Some identities involving derangement polynomials and numbers

The problem of counting derangements was initiated by Pierre Remonde de Motmort in 1708. A derangement is a permutation that has no fixed points and the derangement number Dn is the number of fixed point free permutations on an n element set. Furthermore, the derangement polynomials are natural extensions of the derangement numbers. In this paper, we study the derangement polynomials and numbers, their connections with cosine-derangement polynomials and sine-derangement polynomials and their applications to moments of some variants of gamma random variables.


INTRODUCTION AND PRELIMINARIES
The problem of counting derangements was initiated by Pierre Rémonde de Motmort in 1708 (see [1,2]). A derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, a derangement is a permutation that has no fixed points. The derangement number D n is the number of fixed point free permutations on an n (n ≥ 1) element set.
The aim of this paper is to study derangement polynomials and numbers, their connections with cosine-derangement polynomials and sine-derangement polynomials and their applications to momnets of some variants of gamma random variables. Here the derangement polynomials D n (x) are natural extensions of the derangement numbers.
The outline of our main results is as follows. We show a recurrence relation for derangement polynomials. Then we derive identities involving derangement polynomials, Bell polynomials and Stirling numbers of both kinds. In addition, we also have an identity relating Bell polynomials, derangement polynomials and Euler numbers. Next, we introduce the two variable polynomials, namely cosine-derangement polynomials D (c) n (x, y) and sine-derangement polynomials D (s) n (x, y), in a natural manner by means of derangement polynomials. We obtain, among other things, their explicit expressions and recurrence relations. Lastly, in the final section we show that, if X is the gamma random variable with parameters 1, 1, then D n (p), D (c) n (p, q), D (s) n (p, q) are given by the 'moments' of some variants of X .
In the rest of this section, we recall the derangement numbers, especially their explicit expressions, generating function and recurrence relations. Also, we give the derangement polynomials and give their explicit expressions. Then we recall the gamma random variable with parameters α, λ along with their moments and the Bell polynomials. Finally, we give the definitions of the Stirling numbers of the first and second kinds.
From (5), we have n l D l x n−l t n n! .
For X ∼ Γ(α, λ ), the n-th moment of X is given by It is well known that the Bell polynomials are defined by Bel n (x) t n n! , (see [9]).
When x = 1, Bel n = Bel n (1), (n ≥ 0) are called the Bell numbers. The Stirling numbers of the first kind are defined as As an inversion formula of (13), the Stirling numbers of the second kind are defined by (14) x n = n ∑ l=0 S 2 (n, l)(x) l (n ≥ 0), (see [5,7,18]).

DERANGEMENT POLYNOMIALS AND NUMBERS
From (5), we have On the other hand, Therefore, by (15) and (16), we obtain the following lemma.
Replacing t by 1 − e t in (5), we get From (17), we have It is easy to show that Replacing t by log(1 − t) in (19), we get From (5) and (20), we have Therefore, by (18) and (21), we obtain the following theorem.
Theorem 2. For n ≥ 0, we have Replacing t by −e t in (5), we get On the other hand, we have where E n are the ordinary Euler numbers. Therefore, by (22) and (23), we obtain the following theorem, Now, we observe that where r is a positive integer. On the other hand, Therefore, by (24) and (25), we obtain the following Proposition.
From (5), we note that By (9), (27) and (28), we get From (29) and (30), we can derive the following equations: We define cosine-derangement polynomials and sine-derangement polynomials respectively by Therefore, we obtain the following theorem.
and D (s) From (33), we note that Therefore, by comparing the coefficients on both sides of (37), we obtain the following theorem.
By (33), we get n (x, y) t n n! (38) Thus, we have On the other hand, we also have Therefore, by (39) and (40), we obtain the following theorem. By (33), we get On the other hand, Therefore, by (41) and (42), we obtain the following theorem.
It is not difficult to show that where r is positive integer. By comparing the coefficients on both sides of (39), we get l (x, y)r n−l .
Now, we observe that Form (45), we note that Therefore, we obtain the following theorem.
n (x, y) as a polynomial in x, for each fixed y, and D n (x) are Appell sequences.

From (34), we note that
Therefore, by (46), we obtain the following theorem.
Corollary 14. For n ≥ 1, we have By (46), we see that On the other hand, we also have Therefore, by (47) and (48), we obtain the following theorem.
It is easy to show that ∂ ∂ x D We observe that Comparing the coefficients on both sides of (49), we have the following theorem.
Theorem 16. For n ≥ 1, we have For r ∈ N, we have l (x, y)r n−l , (n ≥ 0).

FURTHER REMARKS
Let X be a gamma random variable with parameters 1,1 which is denoted by X ∼ Γ(1, 1). Then we observe that where f (x) is the density function of X , and p ∈ R.
On the other hand, by Taylor expansion, we get Therefore, by (51) and (52), we obtain the following theorem.
It is easy to show that where X ∼ Γ(1, 1).

CONCLUSION
The introduction of deragement numbers D n goes back to as early as 1708 when Pierre Rémond de Montmort considered some counting problem on derangements. In this paper, we dealt with derangement polynomials D n (x) which are natural extensions of the derangement numbers. We showed a recurrence relation for derangement polynomials. We derived identities involving derangement polynomials, Bell polynomials and Stirling numbers of both kinds. In addition, we also obtained an identity relating Bell polynomials, derangement polynomials and Euler numbers. Next, we introduced the cosine-derangement polynomials D n (x, y), by means of derangement polynomials. Then we derived, among other things, their explicit expressions and recurrence relations. Lastly, as an applications we showed that, if X is the gamma random variable with parameters 1, 1, then D n (p), D We have witnessed that the study of some special numbers and polynomials was done intensively by using several different means, which include generating functions, combinatorial methods, umbral calculus, p-adic analysis, probability theory, special functions and differential equations. Moreover, the same has been done for various degenerate versions of quite a few special numbers and polynomials in recent years with their interests not only in combinatorial and arithmetical properties but also in their applications to symmetric identities, differential equations and probability theories. It would have been nicer if we were able to find abundant applications in other disciplines.
It is one of our future projects to continue to investigate many ordinary and degenerate special numbers and polynomials by various means and find their applications in physics, science and engineering as well as in mathematics.
Author Contributions: T.K. and D.S.K. conceived of the framework and structured the whole paper; D.S.K. and T.K. wrote the paper; L.C.J. checked the errors of the paper; H.L. typed the paper; D.S.K. and T.K. completed the revision of the article. All authors have read and agreed to the published version of the manuscript.