Gegenbauer function: Series representations (formula 07.14.06.0033)

GegenbauerC

$Mathematica Notation$

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/07.14.06.0033.01

Input Form

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