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

In this paper we characterize sequences of orthogonal polynomials on the unit circle whose Caratheodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of matrix Sylvester differential equations. For the particular case of semi-classical orthogonal polynomials on the unit circle, it is derived a characterization in terms of first order linear differential systems. Key words and phrases: Caratheodory function, matrix Riccati differential equations, matrix Sylvester differential equations, measures on the unit circle, semi-classical class.