On Computation of Highly Oscillatory Integrals with Bessel Kernel

In this paper, we introduce a new numerical scheme for approximation of highly oscillatory integrals having Bessel kernel. We transform the given integral to a special form having improper nonoscillatory Laguerre type and proper oscillatory integrals with Fourier kernels. Integrals with Laguerre weights over [0, ∞) will be solved by Gauss-Laguerre quadrature and oscillatory integrals with Fourier kernel can be evaluated by meshless-Levin method. Some numerical examples are also discussed to check the efficiency of proposed method.

The generating function for the Bessel point process and a system of coupled Painlevé V equations

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.