Title: Probabilistic multi-Stirling numbers of the second kind and probabilistic multi-Lah numbers Show Chinese title

Title: 第二类概率多斯特林数和概率多拉赫数

Abstract: Assume that the moment generating function of the random vari able Y exists in a neighborhood of the origin. We introduce the probabilistic multi-Stirling numbers of the second kind associated with Y and the proba bilistic multi-Lah numbers associated with Y, both of indices (k1,k2,...,kr), by means of the multiple logarithm. Those numbers are respectively probabilistic extensions of the multi-Stirling numbers of the second kind and the multi-Lah numbers which, for (k1,k2,...,kr) = (1,1,...,1), boil down respectively to the Stirling numbers of the second and the unsigned Lah numbers. The aim of this paper is to study some properties, related identities, recurrence relations and explicit expressions of those probabilistic extension numbers in connection with several other special numbers Show Chinese abstract

Abstract: 假设随机变量 \( Y \) 的矩生成函数在原点附近的某个邻域内存在。 我们引入与 \( Y \) 相关的第二种概率多斯特林数和与 \( Y \) 相关的概率多拉赫数，其指标为 \( (k_1, k_2, ..., k_r) \)，通过多重对数来定义这些数。 这些数分别是第二种多斯特林数和多拉赫数的概率扩展形式，当 \( (k_1, k_2, ..., k_r) = (1, 1, ..., 1) \) 时，分别退化为斯特林数（第二类）和无符号拉赫数。 本文的目的是研究这些概率扩展数的一些性质、相关恒等式、递推关系以及与若干其他特殊数之间的显式表达式。