Gegenbauer function: Differentiation (formula 07.14.20.0002)

GegenbauerC

$Mathematica Notation$

Hypergeometric Functions

GegenbauerC[nu,lambda,z]

Differentiation

Low-order differentiation

With respect to lambda

http://functions.wolfram.com/07.14.20.0002.01

Input Form

D[GegenbauerC[\[Nu], \[Lambda], z], \[Lambda]] == ((2 \[Nu] (z - 1) Gamma[\[Nu] + 2 \[Lambda]])/((1 + 2 \[Lambda])^2 Gamma[\[Nu] + 1] Gamma[2 \[Lambda]])) ((1 + 2 \[Lambda]) HypergeometricPFQ[ {{1 - \[Nu], 1 + \[Nu] + 2 \[Lambda]}, {1}, {1, \[Nu] + 2 \[Lambda]}}, {{2, 3/2 + \[Lambda]}, {}, {1 + \[Nu] + 2 \[Lambda]}}, (1 - z)/2, (1 - z)/2] - (\[Nu] + 2 \[Lambda]) HypergeometricPFQ[ {{1 - \[Nu], 1 + \[Nu] + 2 \[Lambda]}, {1}, {1, 1/2 + \[Lambda]}}, {{2, 3/2 + \[Lambda]}, {}, {3/2 + \[Lambda]}}, (1 - z)/2, (1 - z)/2]) - ((2^(2 - 2 \[Lambda]) Sqrt[Pi] Gamma[\[Nu] + 2 \[Lambda]])/ (Gamma[\[Nu] + 1] Gamma[\[Lambda]])) Hypergeometric2F1Regularized[-\[Nu], \[Nu] + 2 \[Lambda], 1/2 + \[Lambda], (1 - z)/2] (PolyGamma[2 \[Lambda]] - PolyGamma[\[Nu] + 2 \[Lambda]])

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]