Probabilistic central Bell polynomials

TL;DR

This paper introduces probabilistic central Bell polynomials and related numbers, extending classical combinatorial polynomials to a probabilistic setting and deriving their properties and identities.

Contribution

It presents the first study of probabilistic central Bell polynomials, factorial numbers, and Fubini polynomials associated with a random variable, including their properties and relations.

Findings

Derived explicit expressions and identities for probabilistic central Bell polynomials.

Established recurrence relations for these probabilistic polynomials and numbers.

Extended classical combinatorial polynomials to a probabilistic framework.

Abstract

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. In this paper, we study the probabilistic central Bell polynomials associated with random variable Y, as probabilistic extension of the central Bell polynomials. In addition, we investigate the probabilistic central factorial numbers of the second kind associated with Y and the probabilistic central Fubini polynomials associated with Y. The aim of this paper is to derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers.