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 nnth 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.

Keywords

• bell polynomials and numbers
• degenerate bell polynomials and numbers
• poisson random variable
• degenerate poisson random variable
• the poisson degenerate central moments
• 11b73
• 11b83
• 05a19