In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles

here \(\theta (x)\) is the Heaviside function, where \(A> 0\), \(t>0\), and \(z\in [0,\infty )\). We derive the ladder operators and its interrelated compatibility conditions. By using the ladder operators, we show two auxiliary quantities \(R_n(t)\) and \(r_n(t)\) satisfy the coupled Riccati equations, from which we also prove that \(R_n(t)\) satisfies a particular Painlevé IV equation. Even more, \(\sigma _n(t)\), allied to \(R_n(t)\), satisfies both the discrete and continuous Jimbo–Miwa–Okamoto \(\sigma \)-form of the Painlevé IV equation. Finally, we consider the situation when n gets large, the second order linear differential equation satisfied by the polynomials \(P_n(x)\) orthogonal with respect to the semi-classical weight turn to be a particular bi-confluent Heun equation.

在本文中，我们提出了半经典 Laguerre 酉系综的某个不连续线性统计量的特征

这里\(\theta (x)\)是 Heaviside 函数，其中\(A> 0\)、\(t>0\)和\(z\in [0,\infty )\)。我们推导出梯形运算符及其相关的兼容性条件。通过使用阶梯算子，我们展示了两个辅助量\(R_n(t)\)和\(r_n(t)\)满足耦合Riccati方程，从中我们也证明\(R_n(t)\)满足一个特殊的Painlevé IV 方程。更重要的是，\(\sigma _n(t)\)与\(R_n(t)\) 联合，同时满足离散和连续的 Jimbo–Miwa–Okamoto \(\sigma \)-Painlevé IV 方程的形式。最后，我们考虑当n变大时，由与半经典权重正交的多项式\(P_n(x)\)所满足的二阶线性微分方程变成一个特定的双汇合Heun方程。