https://functions.wolfram.com/HypergeometricFunctions/GegenbauerC3General/20/03/ShowAll.html
Gegenbauer function: Differentiation (subsection 20/03)
Fractional integro-differentiation (1 formula)
gegenbauer functiondifferentiationsubsection
https://functions.wolfram.com/HypergeometricFunctions/GegenbauerC/06/01/07/01/01/0011/
Renormalized Gegenbauer function: Series representations (formula 07.13.06.0064)
Series representations
gegenbauer functionrenormalizedseriesrepresentationsformula
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