Fully degenerate Bell polynomials associated with degenerate Poisson random variables

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mspace width="-0.15em" /> <m:mo>></m:mo> <m:mspace width="-0.15em" /> <m:mn>0</m:mn> </m:math> α {-0.15em}> {-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> α > 0 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>β</m:mi> <m:mspace width="-0.15em" /> <m:mo>></m:mo> <m:mspace width="-0.15em" /> <m:mn>0</m:mn> </m:math> β {-0.15em}> {-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.