Abstract
In this paper, we give the exponential generating functions for the generalized Fibonacci and generalized Lucas quaternions, respectively. Moreover, we give some new formulas for binomial sums of these quaternions by using their Binet forms.
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1 Introduction
The well-known Fibonacci and Lucas sequences are defined by the following recurrence relations: for \(n\geq0\),
and
where \(F_{0}=0\), \(F_{1}=1\), \(L_{0}=2\), and \(L_{1}=1\), respectively. Here, \(F_{n}\) is the nth Fibonacci number and \(L_{n}\) is the nth Lucas number.
The Binet formulas for the Fibonacci and Lucas sequences are given by
and
where
The numerous relations connecting Fibonacci and Lucas numbers have been given by Vajda [1].
Carlitz, Ferns, Layman and Hoggatt studied many properties of binomial sums of these numbers (see [2–5]).
The generalized Fibonacci and Lucas sequences, \(\{U_{n}\}\) and \(\{ V_{n}\}\), are defined by the following recurrence relations: for \(n\geq0\) and any nonzero integer p,
and
where \(U_{0}=0\), \(U_{1}=1\), \(V_{0}=2\), and \(V_{1}=p\), respectively. If we take \(p=1\), then \(U_{n}=F_{n}\) (nth Fibonacci number) and \(V_{n}=L_{n}\) (nth Lucas number).
Let λ and μ be the roots of the characteristic equation \(x^{2}-px-1=0\). Then the Binet formulas for the sequences \(\{U_{n}\}\) and \(\{V_{n}\}\) are given by
and
where \(\lambda= ( p+\sqrt{p^{2}+4} ) /2\) and \(\mu= ( p-\sqrt{p^{2}+4} ) /2\).
From these equalities, by putting \(\Delta=p^{2}+4\), we see that \(1+\lambda ^{2}=\lambda\sqrt{\Delta}\) and \(1+\mu^{2}=-\mu\sqrt{\Delta}\).
The generating function and the exponential generating function of a sequence \(\{ a_{n} \} \) are defined by
and
respectively.
Now, we would like to explain how the generating function of a given sequence \(\{ a_{n} \} \) is derived: Firstly, we write down a recurrence relation which is a single equation expressing \(a_{n}\) in terms of other elements of the sequence. This equation should be valid for all integers n, assuming that \(a_{-1}=a_{-2}=\cdots=0\). Then we multiply both sides of the equation by \(x^{n}\) and sum over all n. This gives, on the left, the sum \(\sum_{n=0}^{\infty}a_{n}x^{n}\), which is the generating function \(a(x)\). The right-hand side should be manipulated so that it becomes some other expression involving \(a(x)\). Lastly, if we solve the resulting equation, we get a closed form for \(a(x)\). For more details and properties related to the generating functions and special functions, we refer to [6–12].
A quaternion p with real components \(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\), and basis 1, i, j, k is an element of the form
where
Throughout this paper, we work in the real division quaternion algebra without specific references being given.
The nth Fibonacci and the nth Lucas quaternions were defined by Horadam in [13] as
and
respectively, where \(F_{n}\) and \(L_{n}\) are the nth Fibonacci number and the nth Lucas number.
Similar to Horadam, the nth generalized Fibonacci quaternion and nth generalized Lucas quaternion are defined by
and
respectively [14]. Here, \(U_{n}\) and \(V_{n}\) are the nth generalized Fibonacci number and the nth generalized Lucas number.
For \(n\geq0\), the following recurrence relations hold:
and
where
In [15], Iakin gave the Binet formulas for the generalized Fibonacci and generalized Lucas quaternions as follows:
and
where \(\underline{\lambda}=1+\lambda \mathbf {i}+\lambda ^{2}\mathbf {j}+\lambda^{3}\mathbf {k}\) and \(\underline{\mu}=1+\mu \mathbf {i}+\mu^{2}\mathbf {j}+\mu^{3}\mathbf {k}\).
Iyer [16] gave some relations connecting the Fibonacci and Lucas quaternions. In [17], Swamy derived the relations of generalized Fibonacci quaternions. Iakin [18, 19] introduced the concept of a higher order quaternion and generalized quaternions with quaternion components. Horadam [20] studied the quaternion recurrence relations. In [21], Halici derived generating functions and many identities for Fibonacci and Lucas quaternions. In [22], Flaut and Shpakivskyi investigated some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions. Very recently, Ramirez [14] has obtained some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions.
Inspired by these results, in the present paper we give the exponential generating functions for the generalized Fibonacci and generalized Lucas quaternions. Moreover, we derive some new formulas for binomial sums of these quaternions by using their Binet forms.
2 Some properties of generalized Fibonacci and Lucas quaternions
Theorem 2.1
([14])
The generating functions for the generalized Fibonacci and generalized Lucas quaternions are
and
respectively.
Proof
Let \(F(t)=\sum_{n=0}^{\infty}\mathcal{Q}_{n}t^{n}\) and \(L(t)=\sum_{n=0}^{\infty}\mathcal{K}_{n}t^{n}\). Then we get the following equation:
Since, for each \(n\geq2\), the coefficient of \(t^{n}\) is zero in the right-hand side of this equation, we obtain
Similarly, we obtain
□
Theorem 2.2
The exponential generating functions for the generalized Fibonacci and generalized Lucas quaternions are
and
Proof
By the Binet formula for the generalized Fibonacci quaternions, we get
Similarly, by the Binet formula for the generalized Lucas quaternions we obtain
□
Now, we give the following theorems in which the first formulas are proved, and the remaining formulas can be obtained similarly.
Theorem 2.3
For \(n,k\geq0\), we have
Proof
If we use the Binet formula for the generalized Fibonacci quaternions, we get
□
Theorem 2.4
([14])
For \(n\geq0\), we have
Proof
By the Binet formula for the generalized Fibonacci quaternions, we have
□
Theorem 2.5
For \(n\geq0\), we have
Proof
By Theorem 2.3, we obtain
□
Theorem 2.6
For \(n\geq0\), we have
Proof
From the quaternion multiplication, we have
If we consider the Binet formula for the generalized Fibonacci quaternions, we have
□
Note that, by taking \(p=1\) in Theorem 2.6, we obtain
and
where \(Q_{i}\) and \(K_{i}\) are the ith Fibonacci quaternion and the ith Lucas quaternion, respectively.
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The authors would like to thank the referees for their valuable suggestions and comments.
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Polatli, E., Kesim, S. On quaternions with generalized Fibonacci and Lucas number components. Adv Differ Equ 2015, 169 (2015). https://doi.org/10.1186/s13662-015-0511-x
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DOI: https://doi.org/10.1186/s13662-015-0511-x