Difference and differential equations for deformed Laguerre-Hahn orthogonal polynomials on the unit circle

Sequences of orthogonal polynomials on the unit circle whose Caratheodory function satisfies a Riccati-type differential equation with polynomial coefficients are studied. We deduce discrete Lax equations which lead to difference equations for the corresponding sequences of reflection parameters, and we analyze the continuous differential equations that arise when deformations through a dependence on a parameter t occur.