Painlevé IV, bi-confluent Heun equations and the Hankel determinant generated by a discontinuous semi-classical Laguerre weight

Abstract: We consider the discontinuous semi-classical Laguerre weight function with a jump $w(x;t,s)=\mathrm{e}^{-x^2+tx}(A+Bθ(x-s))$, where $x\in\mathbf{R}$, $t,s\ge0$, $A\ge0$, $A+B\ge0$, where $θ(x)$ 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 Painlevé 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 $α{n}$ and $β{n}$ 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 $σ{n}(t,s)$, allied to the logarithmic derivative of $D{n}(t,s)$, satisfies the Jimbo-Miwa-Okamoto $σ$-form of Painlevé IV.