Self-inversive polynomial and quasi-orthogonality on the unit circle

TL;DR

This paper explores the properties of self-inversive polynomials and their relation to quasi-orthogonality on the unit circle, highlighting structural analogies with real-line quasi-orthogonal polynomials and using hypergeometric functions.

Contribution

It introduces a class of self-invariant polynomials and connects their properties to quasi-orthogonality on the unit circle, extending the understanding of spectral analogies with real-line polynomials.

Findings

Self-inversive polynomials exhibit quasi-orthogonality on the unit circle.

Representation via reversed Szegő polynomials links to hypergeometric functions.

Structural similarities between unit circle and real-line quasi-orthogonality are established.

Abstract

In this paper we study quasi-orthogonality on the unit circle based on the structural and orthogonal properties of a class of self-invariant polynomials. We discuss a special case in which these polynomials are represented in terms of the reversed Szeg\H{o} polynomials of consecutive degrees and illustrate the results using contiguous relations of hypergeometric functions. This work is motivated partly by the fact that recently cases have been made to establish para-orthogonal polynomials as the unit circle analogues of quasi-orthogonal polynomials on the real line so far as spectral properties are concerned. We show that structure wise too there is great analogy when self-inversive polynomials are used to study quasi-orthogonality on the unit circle.