Abstract
In combinatorics, a derangement is a permutation that has no fixed points. The number of derangements of an n-element set is called the nth derangement number. In this paper, as natural companions to derangement numbers and degenerate versions of the companions we introduce derangement polynomials and degenerate derangement polynomials. We give some of their properties, recurrence relations, and identities for those polynomials which are related to some special numbers and polynomials.
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Keywords
2010 Mathematices Subject Classification
1 Introduction
It is known that the Fubini polynomials are defined by the generating function
Thus, by (1.1), we get
Here \(S_2(n,k)\) is the Stirling number of the second kind which is defined by
where \((x)_0=1, (x)_n = x(x-1) \ldots (x-n+1)\), \((n \ge 1)\).
As is well known, the Bell polynomials are given by the generating function as follows:
When \(x=1\), \(Bel_n = Bel_n(1)\) are called the Bell numbers. For \(\lambda \in \mathbb {R}\), the partially degenerate Bell polynomials were introduced by Kim–Kim–Dolgy as
Note that \(\lim _{\lambda \rightarrow 0} Bel_{n,\lambda }(x) = Bel_n(x)\), \((n \ge 0)\). When \(x=1\), \(Bel_{n,\lambda } = Bel_{l,\lambda }(1)\) are called the partially degenerate Bell numbers.
From (1.5), we have
where \(S_1(n,k)\) is the Stirling number of the first kind given by
In [1], Carlitz introduced the degenerate Bernoulli and Euler polynomials which are defined by
and
When \(x=0\), \(\beta _{n,\lambda }=\beta _{n,\lambda }(0)\), \(\mathcal {E}_{n,\lambda }= \mathcal {E}_{n,\lambda }(0)\) are called the degenerate Bernoulli numbers and degenerate Euler numbers.
Recently, the degenerate Stirling numbers of the second kind are defined by
where \(n \ge 0\) (see [10]).
Note that \(\lim _{\lambda \rightarrow 0}S_{2,\lambda }(n,k) = S_2(n,k)\). For \(\lambda \in \mathbb {R}\), the \(\lambda \)-analogue of falling factorial sequence is defined by
Note that \(\lim _{\lambda \rightarrow 1} (x)_{n,\lambda } = (x)_n\), \((n\ge 0)\), (see [14]).
A derangement is a permutation with no fixed points. In other words, a derangement of a set leaves no elements in the original place. The number of derangements of a set of size n, denoted \(d_n\), is called the nth derangement number (see [9, 15, 16]).
For \(n\ge 0\), it is well known that the recurrence relation of derangement numbers is given by
It is not difficult to show that
From (1.13), we note that
and
In this paper, as natural companions to derangement numbers and degenerate versions of the companions we introduce derangement polynomials and degenerate derangement polynomials. We give some of their properties, recurrence relations, and identities for those polynomials which are related to some special numbers and polynomials.
2 Derangement Polynomials
Now, we define the derangement polynomials which are given by the generating function
When \(x=1\), \(d_n(1)= d_n\) are the derangement numbers.
From (1.1), we note that
On the other hand,
Therefore, by (2.2) and (2.3), we obtain the following lemma.
Lemma 2.1
For \(n \ge 0\), we have
We observe that
From (2.2) and (2.4), we obtain the following theorem.
Theorem 2.2
For \(n \ge 0\), we have
By (2.1), we get
By comparing the coefficients on both sides of (2.5), we obtain the following theorem.
Theorem 2.3
For \(n \ge 0\), we have
From (2.1), we have
On the other hand,
Thus, by (2.6) and (2.7), we get
From (2.8), we note that
Therefore, we obtain the following theorem.
Theorem 2.4
For \(n \ge 1\), we have
In particular, for \(n \ge 2\), we have
Replacing t by \(e^t-1\) in (2.1), we get
By (2.10), we see that
From (1.1), we note that
Therefore, by (2.11) and (2.12), we obtain the following theorem.
Theorem 2.5
For \(n \ge 0\), we have
From (1.1), we can derive the following Eq. (2.13):
On the other hand,
Therefore, by (2.13) and (2.14), we obtain the following theorem.
Theorem 2.6
For \(n \ge 0\), we have
As is known, Bernoulli polynomials are defined by the generating function
When \(x=0\), \(B_n=B_n(0)\) are Bernoulli numbers. By (2.15), we easily get
By Taylor expansion, we get
From (2.16) and (2.17), we get
By Lemma 2.1, we easily get
Therefore, by Theorem 2.2, (2.18), and (2.19), we obtain the following theorem.
Theorem 2.7
For \(m \ge 1\) and \(n \ge 0\), we have
3 Degenerate Derangement Polynomials
Here we consider the degenerate derangement polynomials which are given by
When \(x=1\), \(d_{n,\lambda }=d_{n,\lambda }(1)\) are called the degenerate derangement numbers.
From (3.1), we note that
On the other hand,
Therefore, by (3.2) and (3.3), we obtain the following theorem.
Theorem 3.1
For \(n \ge 0\), we have
Note that \(\lim _{\lambda \rightarrow 0} d_{n,\lambda }(x) = d_n(x)\), \(\lim _{\lambda \rightarrow 0}d_{n,\lambda }=d_n\), \((n \ge 0)\).
From (3.1), we note that
Comparing the coefficients on both sides of (3.4), we obtain the following theorem.
Theorem 3.2
For \(n \ge 0\), we have
In particular, for \(x=1\),
Now, we observe that
On the other hand,
Therefore, by (3.5) and (3.6), we obtain the following theorem.
Theorem 3.3
For \(n \ge 0\), we have
From Theorem 3.1, we have
where \(n \ge 2\).
Therefore, by (3.7), we obtain the following theorem.
Theorem 3.4
For \(n \ge 2\), we have
In particular, \(x=1\),
Note that
By using Taylor expansion, we get
On the other hand,
Therefore, by (3.10), we obtain the following theorem.
Theorem 3.5
For \(n\ge 1\), we have
By (1.13), we get
On the other hand,
Therefore, by (3.11) and (3.12), we obtain the following theorem.
Theorem 3.6
For \(n \ge 0\), we have
From (3.11), we note that
On the other hand,
Therefore, by (3.13) and (3.14), we obtain the following theorem.
Theorem 3.7
For \(n \ge 0\), we have
Indeed,
Thus, by (3.15), we get
Therefore, by (3.16), we obtain the following corollary.
Corollary 3.8
For \(n \ge 0\), we have
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Kim, T., Kim, D.S. (2018). Some Identities on Derangement and Degenerate Derangement Polynomials. In: Agarwal, P., Dragomir, S., Jleli, M., Samet, B. (eds) Advances in Mathematical Inequalities and Applications. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-3013-1_13
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