A basic class of symmetric orthogonal polynomials of a discrete variable

TL;DR

This paper introduces a new class of symmetric orthogonal polynomials in discrete variables, generalizing classical types, with explicit properties, hypergeometric sequences, and applications to moments and classical polynomials.

Contribution

It presents a unified framework for symmetric orthogonal polynomials in discrete variables, extending classical families with explicit formulas and hypergeometric sequences.

Findings

Derived a second order difference equation for the polynomials

Identified two hypergeometric orthogonal sequences with 20 weight functions

Explicitly computed moments for these weight functions

Abstract

By using a generalization of Sturm-Liouville problems in discrete spaces, a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters, which generalizes all classical discrete symmetric orthogonal polynomials, is introduced. The standard properties of these polynomials, such as a second order difference equation, an explicit form for the polynomials, a three term recurrence relation and an orthogonality relation are presented. It is shown that two hypergeometric orthogonal sequences with 20 different weight functions can be extracted from this class. Moreover, moments corresponding to these weight functions can be explicitly computed. Finally, a particular example containing all classical discrete symmetric orthogonal polynomials is studied in detail.