Probabilistic heterogeneous Stirling numbers and Bell polynomials

TL;DR

The paper addresses unifying Stirling, Lah, Bell and Lah-Bell families under a probabilistic heterogeneous framework via a random variable $Y$ with finite moments. It defines probabilistic heterogeneous Stirling numbers $H_{rac{rac{Y}{rac{Y}{}}}}(n,k)$ and probabilistic heterogeneous Bell polynomials $H_{n,\u007frac{rac{Y}{rac{Y}{}}}}(x)$, derived from moment generating functions and degenerate exponentials. It derives explicit formulas, a Dobiński-like identity, and recurrence relations, and establishes connections to partial Bell polynomials; includes specialized results for Poisson and Bernoulli distributions. Limits $\lambda\to0$ and $\lambda\to1$ recover established probabilistic families $ {n \brace k}_{Y}$, $L^{Y}(n,k)$, $\phi^{Y}_n(x)$ and $LB^{Y}_n(x)$, highlighting the unifying power of the framework.

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, Dobiński-like identities, and recurrence relations. We further establish connections to partial Bell polynomials and provide applications for Poisson and Bernoulli distributions.

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Theorem 2.10