An electrostatic model for the roots of discrete classical orthogonal polynomials

Abstract

An electrostatic model is presented to describe the behaviour of the roots of classical discrete orthogonal polynomials. Indeed, this model applies more generally to the roots of polynomial solutions of second-order linear difference equations $A{\Delta }_h{\nabla }_{h}y+B{\Delta }_{h}y+Cy=0$ where $A$, $B$ and $C$ are polynomials and ${\Delta }_{h}f\left(x\right)=f(x+h)-f\left(x\right)\text{and}{\nabla }_{h}f\left(x\right)=f\left(x\right)-f(x-h)$ with $h>0$. The existence of a unique distribution of points which minimizes the energy of the system is guaranteed under some assumptions on $A$ and $B$. Furthermore, interlacing properties and the monotonicity of the roots depending on the parameters which appear in the difference equation are obtained from this electrostatic model.