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 > , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the nth 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 > and β 0 > , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

A Poisson random variable indicates how many events occurred within a given period of time. A random variable X is a real valued function defined on a sample space. If X takes any values in a countable set, then X is called a discrete random variable.
A random variable X taking on one of the values 0, 1, 2,… is said to be the Poisson random variable with parameter α 0 (> ) if the probability mass function of X is given by  Kim et al. [11] considered the degenerate Poisson random variable X X : λ ( ) with parameter α 0 (> ) if the probability mass function of X is given by Let us take an interesting example in which we consider the degenerate Poisson random variable with parameter α 0 (> ). Let us assume that the probability of success in an experiment is p. We wondered if we can say the probability of success in the 20th trial is still p after failing 19 times in 20 trial experiments. Because there is a psychological burden to be successful. It seems plausible that the probability is less than p (see [2]).
Thus, we study a new type of degenerate Bell polynomials associated with the degenerate Poisson random variable with parameters in this paper. In Section 2, 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 (or single variable fully degenerate Bell polynomials). 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. In Section 3, we will introduce the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α 0 > and β 0 > , called the two-variable fully degenerate Bell polynomials. We show that they are equal to the two-variable fully degenerate Bell polynomials and the Poisson degenerate central moments. Also, we derive some explicit expressions for the two-variable degenerate Bell polynomials. Here we note that the two-variable fully degenerate Bell polynomials are generalization of the fully degenerate Bell polynomials associated with degenerate Poisson random variables with one parameter α 0 > . When a pandemic such as Corona virus spreads throughout society, it changes the psychology of people on both an individual and group level, and in a broader sense the psychology of the community as a whole. In this respect, it is expected that relations with the fully degenerate Bell polynomials and moment of the degenerate Poisson random variable will be applied to predict how many people will be infected within a given period when a number of variables interact in a given environment. Now, we give some definitions and properties needed in this paper.
For any nonzero λ ∈ (or ), the degenerate exponential function is defined by     2 Fully degenerate Bell polynomials associated with degenerate Poisson random variable with parameter α 0 > In this section, 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. We give several combinatorial identities related to these polynomials and numbers.
From this section, for λ ∈ , let X X : λ ( ) be the degenerate Poisson random variable with parameter α 0 > if the probability mass function of X is given by For n ∈ , we note that the expectation and the nth moments of X with parameter α 0 > are and respectively.
From (2), we also observe In view of (11), naturally, we can define a new type of degenerate Bell polynomials, called the fully degenerate Bell polynomials as follows: On the other hand, from (2), (11) and (13), we have Therefore, by comparing the coefficients on both sides of (14), we have the desired result. □ (13), we obtain the following Dovinski-like formula for the fully degenerate Bell numbers as follows:  (1) and (5), we have

From Theorem 1 and
Therefore, by comparing the coefficients on both sides of (15), we get the desired result. □    From (20), we can derive the following generating function of the number of derangements of an n-element set  Proof. By using (16) and (23), By using (1), (7) and (27), we get On the other hand, from (10), we observe Therefore, by comparing the coefficients of (34) and (33), we obtain what we want. □ Thus, by comparing the coefficients of (38) and (39), we get the desired result. □

Conclusion
In this paper, we introduced the fully degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α 0 > and the two-variable fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α 0 > and β 0 > . We showed their connections with nth moment of the degenerate Poisson random variables and the Poisson degenerate central moments, respectively. We also expressed those polynomials and numbers in terms of the degenerate Stirling numbers of the second kind; the degenerate Stirling numbers of the first kind; the degenerate derangement numbers and the Stirling numbers of the first kind; and degenerate Frobenius-Euler polynomials.
It is important that the study of the degenerate version is widely applied not only to numerical theory and combinatorial theory, but also to symmetric identity, differential equations and probability theory. The Bell numbers have also been extensively studied in many different context in such branches of Mathematics [16][17][18][19][20][21][22]. With this in mind, as a future project, I would like to continue to study degenerate versions of certain special polynomials and numbers.