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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 on branch cuts > For the function itself





http://functions.wolfram.com/07.14.06.0045.01









  


  










Input Form





GegenbauerC[\[Nu], \[Lambda], z] == ((2^(1 - 2 \[Lambda]) Sin[Pi \[Nu]])/(Sqrt[Pi] Gamma[\[Lambda]])) Sum[(1/(2^k k!)) (Gamma[k - \[Nu]] Gamma[k + 2 \[Lambda] + \[Nu]] Cos[Pi (\[Lambda] + \[Nu])] (2 I E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - x]/(2 Pi)]) Floor[Arg[z - x]/(2 Pi)] - Sec[Pi \[Lambda]] Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]]) Hypergeometric2F1Regularized[k - \[Nu], k + 2 \[Lambda] + \[Nu], 1/2 + k + \[Lambda], (1 + x)/2] + 2^(-(1/2) + k + \[Lambda]) Pi Sec[Pi \[Lambda]] (1 + x)^(1/2 - k - \[Lambda]) Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]] Hypergeometric2F1Regularized[1/2 + \[Lambda] + \[Nu], 1/2 - \[Lambda] - \[Nu], 3/2 - k - \[Lambda], (1 + x)/2]) (z - x)^k, {k, 0, Infinity}] /; !Element[1/2 - \[Lambda], Integers] && Element[x, Reals] && x < -1










Standard Form





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MathML Form







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</ci> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <sec /> <apply> <times /> <pi /> <ci> &#955; </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <ci> &#957; </ci> </apply> <apply> <plus /> <ci> k </ci> <ci> &#955; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> <integers /> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02