Asymptotics of recurrence coefficients for the Laguerre weight with a singularity at the edge

TL;DR

This paper analyzes the asymptotic behavior of recurrence coefficients for Laguerre orthogonal polynomials with singularities at the soft edge, using advanced Riemann-Hilbert techniques and Painlevé equations.

Contribution

It provides new asymptotic formulas for recurrence coefficients with singularities, connecting them to Painlevé equations, advancing understanding of orthogonal polynomials with singular weights.

Findings

Asymptotic formulas expressed via Painlevé equations

Recurrence coefficients characterized for large n

Singularity effects on asymptotics elucidated

Abstract

In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the Deift-Zhou steepest descent method. The asymptotic formulas of the leading coefficients and the recurrence coefficients for large n are described in terms of a class of analytic solutions to the the {\sigma}-form of the Painlev\'e II equation and the Painlev\'e XXXIV equation.