Fully degenerate Bell polynomials associated with degenerate Poisson random variables

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.

A Series transformation formula and related degenerate polynomials

“…As the inversion formula of (5), the degenerate Stirling numbers of the second kind are given by ( 6) [7,9,10]).…”

“…and express several identities involving 'degenerate formal power series' as those including degenerate Stirling numbers of the second kind, degenerate Bell polynomials, degenerate Fubini polynomials and degenerate poly-Bernoulli polynomials (see Theorems 2,5,6,[9][10][11][12]. From (6), we note that…”

“…In recent years, studying degenerate versions of some special polynomials and numbers regained interests of some mathematicians, which include the degenerate Bernoulli numbers of the second kind, the degenerate Stirling numbers of both kinds, the degenerate Cauchy numbers, the degenerate Bell numbers and polynomials, the degenerate complete Bell polynomials and numbers, and so on (see [7,9,10,13,14,16,18] and the references therein). It is remarkable that this study of degenerate versions is not only limited to polynomials and numbers but also extended to transcendental functions like the gamma functions (see [11,12]).…”

“…The aim of this paper is to adopt the ideas in [3] and to express several identities involving 'degenerate formal power series' as those including degenerate Stirling numbers of the second kind, degenerate Bell polynomials, degenerate Fubini polynomials and degenerate poly-Bernoulli polynomials (see Theorems 2,5,6,[9][10][11][12]. Along the way, we also obtain some related identities which involve the λ -falling factorials, the degenerate Stirling numbers of both kinds, the degenerate Bernoulli numbers, the degenerate Fubini polynomials and the degenerate Euler polynomials.…”

Some new properties on degenerate Bell polynomials

“…In [5], Carlitz initiated the exploration of degenerate Bernoulli and Euler polynomials, which are degenerate versions of the ordinary Bernoulli and Euler polynomials. Along the same line as Carlitz's pioneering work, intensive studies have been done for degenerate versions of quite a few special polynomials and numbers (see [1,5,[7][8][9][10][11][12][13] and the references therein). It is worthwhile to mention that these studies of degenerate versions have been done not only for some special numbers and polynomials but also for transcendental functions like gamma functions (see [10]).…”