Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula

TL;DR

This paper explores polynomial solutions of hypergeometric-type differential equations, introducing complementary polynomials with Rodrigues formulas, generating functions, and applications to classical polynomials.

Contribution

It presents a new method for constructing complementary polynomials from Rodrigues representations and derives their generating functions, recursion relations, and differential equations.

Findings

Complementary polynomials satisfy hypergeometric-type differential equations.

Derived a closed-form generating function for these polynomials.

Applied results to classical polynomial families.

Abstract

Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and an addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given.