The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

Abstract

We study the zeros of orthogonal polynomials pn, N, n=0, 1, …, that are generated by recurrence coefficients an, N and bn, N depending on a parameter N. Assuming that the recurrence coefficients converge whenever n, N tend to infinity in such a way that the ratio n/N converges, we show that the polynomials pn, N have an asymptotic zero distribution as n/N tends to t>0 and we present an explicit formula for the limiting measure. This formula contains the asymptotic zero distri- butions for various special classes of orthogonal polynomials that were found earlier by different methods, such as Jacobi polynomials with varying parameters, discrete Chebyshev polynomials, Krawtchouk polynomials, and Tricomi–Carlitz polynomials. We also give new results on zero distributions of Charlier polynomials, Stieltjes–Wigert polynomials, and Lommel polynomials.