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 $w(x;t,s)={\mathrm{e}}^{-x^2+tx}(A+B\theta (x-s))$, where $x\in \mathbf{R}$, $t,s\ge 0$, $A\ge 0$, $A+B\ge 0$, 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 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 ${\alpha }_n$ and ${\beta }_n$ as $n\to \infty$. 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 Painlevé IV.

19 pages