  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.
