Renormalized Gegenbauer function: Series representations (formula 07.13.06.0070)

GegenbauerC

$Mathematica Notation$

Hypergeometric Functions

GegenbauerC[nu,z]

Series representations

Generalized power series

Expansions at z==infinity

For the function itself

Expansions in 1/(1-z)

http://functions.wolfram.com/07.13.06.0070.01

Input Form

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

Standard Form

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MathML Form

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<cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <ci> ν </ci> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 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Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <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> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> GegenbauerC </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <power /> <cn 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</apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> m </ci> <ci> ν </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </list> <list> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> </list> <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 /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> m </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn 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Rule Form

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

2007-05-02

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