On perturbed orthogonal polynomials on the real line and the unit circle via Szeg\H{o}'s transformation

TL;DR

This paper explores the connections between orthogonal polynomials on the real line and the unit circle through Szeg ext{"o}''s transformation, establishing new relations between their recurrence coefficients and parameters.

Contribution

It introduces novel relations linking recurrence coefficients of real-line orthogonal polynomials with Verblunsky parameters of circle polynomials via Szeg ext{"o}''s transformation.

Findings

Derived new formulas connecting recurrence coefficients and Verblunsky parameters.

Established relations between al functions associated with these polynomials.

Enhanced understanding of the transformation's impact on polynomial properties.

Abstract

By using the Szeg\H{o}'s transformation we deduce new relations between the recurrence coefficients for orthogonal polynomials on the real line and the Verblunsky parameters of orthogonal polynomials on the unit circle. Moreover, we study the relation between the corresponding $\mathcal{S}$-functions and $\mathcal{C}$-functions