Advances in Difference Equations
Volume 2010 (2010), Article ID 406231, 18 pages
doi:10.1155/2010/406231

On linear combinations of two orthogonal polynomial sequences on the unit circle

C. Suárez

Departamento de Matemática Aplicada I, E.T.S.I.I., Universidad de Vigo, Campus Lagoas-Marcosende, 36200 Vigo, Spain

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, pnqn0. 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.