Painleve IV asymptotics for orthogonal polynomials with respect to a modified Laguerre weight

TL;DR

This paper analyzes the asymptotic behavior of orthogonal polynomials with a modified Laguerre weight, revealing connections to Painleve IV equations through Riemann-Hilbert problem techniques.

Contribution

It establishes a link between orthogonal polynomial asymptotics and Painleve IV solutions in a double scaling limit, extending previous analyses to include a modified weight.

Findings

Recurrence coefficients described by Painleve IV solutions

Construction of local parametrix using Painleve IV related functions

Application of Deift/Zhou steepest descent method to Riemann-Hilbert problem

Abstract

We study polynomials that are orthogonal with respect to the modified Laguerre weight $z^{-n + \nu} e^{-Nz} (z-1)^{2b}$ in the limit where $n, N \to \infty$ with $N/n \to 1$ and $\nu$ is a fixed number in $\mathbb{R} \setminus \mathbb{N}_0$. With the effect of the factor $(z-1)^{2b}$, the local parametrix near the critical point $z =1$ can be constructed in terms of $\Psi$-functions associated with the Painleve IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painleve IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann-Hilbert problem associated with orthogonal polynomials.