Probabilistic heterogeneous Stirling numbers and Bell polynomials

TL;DR

This paper introduces probabilistic heterogeneous Stirling numbers and Bell polynomials, unifying classical and probabilistic families, with explicit formulas, identities, and applications to Poisson and Bernoulli distributions.

Contribution

It develops a unified framework for probabilistic heterogeneous Stirling numbers and Bell polynomials, extending classical combinatorial structures with probabilistic and heterogeneous elements.

Findings

Derived explicit formulas and identities for the new structures.

Connected the new polynomials to partial Bell polynomials.

Applied the framework to Poisson and Bernoulli distributions.

Abstract

Let Y be a random variable satisfying specific moment conditions. This paper introduces and investigates probabilistic heterogeneous Stirling numbers of the second kind and probabilistic heterogeneous Bell polynomials. These structures unify several classical and probabilistic families, including those of Stirling, Lah, Bell and Lah-Bell. By integrating the heterogeneous framework of Kim and Kim with probabilistic extensions, we derive explicit formulas, Dobi\'nski-like identities, and recurrence relations. We further establish connections to partial Bell polynomials and provide applications for Poisson and Bernoulli distributions.