Gegenbauer function: Series representations (formula 07.14.06.0015)

GegenbauerC

$Mathematica Notation$

Hypergeometric Functions

GegenbauerC[nu,lambda,z]

Series representations

Generalized power series

Expansions at z==-1

For the function itself

General case

http://functions.wolfram.com/07.14.06.0015.01

Input Form

GegenbauerC[\[Nu], \[Lambda], z] == ((Cos[Pi (\[Nu] + \[Lambda])] Sec[Pi \[Lambda]] Gamma[\[Nu] + 2 \[Lambda]])/ (Gamma[\[Nu] + 1] Gamma[2 \[Lambda]])) Hypergeometric2F1[-\[Nu], \[Nu] + 2 \[Lambda], 1/2 + \[Lambda], (1 + z)/2] - ((2^(1/2 - \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Lambda] - 1/2])/ (Sqrt[Pi] Gamma[\[Lambda]])) (z + 1)^(1/2 - \[Lambda]) Hypergeometric2F1[1/2 + \[Nu] + \[Lambda], 1/2 - \[Nu] - \[Lambda], 3/2 - \[Lambda], (1 + z)/2] /; !Element[\[Lambda] + 1/2, Integers]

Standard Form

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MathML Form

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Rule Form

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

2001-10-29

GegenbauerC[nu,z]
GegenbauerC[n,lambda,z]