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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/07.14.06.0028.01









  


  










Input Form





GegenbauerC[\[Nu], \[Lambda], z] == (((-1)^(\[Nu] + \[Lambda] - 1) 2^(1 - \[Nu] - 2 \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/(Pi Gamma[\[Lambda]] Gamma[1 + \[Nu] + \[Lambda]])) (z - 1)^(-\[Nu] - 2 \[Lambda]) Sum[((Pochhammer[1/2 + \[Nu] + \[Lambda], k] Pochhammer[ \[Nu] + 2 \[Lambda], k])/(k! Pochhammer[1 + 2 \[Nu] + 2 \[Lambda], k])) (PolyGamma[k + 1] - PolyGamma[1/2 - k - \[Nu] - \[Lambda]] - PolyGamma[k + \[Nu] + 2 \[Lambda]] + PolyGamma[ k + 2 \[Nu] + 2 \[Lambda] + 1]) (2/(1 - z))^k, {k, 0, Infinity}] + (((-1)^(\[Nu] + \[Lambda] - 1) 2^(1 - \[Nu] - 2 \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/(Pi Gamma[\[Lambda]] Gamma[1 + \[Nu] + \[Lambda]])) (z - 1)^(-\[Nu] - 2 \[Lambda]) Log[(z - 1)/2] Hypergeometric2F1[\[Nu] + 2 \[Lambda], 1/2 + \[Nu] + \[Lambda], 1 + 2 \[Nu] + 2 \[Lambda], 2/(1 - z)] + ((2^(1 - \[Nu] - 2 \[Lambda]) Sqrt[Pi])/(Gamma[\[Nu] + 1] Gamma[\[Lambda]])) (z - 1)^\[Nu] Sum[(((2 \[Nu] + 2 \[Lambda] - k - 1)! Pochhammer[-\[Nu], k])/ (k! Gamma[1/2 - k + \[Nu] + \[Lambda]])) (2/(1 - z))^k, {k, 0, 2 \[Nu] + 2 \[Lambda] - 1}] /; Element[\[Lambda] + \[Nu], Integers] && \[Lambda] + \[Nu] >= 0 && !Element[\[Nu], Integers]










Standard Form





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MathML Form







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</ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <ci> &#957; </ci> </apply> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <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> &#955; 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Rule Form





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





2001-10-29





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