A New Approach to Ratio Asymptotics for Orthogonal Polynomials

TL;DR

This paper introduces a novel method using Saff's non-linear characterization to analyze the asymptotic behavior of orthonormal polynomials, demonstrating that their behavior is primarily governed by leading coefficients and normalization, with applications to measures on the unit disk.

Contribution

It presents a new approach to ratio asymptotics for orthogonal polynomials based on a non-linear characterization, extending understanding to regular measures on the unit disk.

Findings

Ratio asymptotics hold along sequences of asymptotic density 1 for regular measures.

Behavior of orthonormal polynomials is determined by leading coefficient and normalization.

Applicable to measures on the closed unit disk, including the unit circle.

Abstract

We use a non-linear characterization of orthonormal polynomials due to Saff in order to show that the behavior of orthonormal polynomials is determined only by its leading coefficient and its normalization. Several applications of this equivalence are also discussed. One of our main results is that for regular measures on the closed unit disk - including, but not limited to the unit circle - one has ratio asymptotics along a sequence of asymptotic density 1.