Generalizations of generating functions for hypergeometric orthogonal polynomials with definite integrals

TL;DR

This paper extends generating functions for key hypergeometric orthogonal polynomials by series rearrangement and connection relations, also deriving related definite integrals using orthogonality.

Contribution

It introduces generalized generating functions for Jacobi, Gegenbauer, Laguerre, and Wilson polynomials via series rearrangement and connection relations, expanding their analytical framework.

Findings

Derived new generalized generating functions for multiple orthogonal polynomials.

Established corresponding definite integrals using orthogonality relations.

Enhanced analytical tools for hypergeometric orthogonal polynomials.

Abstract

We generalize generating functions for hypergeometric orthogonal polynomials, namely Jacobi, Gegenbauer, Laguerre, and Wilson polynomials. These generalizations of generating functions are accomplished through series rearrangement using connection relations with one free parameter for these orthogonal polynomials. We also use orthogonality relations to determine corresponding definite integrals.