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Algebraic Generating Functions for Gegenbauer Polynomials

Published 2016-07-18Version 1

It is shown that several of Brafman's generating functions for the Gegenbauer polynomials are algebraic functions of their arguments, if the Gegenbauer parameter differs from an integer by one-fourth or one-sixth. Two examples are given, which come from recently derived expressions for associated Legendre functions with tetrahedral or octahedral monodromy. It is also shown that if the Gegenbauer parameter is restricted as stated, the Poisson kernel for the Gegenbauer polynomials can be expressed in terms of complete elliptic integrals. An example is given.

Comments: 20 pages, to appear in the volume `Frontiers of Orthogonal Polynomials and q-Series'

Categories: math.CA

Subjects: 33A50, 33A45

Keywords: gegenbauer polynomials, algebraic generating functions, complete elliptic integrals, gegenbauer parameter differs, brafmans generating functions

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