A note on degenerate derangement polynomials and numbers

In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their explicit expressions, recurrence relations and some identities involving the degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Stirling numbers of both kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions.


INTRODUCTION AND PRELIMINARIES
A derangement is a permutation with no fixed points. In other words, a derangement is a permutation of the elements of a set that leaves no elements in their original places. The number of derangements of a set of size n is called the n-th derangement number and denoted by d n . The first few terms of the derangement number sequence {d n } ∞ n=0 are d 0 = 1, d 1 = 0, d 2 = 1, d 3 = 2, d 4 = 9, . . . . It was Pierre Rémonde de Motmort who initiated the study of counting derangements in 1708 (see [1]).
Carlitz was the first one who studied degenerate versions of some special polynomials and numers, namely the degenerate Bernoulli polynomials and numbers and degenerate Euler polynomials and numbers. In recent years, the study of various degenerate versions of some special polynomials and numbers regained the interests of quite a few mathematicians and yielded many interesting arithmetical and combinatorial results. It is remarkable that the study of degenerate versions is not just limited to polynomials but can be extended to transcendental functions like gamma functions (see [9,14]).
The aim of this paper is to study the degenerate derangement polynomials, which are a degenerate version of the derangement polynomials. Here the derangement polynomials are a natural extension of the derangement numbers. In more detail, we derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degerate Stirling numbers of both kinds. We also introduce the higher-order degenerate derangement polynomials. Then we explore the degenerate gamma distributions as a degenerate version of the gamma distributions and show that the moments of distributions coming from some variants of degenerate gamma distributions are related to the degenerate derangement polynomials or the degenerate derangement numbers or the higher-order degenerate derangement polynomials.
For the rest of this section, we recall the necessary facts about the degenerate derangement polynomials and numbers and the degenerate exponential functions.
The derangement polynomials are defined by the generating function as [12,13]).
By (3), we get Clearly, we have d n (0) = d n .
From the definition of degenerate derangement polynomials, we investigate some properties and recurrence relations and new identities associated with special numbers and polynomials.

DEGENERATE DERANGEMENT POLYNOMIALS
In light of (3), we may consider the degenerate derangement polynomials which are given by When x = 0, d n,λ = d n,λ (0) are called the degenerate derangement numbers. From (5) and (6), we get Comparing the coefficients on both sides of (7), we obtain the following proposition.
Theorem 2. The following identities hold true: Replacing t by 1 − e λ (t) in (6), we get Here S 2,λ (n, l), (n ≥ l), are the degenerate Stirling numbers of the second kind given either by (11) is also given by Therefore, by (11) and (12), we obtain the following theorem.
Let log λ (t) be the compositional inverse function of e λ (t). Recall that the degenerate Stirling numbers of the first kind are defined either by [7,14]).
For r ∈ N, we define the degenerate derangement polynomials of order r which are given by n (0) are called the degenerate derangement numbers of order r.
From (31), we note that Comparing the coefficients on both sides of (32), we obtain the following theorem.

FURTHER REMARKS
Let f (x) be the probability density function of the continuous random variable X , and let g(x) be a real valued function. Then the expectation of g(X ), E[g(X )], is defined by [18]).