Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circle
The paper is devoted to the study of the orthogonal polynomials on the unit circle, whose corresponding Carathéodory function satisfies a Riccati differential equation with polynomial coefficients. It is established the equivalence between this property and the fact that two matrix sequences (related with the orthogonal polynomials, the associated polynomials of the second kind and the functions of the second kind) satisfy the Sylvester differential equations. In particular, the authors obtain a characterization of the semi-classical orthogonal polynomials on the circle in terms of first order linear differential systems.
scientific article; zbMATH DE number 5735940