On Laguerre–Hahn affine orthogonal polynomials on the unit circle from matrix Sylvester equations

Abstract

In this paper are derived recurrences for the reflection coefficients of Laguerre–Hahn affine orthogonal polynomials on the unit circle, including a form of the discrete Painlevé equations dP_V. The technique is based on the knowledge of the first-order differential equation for the Carathéodory function, combined with a re-interpretation, in the formalism of matrix Sylvester equations, of compatibility conditions for the differential systems satisfied by the polynomials.