A note on some identities of derangement polynomials

The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255–258, 1978, Clarke and Sved in Math. Mag. 66(5):299–303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1–11 2018. A derangement is a permutation that has no fixed points, and the derangement number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d_{n}$\end{document}dn is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r-derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.


Introduction
Let C be the complex number field, and let F be the set of all formal power series in the variable t with coefficients in C: (1.1) Let P = C[x], and let P * be the vector space of all linear functionals on P. We denote the action of a linear functional L ∈ P * on polynomials p(x) ∈ P by L | p(x) , and it is known that vector space operations on P * are defined by where c is a complex constant (see [3][4][5]). For f (t) = ∞ k=0 a k t k k! , we define a linear functional on P by setting f (t) | x n = a n (n ≥ 0) (see [6,7]). (1.3) From (1.3), we note that t k | x n = n!δ n,k (n, k ≥ 0) (see [8]), (1.4) where δ n,k is the Kronecker symbol.
The Bernoulli polynomials are given by [3,12,13] E n (x) t n n! (see [10,11]). (1.14) When x = 0, E n = E n (0) are the Euler numbers. The falling factorial sequence is defined as The Stirling numbers of the first kind are defined by (x) n = n l=0 S 1 (n, l)x l (n ≥ 0) (see [8]), (1.16) and the Stirling numbers of the second kind are given by x n = n l=0 S 2 (n, l)(x) l (n ≥ 0) (see [8,14,15]). (1.17) The Stirling numbers of the second kind are also given by the exponential generating function (see [8, p.59 It is well known that the Bell polynomials are defined by the generating function Bel n (x) t n n! (see [9] S 2 (n, k)x k (n ≥ 0) (see [9]). (1.20) 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 element set (see [1][2][3]). The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see [1][2][3]). 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, d 5 = 44, d 6 = 265, d 7 = 1854, . . . . Indeed, d n is given by the closed form formula: [3]). (1.21) From (1.21), we note that the generating function of derangement numbers is given by d n t n n! (see [9] For r ∈ N, the derangement numbers d (r) n of order r (n ≥ 0), are defined by the generating function (see [3]). (1.25) The umbral calculus comes under the heading of combinatorics, the calculus of finite differences, the theory of special functions, and formal solutions to differential equations.
Also, formal power series play a predominant role in the umbral calculus. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Further, we study two generalizations of derangement polynomials, namely higher-order and r-derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.

Some identities of derangement polynomials arising from umbral calculus
Now, we define the derangement polynomials by We note here that, for x = -1, d n = d n (-1) are the derangement numbers.
We observe that By (2.1) and (2.2), we get Therefore we obtain the following lemma. .
Therefore, we obtain the following proposition.
with the usual convention about replacing d n by d n .
From Proposition 2.2, we have By (1.5) and (2.1), we get where a n are the arrangement numbers defined by Replacing t by e t -1 in (2.1), we get On the other hand, Therefore, by (2.10) and (2.11), we obtain the following theorem. For S n (x) ∼ (g(t), t), from (1.5) we have Thus, by (2.12), we get In (2.13), we take g(t) = 1t, then we have (2.14) Now, we observe that From (2.15), we note that From (1.10), we have In particular, Therefore, we obtain the following corollary. For r ∈ N, we define the derangement polynomials of order r by n are the derangement numbers of order r. For 0 ≤ r ≤ n, the r-derangement numbers, denoted by D (r) n , are the number of derangements on n+r elements under the restriction that the first r-elements are in disjoint cycles. It is known that the generating function of the r-derangement numbers is given by (2.23) We consider the r-derangement polynomials given by From (2.24), we note that D (r) n (-1) = D (r) n are the r-derangement numbers. By (2.13) and (2.22), we easily get Comparing the coefficients on both sides of (2.27), we get Now, we observe that Therefore, by (2.43), we obtain the following theorem. For n ≥ 0, let Then P n is an (n + 1)-dimensional vector space over C.
For p(x) ∈ P n , we let Thus, we have Let us take p(x) = d (r) n (x) ∈ P n . Then we have Assume that p(x) = n k=0 C (r) k d (r) k (x) ∈ P n . Then, by (2.25), we get Thus, from (2.49), we note that Therefore, we obtain the following theorem.

Hence
Bel n (x) = n k=0 2S 2 (n, k) -S 2 (n + 1, k The ordered Bell polynomials are defined by the generating function

Results and discussion
In this paper, as a natural companion to derangement numbers, we have investigated derangement polynomials and derived several interesting properties on them which are related to derangement numbers. Also, we have considered two generalizations of derangement polynomials, namely the higher-order and r-derangement polynomials, and showed some relations between them and also with some other special polynomials. In addition, by using umbral calculus, we derived a formula expressing any polynomials as linear combinations of higher-order derangement polynomials and illustrated this with several special polynomials.

Conclusion
The introduction of derangement numbers goes back to as early as 1708 when Pierre Rémond de Montmort considered some counting problem on derangements. However, it seems that the umbral calculus approach to the derangement polynomials and their generalizations has not yet been done. In this paper, we have used umbral calculus in order to study some interesting properties on them, certain relations between them, and some connections with several other special polynomials.