Abstract
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 \(d_{n}\) 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.
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1 Introduction
Let \(\mathbb{C}\) be the complex number field, and let \(\mathcal{F}\) be the set of all formal power series in the variable t with coefficients in \(\mathbb{C}\):
Let \(\mathbb{P} = \mathbb{C}[x]\), and let \(\mathbb{P}^{*}\) be the vector space of all linear functionals on \(\mathbb{P}\). We denote the action of a linear functional \(L \in\mathbb{P}^{*}\) on polynomials \(p(x) \in\mathbb{P}\) by \(\langle L\mid p(x)\rangle\), and it is known that vector space operations on \(\mathbb{P}^{*}\) are defined by
where c is a complex constant (see [3–5]).
For \(f(t)= \sum_{k=0}^{\infty}a_{k} \frac{t^{k}}{k!}\), we define a linear functional on \(\mathbb{P}\) by setting
From (1.3), we note that
where \(\delta_{n,k}\) is the Kronecker symbol.
The order \(o(f(t))\) of a power series \(f(t)(\neq0) \in\mathcal{F}\) is the smallest integer k such that the coefficients of \(t^{k}\) do not vanish. For \(f(t), g(t) \in\mathcal{F}\), with \(o(f(t))=1\) and \(o(g(t))=0\), there exists a unique sequence \(S_{n}(x)\) of polynomials such that \(\langle g(t) f(t)^{k} | S_{n}(x)\rangle = n! \delta_{n,k}\) for \(n,k \geq 0\) (see [5, 8]). The sequence \(S_{n}(x)\) is called the Sheffer sequence for \((g(t), f(t))\), which is denoted by \(S_{n}(x) \sim(g(t), f(t))\). It is known that \(S_{n}(x) \sim(g(t),f(t))\) if and only if
where \(\bar{f}(t)\) is the compositional inverse of \(f(t)\) with
For \(f(t) \in\mathcal{F}\) and \(p(x) \in\mathbb{P}\), by (1.4), we get
From (1.7), we note that
where \(p^{(k)}(x) = (\frac{d}{dx})^{k} p(x)\).
From (1.8), we easily get
Let \(S_{n}(x) \sim(g(t), f(t))\) and \(r_{n}(x) \sim(h(t), l(t))\) (\(n \geq0\)). Then we have
where
For \(u(\neq1) \in\mathbb{C}\), the Frobenius-Euler numbers are defined by the generating function
When \(u=-1\), \(H_{n}(-1)=E_{n}\) are the ordinary Euler numbers.
The Bernoulli polynomials are given by
When \(x=0\), \(B_{n} = B_{n}(0)\) are the Bernoulli numbers.
We know that the Euler polynomials are defined by
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
and the Stirling numbers of the second kind are given by
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
When \(x=1\), \(\mathrm{Bel}_{n} = \mathrm{Bel}_{n}(1)\) (\(n \geq0\)) are the Bell numbers.
From (1.19), we have
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 [1–3]). The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see [1–3]). The first few terms of the derangement number sequence \(\{d_{n}\}_{n=0}^{\infty}\) 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:
From (1.21), we note that the generating function of derangement numbers is given by
By using (1.22), it is not difficult to show that
and
For \(r \in\mathbb{N}\), the derangement numbers \(d_{n}^{(r)}\) of order r (\(n \geq0\)), are defined by the generating function
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.
2 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
and
Therefore we obtain the following lemma.
Lemma 2.1
For \(n \geq0\), we have
and
From (2.1), we have
Therefore, we obtain the following proposition.
Proposition 2.2
For \(n \geq0\), we have
with the usual convention about replacing \(d^{n}\) by \(d_{n}\).
From Proposition 2.2, we have
That is, \(d_{n}(x)\) (\(n \geq0\)) is an Appell sequence.
Now, we note that
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.
Theorem 2.3
For \(n \geq0\), we have
For \(S_{n}(x) \sim(g(t), t)\), from (1.5) we have
Thus, by (2.12), we get
In (2.13), we take \(g(t)= 1-t\), then we have
Now, we observe that
From (2.15), we note that
From (2.17), we can derive the following equation.
Theorem 2.4
For \(n \geq0\), we have
From (1.10), we have
In particular,
Comparing the coefficients on both sides of (1.17), we have
Therefore, we obtain the following corollary.
Corollary 2.5
For \(n \geq0\), we have
and
For \(r \in\mathbb{N}\), we define the derangement polynomials of order r by
When \(x=-1\), \(d_{n}^{(r)}(-1)=d_{n}^{(r)}\) are the derangement numbers of order r.
For \(0 \leq r \leq n\), the r-derangement numbers, denoted by \(D_{n}^{(r)}\), 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
We consider the r-derangement polynomials given by
From (2.24), we note that \(D_{n}^{(r)}(-1)=D_{n}^{(r)}\) are the r-derangement numbers. By (2.13) and (2.22), we easily get
and
From (2.22) and (2.24), we have
Comparing the coefficients on both sides of (2.27), we get
From (2.22), we have
Thus, by (2.29), we get
From (2.22) and (2.24), we have
Therefore, by (2.27) and (2.31), we obtain the following theorem.
Theorem 2.6
For \(n \geq r\), we have
Now, we observe that
On the other hand, by (2.24), we get
From (2.32) and (2.33), we have
In particular, for \(x=-1\), we get
Therefore, by (2.34) and (2.35), we obtain the following theorem.
Theorem 2.7
For \(n \geq0\), we have
Moreover,
By (2.22), we easily get
Comparing the coefficients on both sides of (2.36), we have
with the usual convention about replacing \((d^{(r)})^{l}\) by \(d_{l}^{(r)}\). Thus, by (2.37), we get
From (2.22), we can derive the following equation:
Thus, by (2.39), we get
For \(x=-2\), from (2.37) and (2.40) we have
From (2.17), we have
Therefore, by (2.43), we obtain the following theorem.
Theorem 2.8
For \(n \geq0\), we have
For \(n \geq0\), let
Then \(\mathbb{P}_{n}\) is an \((n+1)\)-dimensional vector space over \(\mathbb{C}\).
For \(p(x) \in\mathbb{P}_{n}\), we let
From (1.4), we have
Thus, we have
Therefore, by (2.44) and (2.46), we obtain the following theorem.
Theorem 2.9
For \(p(x) \in\mathbb{P}_{n}\), we have
where \(C_{k} = \frac{1}{k!} \langle (1-t)t^{k}|p(x)\rangle = \frac {1}{k!} \langle(1-t)|p^{(k)}(x)\rangle\).
Let us take \(p(x) = d_{n}^{(r)}(x) \in\mathbb{P}_{n}\). Then we have
where
Hence, by (2.47) and (2.48), we get
Assume that \(p(x) = \sum_{k=0}^{n} C_{k}^{(r)}d_{k}^{(r)}(x) \in\mathbb {P}_{n}\). Then, by (2.25), we get
Thus, from (2.49), we note that
Therefore, we obtain the following theorem.
Theorem 2.10
For \(n \geq0\), we have
where
Example 1
For \(p(x)=d_{n}(x)\in\mathbb{P}_{n}\), we have
where
Thus, we note that
Example 2
For \(p(x)=B_{n}(x)\) (\(n \geq0\)), we have
where
Hence
Example 3
For \(p(x)=E_{n}(x)\in\mathbb{P}_{n}\) (\(n \geq0\)), we have
where
Thus, we get
Example 4
For \(p(x)=\mathrm{Bel}_{n}(x)\in\mathbb{P}_{n}\), we have
where
Hence
The ordered Bell polynomials are defined by the generating function
When \(x=0\), \(b_{n}=b_{n}(0)\) (\(n \geq0\)) are the ordered Bell numbers. From (2.12) and (2.51), we note that \(b_{n}(x) \sim (2-e^{t},t)\) (\(n \geq0\)). For \(b_{n}(x) \sim(2-e^{t},t)\), \(d_{n}(x) \sim(1-t,t)\), by (2.7) and (2.13), we get
where
Therefore, we obtain the following theorem.
Theorem 2.11
For \(n \geq0\), we have
For \(d_{n}(x) \sim(1-t,t)\), \((x)_{n} \sim(1,e^{t}-1)\), we have
where
Therefore, by (2.54) and (2.55), we obtain the following theorem.
Theorem 2.12
For \(n \geq0\), we have
3 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.
4 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.
References
Carlitz, L.: The number of derangements of a sequence with given specification. Fibonacci Q. 16(3), 255–258 (1978)
Clarke, R.J., Sved, M.: Derangements and Bell numbers. Math. Mag. 66(5), 299–303 (1993)
Kim, D.S., Kim, T., Kwon, H.-I.: Fourier series for r-derangement and higher-order derangement functions. Adv. Stud. Contemp. Math. (Kyungshang) 28(1), 1–11 (2018)
Dere, R., Simsek, Y.: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 22(3), 433–438 (2012)
Kim, D.S., Kim, T., Ryoo, C.S.: Sheffer sequences for the powers of Sheffer pairs under umbral composition. Adv. Stud. Contemp. Math. (Kyungshang) 23(2), 275–285 (2013)
Kim, T., Kim, D.S.: On λ-Bell polynomials associated with umbral calculus. Russ. J. Math. Phys. 24(1), 69–78 (2017)
Kim, T., Kim, D.S., Jang, G.-W., Jang, L.-C.: Degenerate ordered Bell numbers and polynomials associated with umbral calculus. J. Nonlinear Sci. Appl. 10(10), 5142–5155 (2017)
Roman, S.: The Umbral Calculus. Pure and Applied Mathematics, vol. 111, Academic Press, New York (1984) x+193 pp.
Kim, T., Kim, D.S., Jang, G.-W.: Some formulas of ordered Bell numbers and polynomials arising from umbral calculus. Proc. Jangjeon Math. Soc. 20(4), 659–670 (2017)
Bayad, A., Kim, T.: Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 20(2), 247–253 (2010)
Ding, D., Yang, J.: Some identities related to the Apostol–Euler and Apostol–Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 20(1), 7–21 (2010)
He, Y., Kim, T.: General convolution identities for Apostol–Bernoulli, Euler and Genocchi polynomials. J. Nonlinear Sci. Appl. 9(6), 4780–4797 (2016)
Simsek, Y.: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. (Kyungshang) 16(2), 251–278 (2008)
Kwasniewski, A.K.: On Ψ-umbral extensions of Stirling numbers and Dobinski-like formulas. Adv. Stud. Contemp. Math. (Kyungshang) 12(1), 73–100 (2006)
Kim, T.: Identities involving Laguerre polynomials derived from umbral calculus. Russ. J. Math. Phys. 21(3), 584–589 (2014)
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1E1A1A03070882).
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Kim, T., Kim, D.S., Jang, GW. et al. A note on some identities of derangement polynomials. J Inequal Appl 2018, 40 (2018). https://doi.org/10.1186/s13660-018-1636-8
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DOI: https://doi.org/10.1186/s13660-018-1636-8