In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators to our problem, we obtain two auxiliary quantities [Formula: see text] and [Formula: see text] and show that they satisfy the coupled Riccati equations, from which we find that [Formula: see text] satisfies the Painlevé V equation. Furthermore, we prove that [Formula: see text], a quantity related to the logarithmic derivative of the Hankel determinant, satisfies both the continuous and discrete Jimbo–Miwa–Okamoto [Formula: see text]-form of the Painlevé V. In the end, by using Dyson’s Coulomb fluid approach, we consider the large [Formula: see text] asymptotic behavior of our problem at the soft edge, which gives rise to the Painlevé XXXIV equation.

中文翻译：

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Painlevé V、Painlevé XXXIV 和退化的拉盖尔酉合奏

在本文中，我们研究了与简并拉盖尔酉系综（dLUE）相关的汉克尔行列式。这个问题源于 dLUE 的最大或最小特征值分布。我们推导出梯形算子及其关于一般扰动权重函数的相容条件。通过将梯形算子应用于我们的问题，我们得到了两个辅助量[公式：见文本]和[公式：见文本]，并证明它们满足耦合的里卡蒂方程，从中我们发现[公式：见文本]满足Painlevé V 方程。此外，我们证明了[公式：见文本]，与汉克尔行列式的对数导数有关的量，满足 Painlevé V 的连续和离散的 Jimbo-Miwa-Okamoto [公式：见文本]-形式。结尾，