Probabilistic Lah numbers and Lah-Bell polynomials

TL;DR

This paper introduces probabilistic Lah numbers and Lah-Bell polynomials linked to a random variable Y, deriving properties, explicit formulas, and identities, with special cases for Poisson and Bernoulli distributions.

Contribution

It develops a new probabilistic framework for Lah numbers and Lah-Bell polynomials, extending their classical definitions to random variables.

Findings

Derived properties, explicit formulas, and identities for probabilistic Lah numbers and Lah-Bell polynomials.

Analyzed special cases where Y is Poisson or Bernoulli distributed.

Established recurrence relations and probabilistic interpretations.

Abstract

Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. The aim of this paper is to study the probabilistic Lah numbers associated with Y and the probabilistic Lah-Bell polynomials associated with Y, as probabilistic versions of the Lah numbers and the Lah-Bell polynomials, respectively. We derive some properties, explicit expressions, recurrence relations and certain identities for those numbers and polynomials. In addition, we treat the special cases that Y is the Poisson random variable with parameter {\alpha} > 0 and the Bernoulli random variable with probability of success p.