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

TL;DR

This paper derives differential and difference equations for a class of semiclassical orthogonal polynomials on the unit circle, including generalizations of Jacobi and Bessel polynomials, and explores their connection to Painlevé equations.

Contribution

It introduces structure relations and explicit differential equations for semiclassical orthogonal polynomials on the unit circle, expanding understanding of their properties and applications.

Findings

Derived explicit differential equations for generalized Jacobi and Bessel polynomials on the unit circle.

Established difference equations and structure relations for these polynomials.

Connected Verblunsky coefficients to discrete Painlevé II equations.

Abstract

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.