Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circle

Abstract

In this paper we characterize sequences of polynomials on the unit circle, orthogonal with respect to a Hermitian linear functional such that its corresponding Carath´eodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of matrix Sylvester differential equations. Furthermore, under certain conditions, we give a representation of such sequences in terms of semi-classical orthogonal polynomials on the unit circle. For the particular case of semi-classical orthogonal polynomials on the unit circle, a characterization in terms of first order differential systems is established.