On a class of Rainville type generating functions for classical orthogonal polynomials

TL;DR

This paper characterizes all generating functions of a specific form for classical orthogonal polynomials and derives additional generating functions through differentiation, expanding the understanding of their structure.

Contribution

It provides a complete characterization of a class of generating functions for classical orthogonal polynomials and introduces new generating functions via derivation.

Findings

Complete characterization of Rainville type generating functions.

Derivation of new generating functions for classical orthogonal polynomials.

Enhanced understanding of the structure of generating functions.

Abstract

The aim of this work is to characterize all generating functions of the form $A(t)F(xtA(t)-R(t))$ for the classical orthogonal polynomials. Further generating functions are also provided by derivation.