Algebraic Generating Functions for Gegenbauer Polynomials

TL;DR

This paper demonstrates that certain generating functions for Gegenbauer polynomials are algebraic under specific parameter conditions, linking them to elliptic integrals and special functions, with implications for their algebraic and analytical properties.

Contribution

It reveals algebraic forms of Brafman's generating functions for Gegenbauer polynomials when parameters differ from integers by one-fourth or one-sixth, and connects these to elliptic integrals.

Findings

Certain generating functions are algebraic under specific parameter conditions.

Poisson kernel can be expressed using complete elliptic integrals.

Examples relate to associated Legendre functions with special monodromy.

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.