The Gegenbauer polynomials are solutions to the Gegenbauer
differential equation for integer . They are generalizations of the associated Legendre
polynomials to -D space, and are proportional to (or, depending
on the normalization, equal to) the ultraspherical polynomials .
Following Szegö, in this work, Gegenbauer polynomials are given in terms of the Jacobi polynomials with by
(1)
(Szegö 1975, p. 80), thus making them equivalent to the Gegenbauer polynomials implemented in the Wolfram Language
as GegenbauerC[n,
lambda, x]. These polynomials are also given by the generating
function
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