Ratio asymptotics and zero density for orthogonal polynomials with varying Verblunsky coefficients

TL;DR

This paper investigates the asymptotic behavior and zero distribution of orthogonal polynomials on the unit circle with varying Verblunsky coefficients, providing streamlined proofs and extending results to different regimes of coefficient variation.

Contribution

It offers a simplified proof of ratio asymptotics and characterizes zero distributions for orthogonal polynomials with varying coefficients in multiple regimes.

Findings

Established ratio asymptotics for asymptotically constant and periodic coefficients.

Determined zero distribution for paraorthogonal polynomials in locally constant and periodic regimes.

Extended results to orthogonal polynomials under mild conditions.

Abstract

We study asymptotic behavior of orthogonal polynomials on the unit circle with varying Verblunsky coefficients $\alpha_{n,N}$ when the ratio $n/N$ converges as $n,N\to\infty$. First, we give a streamlined proof of ratio asymptotics for orthogonal and paraorthogonal polynomials in the case of asymptotically constant and asymptotically periodic coefficients $\alpha_{n,N}$. Second, we determine the asymptotic zero distribution of paraorthogonal polynomials in the locally constant and locally periodic regimes. Analogous results are obtained for orthogonal polynomials under a mild additional condition on the varying coefficients.