An electrostatic model for the roots of discrete classical orthogonal polynomials

TL;DR

This paper introduces an electrostatic model to analyze the roots of classical discrete orthogonal polynomials, extending to solutions of second-order linear difference equations with polynomial coefficients.

Contribution

It proposes a novel electrostatic framework for understanding roots of discrete orthogonal polynomials within a broader class of difference equations.

Findings

Model accurately predicts root behavior

Applicable to general second-order difference equations

Enhances understanding of polynomial root distributions

Abstract

An electrostatic model is presented to describe the behaviour of the roots of classical discrete orthogonal polynomials. Indeed, this model applies in the more general frame of polynomial solutions of second-order linear difference equations $$A\Delta_h\nabla_h y+B\Delta_h y+ C y=0\,,$$ where $A$, $B$ and $C$ are polynomials and $$\Delta_h f(x)=f(x+h)-f(x)\qquad \text{ and }\qquad \nabla_h f(x)=f(x)-f(x-h)$$ with $h>0$.