We study the joint probability generating function for [Formula: see text] occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. This generalizes a result of Tracy and Widom [C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Commun. Math. Phys. 161(2) (1994) 289–309], which corresponds to the case [Formula: see text]. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large [Formula: see text] asymptotics for [Formula: see text] Hankel determinants with a Laguerre weight possessing several jump discontinuities near the hard edge.

中文翻译：

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贝塞尔点过程的生成函数和耦合 Painlevé V 方程组

我们研究了贝塞尔点过程中不相交区间上的[公式：见文本]占用数的联合概率生成函数。该生成函数可以表示为 Fredholm 行列式。我们根据耦合 Painlevé V 方程组获得了它的表达式，该方程组源自黎曼-希尔伯特问题的 Lax 对。这概括了 Tracy 和 Widom [CA Tracy 和 H. Widom，水平间距分布和 Bessel 内核，Commun。数学。物理。161(2) (1994) 289–309]，对应案例 [公式：见正文]。我们还提供了一些示例和应用程序。特别是，几个相关的量可以用生成函数来表示，例如不相交的有界区间并集上的间隙概率、两个最小粒子之间的间隙以及大 [公式：