On classical orthogonal polynomials on lattices and some characterization theorems

TL;DR

This paper establishes conditions for solutions of functional equations related to classical orthogonal polynomials, providing formulas and characterizations that advance understanding of their structure and properties.

Contribution

It offers new necessary and sufficient conditions for regular solutions, along with explicit formulas for Rodrigues and recurrence coefficients, enhancing the theoretical framework of orthogonal polynomials.

Findings

Derived conditions for regularity of solutions

Provided Rodrigues and recurrence formulas

Solved characterization problems in orthogonal polynomial theory

Abstract

In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues formula and a closed formula for the recurrence coefficients. We finally used these results to solve some interesting research problems concerning characterization theorems.