Generalized semiclassical orthogonal polynomials on the unit circle: A Riemann-Hilbert perspective

TL;DR

This paper uses Riemann-Hilbert analysis to derive differential equations and properties of generalized orthogonal polynomials on the unit circle, focusing on modified Jacobi and Bessel weights.

Contribution

It introduces a Riemann-Hilbert approach to analyze generalized semiclassical orthogonal polynomials on the unit circle, deriving new differential equations and recurrence relations.

Findings

Derived first and second order differential equations for orthogonal polynomials.

Established properties of recurrence coefficients from weight differential properties.

Applied methodology to generalized modified Jacobi and Bessel weights.

Abstract

In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain first and second order differential equations for the orthogonal polynomials and associated functions with a weight on the unit circle. We deduce properties for the recurrence relation coefficients from differential properties of the weight. We take the so called generalized modified Jacobi and Bessel weights as a case study.