Painlevé 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, Filipuk et al., and Clarkson and Jordaan. Filipuk et al. proved that the diagonal recurrence coefficient satisfies the fourth Painlevé equation. In this paper, we show that the off‐diagonal recurrence coefficient 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 ‐form of Painlevé IV. By using Dyson's Coulomb fluid approach together with the discrete system for and , we obtain the large asymptotic expansions of the recurrence coefficients and the sub‐leading coefficient. The large asymptotics of the associated Hankel determinant (including the constant term) is derived from its integral representation in terms of the sub‐leading coefficient.