Strong asymptotics for the Pollaczek multiple orthogonal polynomials ensembles

TL;DR

This paper derives strong asymptotic formulas for a class of multiple orthogonal polynomials associated with Nikishin systems, extending previous weak asymptotics results using Riemann-Hilbert analysis.

Contribution

It provides the first derivation of strong asymptotics for these polynomials, building on prior weak asymptotics and employing Riemann-Hilbert techniques.

Findings

Established strong asymptotics for the polynomials

Derived asymptotics for the reproducing kernel

Extended previous weak asymptotics results

Abstract

We study the asymptotic properties of a class of multiple orthogonal polynomials with respect to a Nikishin system generated by two measures $(\sigma_1, \sigma_2)$ with unbounded supports (${supp}(\sigma_1) \subset \mathbb{R}_+$, ${supp}(\sigma_2) \subset \mathbb{R}_-$), and such that the second measure $\sigma_2$ is discrete. The weak asymptotics for these polynomials was obtained previously by V. Sorokin. We use his result and the Riemann-Hilbert analysis to derive the strong asymptotics of these polynomials and of the reproducing kernel.