Probabilistic degenerate poly-Bell polynomials associated with random variables

TL;DR

This work develops the theory of probabilistic degenerate poly-Bell polynomials associated with a random variable $Y$ whose mgf exists near the origin. The authors define $\mathrm{Bel}_{n,\lambda}^{(k,Y)}(x)$ via $\mathrm{Ei}_{k,\lambda}(x(E[e_{\lambda}^{Y}(t)]-1))$ and establish explicit finite-sum expressions in terms of $ {n \brace l}_{Y,\lambda}$ and $E[(S_{m})_{n,\lambda}]$, along with several structural identities. Special cases where $Y$ is Bernoulli$(p)$ or $Y\sim \Gamma(1,1)$ yield concrete formulas, such as $\mathrm{Bel}_{n,\lambda}^{(k,Y)}(x)=\sum_{m=1}^{n}\frac{(1)_{m,\lambda}}{m^{k-1}}p^{m}{n \brace m}_{\lambda}x^{m}$ and $\mathrm{Bel}_{n,\lambda}^{(k,Y)}(x)=\sum_{m=1}^{n}\sum_{l=m}^{n}\frac{(1)_{m,\lambda}}{m^{k-1}}\lambda^{n-l}L(l,m)S_{1}(n,l)x^{m}$. A key identity, $\sum_{m=0}^{l}\binom{l}{m}(-1)^{l-m}E[(S_{m})_{n,\lambda}]=0$, is highlighted under certain $\lambda$, indicating deep combinatorial structure. The results provide computable formulas and pave the way for connections to other distributions (e.g., Poisson) and further probabilistic-degenerate polynomial theory.

Abstract

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to study the probabilistic degenerate poly-Bell polynomials associated with the random variable Y, arising from the degenerate polyexponential functions, which are probabilistic extensions of degenerate versions of the poly-Bell polynomials. We derive several explicit expressions and some related identities for them. In addition, we consider the special cases that Y is the Bernoulli random variable with probability of success p or the gamma random variable with parameters 1,1.

Theorems & Definitions (9)

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