Generalizations of generating functions for higher continuous hypergeometric orthogonal polynomials in the Askey scheme

TL;DR

This paper generalizes generating functions for higher continuous hypergeometric orthogonal polynomials in the Askey scheme using connection relations and series rearrangement, and derives related definite integrals based on orthogonality.

Contribution

It introduces new generalized generating functions for Wilson, continuous dual Hahn, continuous Hahn, and Meixner-Pollaczek polynomials, expanding the analytical tools available for these polynomials.

Findings

Derived generalized generating functions for multiple orthogonal polynomials.

Established new definite integrals using orthogonality relations.

Enhanced understanding of the structure of hypergeometric orthogonal polynomials.

Abstract

We use connection relations and series rearrangement to generalize generating functions for several higher continuous orthogonal polynomials in the Askey scheme, namely the Wilson, continuous dual Hahn, continuous Hahn, and Meixner-Pollaczek polynomials. We also determine corresponding definite integrals using the orthogonality relations for these polynomials.