On a generalization of the generating function for Gegenbauer polynomials

TL;DR

This paper introduces a generalized generating function for Gegenbauer polynomials using associated Legendre functions, extending known formulas and enabling hyperspherical harmonic expansions for polyharmonic equations.

Contribution

It presents a new generalized generating function for Gegenbauer polynomials involving associated Legendre functions, broadening the scope of existing formulas.

Findings

Generalized generating function involving Legendre functions.

Extension of Heine's formulas and identities.

Application to hyperspherical harmonic expansions.

Abstract

A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of several previously derived formulae such as Heine's formula and Heine's reciprocal square-root identity. We also show how this expansion can be used to compute hyperspherical harmonic expansions for power-law fundamental solutions of the polyharmonic equation.