Fully degenerate Bell polynomials associated with degenerate Poisson random variables

Hye Kyung Kim

doi:10.1515/math-2021-0022

Abstract

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

Highlights

Carlitz [1] initiated a study of degenerate versions of some special polynomials and numbers, called the degenerate Bernoulli and Euler polynomials and numbers
We introduce a new type of degenerate Bell polynomials and numbers associated with the degenerate Poisson random variable with parameter α > 0, called the fully degenerate Bell polynomials and numbers. We show their connections with nth moment of the degenerate Poisson random variable with parameter α > 0, and give several identities related to these polynomials including the degenerate Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, degenerate derangement numbers, degenerate Frobenius-Euler polynomials and numbers, etc
As one of the generalizations of the fully degenerate Bell polynomials in Section 2, we will introduce the two-variable fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 and β > 0, and show their connection with the degenerate Poisson central moments

Summary

Introduction

Carlitz [1] initiated a study of degenerate versions of some special polynomials and numbers, called the degenerate Bernoulli and Euler polynomials and numbers. Kim et al [11] considered the degenerate Poisson random variable X(:Xλ) with parameter α (>0) if the probability mass function of X is given by pλ (i). We study a new type of degenerate Bell polynomials associated with the degenerate Poisson random variable with parameters in this paper. We introduce a new type of degenerate Bell polynomials and numbers associated with degenerate Poisson random variable with parameter α > 0, called the fully degenerate Bell polynomials and numbers. For λ ∈ , let X (: Xλ) be the degenerate Poisson random variable with parameter α > 0 if the probability mass function of X is given by. By comparing the coefficients of (29) and (30), we get the desired result

Two-variable fully degenerate Bell polynomials

Conclusion