On Co-polynomials on the Real Line and the Unit Circle

Abstract

In this paper, we present an overview about algebraic and analytic aspects of orthogonal polynomials on the real line when finite modifications of the coefficients of the three-term recurrence relation they satisfy, the so-called co-polynomials on the real line, are considered. We investigate the behavior of their zeros, mainly interlacing and monotonicity properties. Furthermore, using a transfer matrix approach we obtain new structural relations, combining theoretical and computational advantages. In the case of orthogonal polynomials on the unit circle, we analyze the effects of finite modifications of Verblunsky coefficients on Szegő recurrences. More precisely, we study the structural relations and the corresponding \(\mathcal{C}\)-functions of the orthogonal polynomials with respect to these modifications from the initial ones. By using the Szegő’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 as well as the relation between the corresponding \(\mathcal{S}\)-functions and \(\mathcal{C}\)-functions is studied.