Some Identities of the Degenerate Higher Order Derangement Polynomials and Numbers

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials without using degenerate umbral calculus. In this paper, the y the degenerate derangement polynomials of order s (s ∈ N) and give a combinatorial meaning about higher order derangement numbers. In addition, the author gives some interesting identities related to the degenerate derangement polynomials of order s and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derive the inversion formulas of these identities.


Introduction
Beginning with Carlitz's degenerate Bernoulli polynomial and degenerate Euler polynomial [1], many scholars in the field of mathematics have been working on degenerate versions of special polynomials and numbers which include the degenerate Stirling numbers of the first and second kinds, the degenerate Bernstein polynomials, the degenerate Bell numbers and polynomials, the degenerate gamma function, the degenerate gamma random variables, degenerate coloring and so on [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. They have been studied by various ways like combinatorial methods, umbral calculus techniques, generating functions, differential equations and probability theory, etc. We can find the motivation to study degenerate polynomials and numbers in the following real world examples. Suppose the probability of a baseball player getting a hit in a match is p. We wonder if the probability that the player will succeed in the 6th trial after failing 4 times in 5 trials is still p. We can see cases where the probability is less than p because of the psychological burden that the player must succeed in the 6th trial.
In particular, the umbral calculus, based on the modern idea of linear functions, linear operators and adjoints, began to build a rigorous foundation by Rota in the 1970s, primarily as symbolic techniques for the manipulation of numerical and polynomial sequences [18]. One of the important tools in the study of degenerate polynomials and numbers is the umbral calculus [16][17][18][19][20][21]. Kim Kim recently introduced the degenerate umbral calculus [15]. Furthermore, Kim et al. [13] studied the degenerate derangement polynomials of order s (s ∈ N) and numbers, investigate some properties of those polynomials without using degenerate umbral calculus. Motivated by Kim et al.'s work, the author considers the degenerate derangement polynomials of order s (s ∈ N) and give an example of the derangement number of order s in real-life. In addition, the author gives their connections with the degenerate derangement polynomials of order s and the well-known special polynomials and numbers.
First, we provide the definitions and properties required for this paper. Let n objects be labelled 1, 2, . . . , n. An arrangement or permutation in which object i is not placed in the i-th place for any i is called a derangement. The number of derangements of an n-element set is called the nth derangement number and denoted by d n . The nth derangement number is given by [13,[22][23][24] We note that the generating function of the nth derangement number is given by [12,13,25] From (1), Kim et al. naturally considered the derangement polynomials and degenerate derangement polynomials, respectively, which are given by [13,24] and When x = 0, d n (0) = d n , n ≥ 0 is the n-th derangement numbers. When x = 0, d n,λ (0) := d n,λ is called the degenerate derangement numbers and d 0,λ = 1.
Then P n is an (n + 1)-dimensional vector space over C.
Recently, Kim-Kim [15] considered λ-linear functional and λ-differential operator as follows: a k t k k! ∈ F and a fixed nonzero real number λ, each λ gives rise to the linear functional f (t) | · λ on P, called λ-linear functional given by f (t), which is defined by f (t) | (x) n,λ λ = a n , for all n ≥ 0.
For λ = 0, we observe that the linear functional f (t) | · 0 agrees with the one in ) λ (x) n,λ λ and (10), we note that for all n, k ≥ 0, where δ n,k is the Kronecker's symbol. From (11), for each λ ∈ R, and each nonnegative integer k, the differential operator on P is given by [15] and for any power series The order o( f (t)) of a power series f (t)( = 0) is the smallest integer k for which the coefficient of t k does not vanish. The series Let f (t) and g(t) be a delta series and an invertible series, respectively, and s n,λ (x) be a degenerate polynomial of a degree n. Then there exists a unique sequences s n,λ (x) such that the orthogonality conditions [15] The sequences s n,λ (x) are called the λ-Sheffer sequences for (g(t), f (t)), which are denoted by s n,λ ( Assume that for each λ ∈ R * of the set of nonzero real numbers, s n,λ (x) is λ-Sheffer for (g λ (t), f λ (t)). Assume also that lim λ→0 f λ (t) = f (t) and lim λ→0 g λ (t) = g(t), for some delta series f (t) and an invertible series g(t) .
In this case, Kim-Kim called that the family {s n,λ (x)} λ∈R−{0} of λ-Sheffer sequences s n,λ are the degenerate (Sheffer) sequences for the Sheffer polynomial s n (x). Let

Degenerate Derangement Polynomials Order s Arising from Degenerate Sheffer Sequences
In this section, we consider the degenerate derangement polynomials of order s, and give a combinatorial meaning of these numbers and noble identities related to these polynomials and the well-known special polynomials and numbers arising from degenerate Sheffer sequences.
From (3), naturally, we can consider the degenerate derangement polynomials of order s (c.f. [13]) which is given by When n,λ is called the degenerate derangement numbers of order s and d (s) It is known (see [15]) that for f (t), g(t) ∈ F , From (18), by the mathematical induction, for where Therefore, from (19), we get From (20), we have From (21), when λ → 0, we can give a combinatorial meaning about derangement numbers of order s in real-life. Example 1. Suppose n players play a card game randomly divided into s rooms. In addition, assume everyone wears a hat and hangs it on the entrance wall when entering a room. If all the lights suddenly turn out during the game, how many ways no one takes his hat when everyone comes out at same time?

Connection with the Degenerate r-Extended Lah-Bell Polynomials
The r-Lah number L r (n, k) counts the number of partitions of a set with n + r elements into k + r ordered blocks such that r distinguished elements have to be in distinct ordered blocks and an explicit formula of L r (n, k) (see [8,24,26,[28][29][30]) given by From (32), we have the generating function of L r (n, k) given by [28][29][30] Recently, Kim-Kim introduced the r-extended Lah-Bell polynomials, respectively, as follows [30]: When x = 1, B L n = B L n (1) and B L r,n = B L r,n (1) are called the Lah-Bell numbers and r-extended Lah-Bell numbers respectively.

Connection with the Degenerate Bernoulli Polynomials of Higher Order r
The degenerate Bernoulli polynomials of order r are given by the generating function [1,6,21] We note that β (r) n,λ (0) are called the degenerate Bernoulli numbers of order r. From (20), we observe that Theorem 3. For n ∈ N ∪ {0} and r, s ∈ N, we have In particular, when s = 1, as the inversion formula of (45), we have Proof. From (14), (16) and (43), we consider the following two degenerate Sheffer sequences.
From (15) and (47), we have where For r > n, from (9), we note that Now, by using (4), (16) and (50), we have For r > n and 0 ≤ k < r, we have the same result when r > n.
When u = −1, the degenerate Euler polynomials of order r respectively are given by the generating function [1,21] to be 2 e λ (t) + 1 We note that E (r) n,λ (0) (n ≥ 0), are called the degenerate Euler numbers of order r.
When x = 0 and r = 1, E n,λ = E n,λ (0) are called the degenerate Euler numbers. From (20), in the same way as (44), we have and In particular, when s = 1, as the inversion formula of (59), we have Proof. From (15), (16) and (55), we consider the following two degenerate Sheffer sequences.