In this paper we characterize sequences of orthogonal polynomials on the unit circle whose Carathéodory 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.
A. Branquinho. M.N. Rebocho. "Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circle." Bull. Belg. Math. Soc. Simon Stevin 17 (2) 355 - 376, april 2010. https://doi.org/10.36045/bbms/1274896211