Weights with both absolutely continuous and discrete components: Asymptotics via the Riemann-Hilbert approach

TL;DR

This paper develops uniform asymptotic formulas for orthogonal polynomials with weights combining continuous and discrete parts, using Riemann-Hilbert techniques, and introduces a new band structure in the analysis.

Contribution

It extends the Riemann-Hilbert approach to orthogonal polynomials with mixed weights, revealing a novel band structure involving both continuous and discrete support.

Findings

Derived Plancherel-Rotach asymptotics for sieved Pollazek Polynomials

Identified a new band structure with two adjacent intervals in the support

Applied Riemann-Hilbert method to complex asymptotic analysis

Abstract

We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollazek Polynomials as an example. The Plancherel-Rotach type asymptotics of the sieved Pollazek Polynomials are obtained in the whole complex plane. The Riemann-Hilbert method is applied to derive the results. A main feature of the treatment is the appearance of a new band consisting of two adjacent intervals, one of which is a portion of the support of the absolutely continuous measure, the other is the discrete band.