Painlev\'{e} IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants

TL;DR

This paper explores the connections between semi-classical Laguerre polynomials, Painlevé IV, and Chazy II equations, providing asymptotic formulas for recurrence coefficients and Hankel determinants using advanced analytical methods.

Contribution

It establishes new links between recurrence coefficients and integrable systems, deriving asymptotics for these coefficients and determinants in the semi-classical Laguerre polynomial context.

Findings

Recurrence coefficient α_n(t) satisfies Painlevé IV.

Off-diagonal coefficient β_n(t) fulfills Chazy II system.

Asymptotic expansion of Hankel determinant obtained.

Abstract

This paper studies the monic semi-classical Laguerre polynomials based on previous work by Boelen and Van Assche \cite{Boelen}, Filipuk et al. \cite{Filipuk} and Clarkson and Jordaan \cite{Clarkson}. Filipuk, Van Assche and Zhang proved that the diagonal recurrence coefficient $\alpha_n(t)$ satisfies the fourth Painlev\'{e} equation. In this paper we show that the off-diagonal recurrence coefficient $\beta_n(t)$ fulfills the first member of Chazy II system. We also prove that the sub-leading coefficient of the monic semi-classical Laguerre polynomials satisfies both the continuous and discrete Jimbo-Miwa-Okamoto $\sigma$-form of Painlev\'{e} IV. By using Dyson's Coulomb fluid approach together with the discrete system for $\alpha_n(t)$ and $\beta_n(t)$, we obtain the large $n$ asymptotic expansions of the recurrence coefficients and the sub-leading coefficient. The large $n$ asymptotics of the associate Hankel determinant (including the constant term) is derived from its integral representation in terms of the sub-leading coefficient.