Generalizations and specializations of generating functions for Jacobi, Gegenbauer, Chebyshev and Legendre polynomials with definite integrals

TL;DR

This paper extends and specializes generating functions for classical orthogonal polynomials like Jacobi, Gegenbauer, Chebyshev, and Legendre, providing new formulas and related definite integrals.

Contribution

It introduces a generalized generating function for Gegenbauer polynomials based on Jacobi polynomial generating functions and re-expresses hypergeometric functions for specialization.

Findings

Derived a generalized generating function for Gegenbauer polynomials.

Re-expressed hypergeometric functions for polynomial specialization.

Provided definite integrals corresponding to polynomial series expansions.

Abstract

In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given.