Asymptotic distribution of zeros of polynomials satisfying difference equations

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+δ)−C(x,n)p n(x)+D(x)p n(x−δ)=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 technique to the approach based on the Nevai–Dehesa–Ullman distribution.