Gegenbauer function: Series representations (formula 07.14.06.0051)

GegenbauerC

$Mathematica Notation$

Hypergeometric Functions

GegenbauerC[nu,lambda,z]

Series representations

Generalized power series

Expansions at z==0

For the function itself

General case

http://functions.wolfram.com/07.14.06.0051.01

Input Form

GegenbauerC[\[Nu], \[Lambda], z] == Subscript[F, Infinity][z, \[Nu], \[Lambda]] /; Subscript[F, m][z, \[Nu], \[Lambda]] == ((2^\[Nu] Sqrt[Pi] Gamma[\[Lambda] + \[Nu]/2])/(Gamma[\[Lambda]] Gamma[(1 - \[Nu])/2] Gamma[1 + \[Nu]])) Sum[((Pochhammer[-(\[Nu]/2), k] Pochhammer[\[Lambda] + \[Nu]/2, k])/ (Pochhammer[1/2, k] k!)) z^(2 k), {k, 0, m}] + ((2^\[Nu] Sqrt[Pi] z Gamma[\[Lambda] + (1 + \[Nu])/2])/ (Gamma[\[Lambda]] Gamma[1 - \[Nu]/2] Gamma[\[Nu]])) Sum[((Pochhammer[(1 - \[Nu])/2, k] Pochhammer[\[Lambda] + (\[Nu] + 1)/2, k])/(Pochhammer[3/2, k] k!)) z^(2 k), {k, 0, m}] == GegenbauerC[\[Nu], \[Lambda], z] - ((2^\[Nu] Sqrt[Pi] z^(2 m + 2) Gamma[\[Lambda] + \[Nu]/2])/ ((1 + m)! Gamma[\[Lambda]] Gamma[(1 - \[Nu])/2] Gamma[1 + \[Nu]] Pochhammer[1/2, 1 + m])) Pochhammer[\[Lambda] + \[Nu]/2, 1 + m] Pochhammer[-(\[Nu]/2), 1 + m] HypergeometricPFQ[ {1, 1 + m - \[Nu]/2, 1 + m + \[Lambda] + \[Nu]/2}, {3/2 + m, 2 + m}, z^2] - ((2^\[Nu] Sqrt[Pi] z^(2 m + 3) Gamma[\[Lambda] + (1 + \[Nu])/2])/ ((1 + m)! Gamma[\[Lambda]] Gamma[1 - \[Nu]/2] Gamma[\[Nu]] Pochhammer[3/2, 1 + m])) Pochhammer[(1 - \[Nu])/2, 1 + m] Pochhammer[\[Lambda] + (1 + \[Nu])/2, 1 + m] HypergeometricPFQ[{1, 3/2 + m - \[Nu]/2, 3/2 + m + \[Lambda] + \[Nu]/2}, {2 + m, 5/2 + m}, z^2] && Element[m, Integers] && m >= 0

Standard Form

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MathML Form

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

2007-05-02

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