\[Nu] Pi])/(Pi Gamma[\[Lambda]])) Gamma[-\[Nu] - \[Lambda]] Gamma[\[Nu] + 2 \[Lambda]] (z - 1)^(-\[Nu] - 2 \[Lambda]) Sum[((Pochhammer[\[Nu] + 2 \[Lambda], k] Pochhammer[ 1/2 + \[Nu] + \[Lambda], k])/(Pochhammer[1 + 2 \[Nu] + 2 \[Lambda], k] k!)) (2/(1 - z))^k, {k, 0, m}] == GegenbauerC[\[Nu], \[Lambda], z] + ((2^(1 + m + \[Nu]) (-1)^m Gamma[\[Lambda] + \[Nu]] Pochhammer[1/2 - \[Lambda] - \[Nu], 1 + m] Pochhammer[-\[Nu], 1 + m])/ ((1 + m)! Gamma[\[Lambda]] Gamma[1 + \[Nu]] Pochhammer[1 - 2 \[Lambda] - 2 \[Nu], 1 + m])) (-1 + z)^(\[Nu] - m - 1) HypergeometricPFQ[{1, 1 + m - \[Nu], 3/2 + m - \[Lambda] - \[Nu]}, {2 + m, 2 + m - 2 \[Lambda] - 2 \[Nu]}, 2/(1 - z)] - ((2^(1 + m - 2 \[Lambda] - \[Nu]) (-1)^m Sin[Pi \[Nu]] Gamma[-\[Lambda] - \[Nu]] Gamma[2 \[Lambda] + \[Nu]])/ (Pi (1 + m)! Gamma[\[Lambda]] Pochhammer[1 + 2 \[Lambda] + 2 \[Nu], 1 + m])) Pochhammer[1/2 + \[Lambda] + \[Nu], 1 + m] Pochhammer[2 \[Lambda] + \[Nu], 1 + m] (-1 + z)^(-2 \[Lambda] - \[Nu] - m - 1) HypergeometricPFQ[ {1, 3/2 + m + \[Lambda] + \[Nu], 1 + m + 2 \[Lambda] + \[Nu]}, {2 + m, 2 + m + 2 \[Lambda] + 2 \[Nu]}, 2/(1 - z)] && Element[m, Integers] && m >= 0) && !Element[2 \[Lambda] + 2 \[Nu], Integers]










Standard Form





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MathML Form







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</mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[Lambda]&quot;]], &quot;+&quot;, &quot;\[Nu]&quot;]], &quot;)&quot;]], RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;2&quot;, &quot; 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Rule Form





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





2007-05-02





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