Zeros of non-Baxter paraorthogonal polynomials on the unit circle

TL;DR

This paper analyzes the asymptotic behavior of zeros of certain paraorthogonal polynomials on the unit circle, focusing on the size of gaps around 1 under specific conditions on their Verblunsky coefficients.

Contribution

It provides leading order asymptotics for zero gaps of paraorthogonal polynomials with Verblunsky coefficients decaying slowly within (-1,0).

Findings

Asymptotic estimates for zero gaps around 1.

Results under both strict and relaxed conditions on coefficients.

Insights into zero distribution for these polynomials.

Abstract

We provide leading order asymptotics for the size of the gap in the zeros around 1 of paraothogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (-1,0). We also include related results that impose less restrictive conditions on the Verblunsky coefficients.