Copyright © 2010 C. Suárez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let {Φn} be a monic orthogonal polynomial sequence on the unit circle. We define recursively a new sequence {Ψn} of polynomials by the following linear combination: Ψn(z)+pnΨn-1(z)=Φn(z)+qnΦn-1(z), pn,qn∈ℂ, pnqn≠0. In this paper, we give necessary and sufficient conditions in order to make {Ψn} be an orthogonal polynomial sequence too. Moreover, we obtain an explicit representation for the Verblunsky coefficients {Φn(0)} and {Ψn(0)} in terms of pn and qn. Finally, we show the relation between their corresponding Carathéodory functions and their associated linear functionals.