We consider the discontinuous semi-classical Laguerre weight function with a jump $w(x;t,s)=\mathrm{ e}^{ -x^2+tx}(A+B\theta(x-s))$, where $x\in\mathbf{ R}$, $t,s\ge0$, $A\ge0$, $A+B\ge0$, where $\theta(x)$ is 1 for $x > 0$ and 0 otherwise. Based on the ladder operator approach, we obtain some important difference and differential equations about the auxiliary quantities and the recurrence coefficients. By proper tranformation, It is shown that $R_{ n}(t,s)$ is related to Painlev\'{ e} IV equations and $r_{ n}(t,s)$ satisfies the Chazy II equations. With the aid of Dyson's Coulomb fluid approach, we derive the asymptotic expansions for $\alpha_{ n}$ and $\beta_{ n}$ as $n\rightarrow\infty$. Furthermore, This enables us to obtain the lagre $n$ behavior of the orthogonal polynomials and derive that they satisfy the biconfluent Heun equation. We also consider the Hankel determinant $D_{ n}(t,s)$ generated by the discountinuous semi-classical Laguerre weight. We find that the quantity $\sigma_{ n}(t,s)$, allied to the logarithmic derivative of $D_{ n}(t,s)$, satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of Painlev\'{ e} IV.
Categories: math.CA