GegenbauerC[\[Nu], \[Lambda], z] == Subscript[F, Infinity][z, \[Nu], \[Lambda]] /; (Subscript[F, m][z, \[Nu], \[Lambda]] == ((Cos[Pi (\[Nu] + \[Lambda])] Sec[Pi \[Lambda]] Gamma[\[Nu] + 2 \[Lambda]])/(Gamma[\[Nu] + 1] Gamma[2 \[Lambda]])) Sum[((Pochhammer[-\[Nu], k] Pochhammer[\[Nu] + 2 \[Lambda], k])/ (Pochhammer[1/2 + \[Lambda], k] k!)) ((z + 1)/2)^k, {k, 0, m}] - ((2^(1/2 - \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Lambda] - 1/2])/ (Sqrt[Pi] Gamma[\[Lambda]])) (z + 1)^(1/2 - \[Lambda]) Sum[((Pochhammer[1/2 + \[Nu] + \[Lambda], k] Pochhammer[ 1/2 - \[Nu] - \[Lambda], k])/(Pochhammer[3/2 - \[Lambda], k] k!)) ((z + 1)/2)^k, {k, 0, m}] == GegenbauerC[\[Nu], \[Lambda], z] - ((2^(-1 - m) Sec[Pi \[Lambda]] Cos[Pi (\[Lambda] + \[Nu])] Gamma[2 \[Lambda] + \[Nu]] Pochhammer[-\[Nu], 1 + m] Pochhammer[2 \[Lambda] + \[Nu], 1 + m])/((1 + m)! Gamma[2 \[Lambda]] Gamma[1 + \[Nu]] Pochhammer[1/2 + \[Lambda], 1 + m])) (1 + z)^(1 + m) HypergeometricPFQ[{1, 1 + m - \[Nu], 1 + m + 2 \[Lambda] + \[Nu]}, {2 + m, 3/2 + m + \[Lambda]}, (1 + z)/2] + ((2^(-(1/2) - m - \[Lambda]) Sin[Pi \[Nu]] Gamma[-(1/2) + \[Lambda]] Pochhammer[1/2 - \[Lambda] - \[Nu], 1 + m] Pochhammer[1/2 + \[Lambda] + \[Nu], 1 + m])/(Sqrt[Pi] (1 + m)! Gamma[\[Lambda]] Pochhammer[3/2 - \[Lambda], 1 + m])) (1 + z)^(3/2 + m - \[Lambda]) HypergeometricPFQ[ {1, 3/2 + m - \[Lambda] - \[Nu], 3/2 + m + \[Lambda] + \[Nu]}, {2 + m, 5/2 + m - \[Lambda]}, (1 + z)/2] && Element[m, Integers] && m >= 0) && !Element[\[Lambda] + 1/2, Integers]

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</mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mi> &#957; </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> </mrow> <mrow> <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> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#955; 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2007-05-02