Probabilistic Stirling numbers associated with sequences

TL;DR

This work addresses the lack of orthogonality and inverse relations in existing probabilistic Stirling numbers associated with a random variable $Y$ by redefining the first-kind numbers as $S_{1}^{Y}(n,k)$ using the compositional inverse of $e_{Y}(t)=E[e^{Yt}]-1$, ensuring orthogonality with $S_{2}^{Y}(n,k)$ and recovering classical Stirling numbers when $Y=1$. It extends this framework to degenerate versions $S_{2,\lambda}^{Y}(n,k)$ and $S_{1,\lambda}^{Y}(n,k)$, proving they share the same orthogonality/inversion structure and coincide with the standard degenerate numbers when $Y=1$. The paper provides explicit computations of these quantities for a wide range of distributions (Bernoulli, Binomial, Poisson, Geometric, Exponential, Gamma, Normal, Uniform) and develops comprehensive inversion formulas to represent polynomials in terms of probabilistic Euler polynomials $\mathcal{E}_{n}^{Y}(x)$ and $\mathcal{E}_{n,\lambda}^{Y}(x)$. Overall, the work delivers a consistent probabilistic generalization of Stirling numbers that preserves essential algebraic relations, enabling new applications in polynomial expansions and probabilistic combinatorics.

Abstract

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. Recently, probabilistic Stirling numbers of the first kind and of the second kind associated with Y have been introduced. However, probabilistic stirling number of the first kind based on the cumulant generating function of Y, and probabilistic stirling number of the second kind do not satisfy orthogonality and inverse relations. This paper aims to redefine the probabilistic stirling numbers of the first kind associated with Y such that probabilistic stiirling number of the first kind and probabilistic stirling number of the second kind do satisfy these crucial relations. Furthermore, we investigate their degenerate counterparts, the probabilistic degenerate Stirling numbers of both kinds. We explicitly compute probabilistic stiirling number of the first kind and probabilistic stirling number of the second kind .

Theorems & Definitions (16)

• Proposition 1.1
• Proposition 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8