Painlev\'e IV, bi-confluent Heun equations and the Hankel determinant generated by a discontinuous semi-classical Laguerre weight

TL;DR

This paper explores the connections between a discontinuous semi-classical Laguerre weight, Painlevé IV equations, and biconfluent Heun equations, providing asymptotic analysis and differential equations for related orthogonal polynomials and Hankel determinants.

Contribution

It establishes new links between orthogonal polynomials with discontinuous weights, Painlevé IV, and Heun equations, including asymptotic expansions and differential equations for associated quantities.

Findings

Relation of $R_{n}(t,s)$ to Painlevé IV equations

Asymptotic expansions of recurrence coefficients as $n\to\infty$

Derivation of the biconfluent Heun equation for orthogonal polynomials

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 $x\in\mathbf{R}$, $t,s\ge0$, $A\ge0$, $A+B\ge0$, 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.