On Linear Combinations of Two Orthogonal Polynomial Sequences on the Unit Circle

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∈ , 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.