Asymptotic distribution of zeros of polynomials satisfying difference equations

TL;DR

This paper develops a method to determine the asymptotic distribution of zeros for orthogonal polynomials satisfying specific difference equations, with applications to Meixner and Meixner-Pollaczek polynomials.

Contribution

It introduces a new approach to analyze the asymptotic zero distribution of polynomials satisfying difference equations, connecting to WKB and Nevai-Ullman methods.

Findings

Calculated asymptotic zero distribution for Meixner and Meixner-Pollaczek polynomials.

Identified delicate features in the zero distribution of Meixner polynomials.

Linked the approach to existing techniques like WKB and Nevai-Ullman distribution.

Abstract

We propose a way to find the asymptotic distribution of zeros of orthogonal polynomials p_n(x) satisfying a difference equation of the form B(x)p_n(x+\delta)-C(x,n)p_n(x)+D(x)p_n(x-\delta)=0. We calculate the asymptotic distribution of zeros and asymptotics of extreme zeros of the Meixner and Meixner-Pollaczek polynomials. The distribution of zeros of Meixner polynomials shows some delicate features. We indicate the relation of our approach to the WKB technique and to the approach based on the Nevai-Ullman distribution.