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http://functions.wolfram.com/07.14.06.0018.01
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GegenbauerC[\[Nu], \[Lambda], z] ==
((Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/(Pi Gamma[\[Nu] + 1]
Gamma[2 \[Lambda]])) Log[(z + 1)/2] Hypergeometric2F1[-\[Nu],
\[Nu] + 2 \[Lambda], 1/2 + \[Lambda], (z + 1)/2] -
((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])/(k! Pochhammer[3/2 - \[Lambda], k]))
((z + 1)/2)^k, {k, 0, \[Lambda] - 3/2}] -
((2^(1 - 2 \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/
(Sqrt[Pi] Gamma[\[Nu] + 1] Gamma[\[Lambda]]))
Sum[((Pochhammer[-\[Nu], k] Pochhammer[\[Nu] + 2 \[Lambda], k])/
(k! Gamma[1/2 + k + \[Lambda]])) (PolyGamma[k + 1] -
PolyGamma[k + \[Nu] + 2 \[Lambda]] - PolyGamma[k - \[Nu]] +
PolyGamma[1/2 + k + \[Lambda]]) ((1 + z)/2)^k, {k, 0, Infinity}] /;
Element[\[Lambda] - 3/2, Integers] && \[Lambda] - 3/2 >= 0 &&
!(Element[\[Nu], Integers] && \[Nu] >= 0)
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ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <ci> ν </ci> </apply> <apply> <plus /> <ci> λ </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> 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<apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> ν </ci> <pi /> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> 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