Advances in Difference Equations
Volume 2010 (2010), Article ID 406231, 18 pages
Research Article

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

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

Received 1 August 2009; Revised 1 December 2009; Accepted 5 March 2010

Academic Editor: Panayiotis Siafarikas

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.


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.