Probabilistic multi-Stirling numbers of the second kind and probabilistic multi-Lah numbers

TL;DR

This paper introduces probabilistic multi-Stirling and multi-Lah numbers linked to a random variable Y, extending classical combinatorial numbers through probabilistic methods and analyzing their properties and identities.

Contribution

It presents the first probabilistic extensions of multi-Stirling and multi-Lah numbers using the moment generating function of Y, along with their properties and relations.

Findings

Derived recurrence relations for probabilistic multi-Stirling and multi-Lah numbers.

Established identities connecting these probabilistic numbers with classical special numbers.

Provided explicit formulas and properties of the probabilistic extension numbers.

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