We consider the limiting process that arises at the hard edge of Muttalib–Borodin ensembles. This point process depends on \(\theta > 0\) and has a kernel built out of Wright’s generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form

where the constants \(\rho \), a, and b have been derived explicitly via a differential identity in s and the analysis of a Riemann–Hilbert problem. Their method can be used to evaluate c (with more efforts), but does not allow for the evaluation of C. In this work, we obtain expressions for the constants c and C by employing a differential identity in \(\theta \). When \(\theta \) is rational, we find that C can be expressed in terms of Barnes’ G-function. We also show that the asymptotic formula can be extended to all orders in s.

我们考虑在 Muttalib-Borodin 集合的硬边缘出现的限制过程。这个点过程依赖于\(\theta > 0\)并且有一个由 Wright 的广义 Bessel 函数构建的内核。在最近的一篇论文中，Claeys、Girotti 和 Stivigny 已经为这些集合中的大间隙概率建立了一阶和二阶渐近法。这些渐近线的形式为

其中常数\(\rho \)、a和b已通过s 中的微分恒等式和黎曼-希尔伯特问题的分析明确导出。他们的方法可用于评估c（需要更多努力），但不允许评估C。在这项工作中，我们通过在\(\theta \) 中使用微分恒等式来获得常数c和C的表达式。当\(\theta \)是有理数时，我们发现C可以用 Barnes' G 表示-功能。我们还表明渐近公式可以扩展到s 中的所有阶数。