Fine Structure of the Zeros of Orthogonal Polynomials, II. OPUC With Competing Exponential Decay

TL;DR

This paper develops a comprehensive theory describing the asymptotic behavior of zeros of orthogonal polynomials on the unit circle with Verblunsky coefficients exhibiting competing exponential decay, extending understanding of their zero distribution.

Contribution

It introduces a complete asymptotic analysis framework for OPUC with Verblunsky coefficients having multiple exponential decay components, a novel extension in the field.

Findings

Provides explicit asymptotic formulas for zeros of OPUC

Characterizes zero distribution in the presence of multiple exponential decay terms

Extends classical results to more complex coefficient structures

Abstract

We present a complete theory of the asymptotics of the zeros of OPUC with Verblunsky coefficients $\alpha_n = \sum_{\ell=1}^L C_\ell b_\ell^n + O((b\Delta)^n)$ where $\Delta <1$ and $\abs{b_\ell} = b<1$.