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

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.