Algebraic Generating Functions for Gegenbauer Polynomials
Classical Analysis and ODEs
2018-02-02 v3
Abstract
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 octahedral or tetrahedral 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.
Cite
@article{arxiv.1607.05215,
title = {Algebraic Generating Functions for Gegenbauer Polynomials},
author = {Robert S. Maier},
journal= {arXiv preprint arXiv:1607.05215},
year = {2018}
}
Comments
20 pages, final version, typos corrected, to appear in the volume `Frontiers of Orthogonal Polynomials and q-Series' (World Scientific)