Universality at an Endpoint for Orthogonal Polynomials with Geronimus-Type Weights

TL;DR

This paper derives a new closed-form expression for Geronimus polynomials on the unit circle and establishes a universality result at an arc endpoint, advancing understanding of orthogonal polynomials with Geronimus-type weights.

Contribution

It introduces a novel closed-form expression for Geronimus polynomials and proves a universality result at an arc endpoint for these polynomials.

Findings

New closed-form expression for Geronimus polynomials

Universality result at an arc endpoint for orthogonal polynomials

Application of McLaughlin's matrix power formula to derive polynomial formulas

Abstract

We provide a new closed form expression for the Geronimus polynomials on the unit circle and use it to obtain new results and formulas. Among our results is a universality result at an endpoint of an arc for polynomials orthogonal with respect to a Geronimus type weight on an arc of the unit circle. The key tool is a formula of McLaughlin for powers of a two-by-two matrix, which we use to derive convenient formulas for Geronimus polynomials.