Painleve IV, Chazy II, and asymptotics for recurrence coefficients of semi-classical Laguerre polynomials and their Hankel determinants

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 alpha n(t)$$ {\alpha}_n(t) $$ satisfies the fourth Painleve equation. In this paper, we show that the off-diagonal recurrence coefficient beta n(t)$$ {\beta}_n(t) $$ 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$$ \sigma $$-form of Painleve IV. By using Dyson's Coulomb fluid approach together with the discrete system for alpha n(t)$$ {\alpha}_n(t) $$ and beta n(t)$$ {\beta}_n(t) $$, we obtain the large n$$ n $$ asymptotic expansions of the recurrence coefficients and the sub-leading coefficient. The large n$$ n $$ asymptotics of the associated Hankel determinant (including the constant term) is derived from its integral representation in terms of the sub-leading coefficient.