Differential equations for a class of semiclassical orthogonal polynomials on the unit circle

We consider semiclassical orthogonal polynomials on the unit circle associated with a weight function that satisfy a Pearson-type differential equation involving two polynomials of degree at most three. Structure relations and difference equations for these orthogonal polynomials are found, and, as a consequence, explicit first and second order differential equations are derived. Among the applications, differential equations for a family of polynomials that generalizes the Jacobi polynomials on the unit circle and the modified Bessel polynomials are established. It is also shown that in some cases the Verblunsky coefficients satisfy a discrete Painlev\'e II equation.