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Generalized k-Order Fibonacci and Lucas Quaternions

Suleyman Aydinyuz*, Mustafa Asci

Department of Mathematics, Pamukkale University, Denizli, Turkey

Abstract: In this study, we define a new interesting generalization of quaternions called as generalized k-order Fibonacci and Lucas quaternions. We give some important results with specific choices. Depending on the di and q choices, we obtain k-order Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas quaternions. For k=2, we obtain the recurrence relations of known special numbers such as Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas quaternions. By multiplying the choices we made, we can obtain the quaternion definitions for other special numbers. We give generating functions for these quaternions. Also, we identify and prove the matrix representations for generalized k-order Fibonacci and Lucas quaternions. In this way, we obtain the matrix representations for usual Fibonacci, Lucas, Pell and the other special numbers known with the di and q values we chose and give some properties about matrix representations for generalized k-order Fibonacci and Lucas quaternions.

Keywords: Fibonacci Numbers, Lucas Numbers, Generalized k-Order Fibonacci and Lucas Numbers, Quaternions, Generalized k-Order Fibonacci and Lucas Quaternions, Matrix Representations

Received March 04, 2023; Revised April 08, 2023; Accepted April 17, 2023

Citation: Suleyman Aydinyuz, Mustafa Asci, Generalized k-Order Fibonacci and Lucas Quaternions. Turkish Journal of Analysis and Number Theory, 11 (1). 1-6. January 2023. doi:10.12691/tjant-11-1-1

1. Introduction

Quaternion is a number system that expands complex figures into one real and three imaginary confines in mathematics. Irish mathematician Sir William Rowan Hamilton defined quaternion algebra in 1843 in [1] and applied it to mathematics in 3D space. Quaternions do not have the property of dicker. Although vectors and matrices have replaced quaternions in numerous operations, they are still used in theoretical and applied mathematics. Its main use is the calculation of rotational motion in 3D space. The family of quaternion computation plays an important role in mathematics and countless fields similar to algebraic systems, dispose fields or non-commutative division algebras and matrices in commutative rings and geometry. These studies are seen in [2].

Quaternion algebra is defined by (Hamilton). They are also defined as Clifford algebra classification algebra has a significant place in analysis. As a consequence of the Frobenius theorem, it is one of the four finite-dimensional quotient algebras containing the field of real numbers as subrings (the others being real numbers, complex numbers and octanions).

The definition of the quaternion family is given below:

Quaternions are the four-dimensional vector space over R having basis The multiplication table for the basis of is in Table 1 as the following:

Table 1. The multiplication table

Let be a quaternion. It shows as follows:

Quaternions consists two parts. The first part is called as a scalar part and the other part is vectorial part of . Then we can write as .

The conjugate of quaternion is defined as .

Let and be two quaternions such that;

The quaternion arithmetic is defined for and by the following:

•

•

• for

• , where and

The norm of quaternion is defined

For results on quaternion theory, one can see in [1,2,3].

Fibonacci and Lucas quaternions were defined by Horadam in 1963 in [4] and introduced the recurrence relations of quaternions in [5] in 1993. Iyer studied some properties about Fibonacci and Lucas quaternions in [6]. Halici gave Binet’s formula, generating function of Fibonacci quaternion and some properties in [7]. With a description similar to Fibonacci quaternions, Cimen and Ipek defined Pell and Pell-Lucas quaternions in [8]. Liana and Wloch defined a new kind of quaternion called Jacobsthal and Jacobsthal-Lucas in [9] and they provided interesting properties of these quaternions. Also, Polatli, Kizilates and Kesim defined a new type of quaternion on split k-Fibonacci and k-Lucas numbers in [10]. Cerda-Morales defined the Tribonacci quaternion by generalizing the Fibonacci quaternions one step in [11]. Tasci and Yalcin generalized the Fibonacci quaternions as Fibonacci p-quaternions in 2015 in [12]. Also, Tasci defined the Padovan and Pell-Padovan quaternions in [13]. Asci and Aydinyuz generalized all of these studies and defined k-order Fibonacci quaternions in [14]. They obtained Fibonacci, Tribonacci, Tetranacci and similar quaternions for special cases.

In this paper, we define a new generalization and move the Fibonacci and Lucas quaternions to order k. With this generalization, we can obtain quaternions of special numbers such as Fibonacci, Lucas, Pell and Jacobsthal depending on special cases and give the recurrence relations, generating functions and some properties of these quaternions. In the last part of this study, we employ the quaternion family to matrix theory by giving the matrix representations of the quaternions we have defined.

2. Generalized k-Order Fibonacci and Lucas Numbers

In this section, we recall the generalized k-order Fibonacci and Lucas numbers defined by Asci and Aydinyuz in [15] in 2021.

Definition 1. Asci and Aydinyuz defined the generalized order Fibonacci and Lucas numbers in [15] by the recurrence relation

with the initial conditions for,

Let’s look at what strings of numbers the relation turns into for some special choices.

• If we choose we obtain in Table 2 as follows:

Table 2. Some special cases of Vn(k)

By increasing these special selections, we can obtain other special numbers.

• For and the Tribonacci sequence is obtained.

• For and the order Fibonacci sequence is obtained.

• For and ; the order Lucas sequence is obtained.

• For and ; the order Pell sequence is obtained.

• For and ; the order Pell-Lucas sequence is obtained.

• For and the k-order Jacobsthal sequence is obtained.

• For and ; the order Jacobsthal-Luas sequence is obtained.

3. Generalized k-Order Fibonacci and Lucas Quaternions

In this section, we firstly define the generalized k-order Fibonacci and Lucas quaternions. Also, we give the generating functions of these quaternions and obtain some interesting properties. Finally, we identify and prove the matrix representations.

Definition 2. The nth generalized k-order Fibonacci and Lucas quaternions are defined as

[picture] | (3.1)

where is generalized k-order Fibonacci and Lucas numbers.

Corollary 1. For, we obtain these quaternions in Table 3 as follows:

Table 3. Some special cases of QVn(k)

Corollary 2. If we choose and , the Tribonacci quaternion is obtained in [11].

Corollary 3. If we choose and , the k- order Fibonacci quaternions are obtained in [14].

Quaternion definitions of the other special numbers can be reached by making similar choices.

Definition 3. The conjugate of generalized k-order Fibonacci and Lucas quaternions is defined by

Definition 4. The norm of generalized k-order Fibonacci and Lucas quaternions is defined by

Proposition 1. For and , the following properties are obtained:

1.

2.

3.

4. .

Theorem 1. The recurrence relation of the generalized k-order Fibonacci and Lucas quaternions is given by

Proof: From the definition of generalized k-order Fibonacci and Lucas numbers and quaternions, we obtain as

Theorem 2. The generating function for the generalized k-order Fibonacci and Lucas quaternions is defined as

[picture] | (3.2)

Proof: We can give the following proof, with being the generating function of the generalized k-order Fibonacci and Lucas quaternions

By doing the necessary operations, we can obtain the generating function as follows:

Corollary 4. Let’s consider the generating functions with special choices given in (3.2) as follows:

1. For and , the generating function of k-order Fibonacci quaternions is obtained in [14] as

2. For and , the generating function of order Pell quaternions is obtained as

3. For and , the generating function of order Jacobsthal quaternions is obtained as

By changing our choices for and we can get the generating functions of other special numbers.

Corollary 5. Some special cases of generating functions are obtained given in (3.2) for in Table 4 as follows:

Table 4. Generating Functions

Now let’s identify the matrix of the generalized k-order Fibonacci and Lucas quaternions. We introduce the matrices and Let and determine as

and

Lemma 1. Let Then, we get as

Theorem 3. For , we get as

[picture] | (3.3)

Proof: The proof is provided by the induction method on If we get as

Suppose that it is true for

Then for, we get the proof of the theorem as

Corollary 6. For

1. The matrix representation of the Horadam Quaternions is obtained in [5] as follows:

2. For and the matrix representation of the Fibonacci quaternions is obtained in [7] as follows:

3. For and the matrix representation of the Lucas quaternions is obtained in [6] as follows:

4. For and the matrix representation of the Pell quaternions is obtained in [8] as follows:

5. For and the matrix representation of the Pell-Lucas quaternions is obtained in [8] as follows:

6. For and the matrix representation of the Jacobsthal quaternions is obtained in [9] as follows:

7. For and the matrix representation of the Jacobsthal-Lucas quaternions is obtained in [9] as follows:

Corollary 7. For and the matrix representation of the Tribonacci quaternions is obtained in [11] as follows:

Corollary 8. For and ; the matrix representation of the k-order Fibonacci quaternions are shown in [14].

We can obtain matrix representations of other special quaternions such as k-order Pell, Jacobsthal by making similar choices.

Theorem 4. Let be integer. Then, we get as

Theorem 5. Let m and n be an integer. Then, we get as

where is the generalized k-order Fibonacci and Lucas numbers.

Proof: For , the matrix is defined in [15] as

If we use (3.3), we get as follows

and

Then, we have

If the equality of matrices is used, we get for

4. Conclusion

In this study, we defined the generalized k-order Fibonacci and Lucas quaternion family by making a new generalization. Depending on the di and q choices, we gained k-order Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas quaternions. For, k=2, we obtained the recurrence relations of known special numbers as Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas quaternions. By multiplying the choices we have made, we obtained the quaternion definitions for other special numbers. We gave the generating functions for these quaternions and we obtained the generating functions of special numbers. Also, we identified and proved the matrix representations for generalized k-order Fibonacci and Lucas quaternions. In this way, we obtained the matrix representations for usual Fibonacci, Lucas, Pell and the other special numbers known with the diand q values we have chosen and gave some properties about matrix representations for generalized k-order Fibonacci and Lucas quaternions.

References

[1]   | Hamilton, W. R., (1866). Elements of Quaternions Longmans. Green and Co., London.
[2]   | Gürlebeck, K., & Sprössig, W. (1997). Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York.
[3]   | Lounesto, P. (2001). Clifford algebras and spinors. Cambridge Univ Press.
[4]   | Horadam, A. F. (1963). Complex Fibonacci Numbers and Fibonacci Quaternions. American Math. Monthly, 70, 289-291.
[5]   | Horadam, A. F. (1993). Quaternion Recurrence Relations. Ulam Quarterly, 2, 23-33.
[6]   | Iyer, M. R. (1969). A Note on Fibonacci Quaternions. The Fibonacci Quarterly, 3, 225-229.
[7]   | Halici, S., (2012). On Fibonacci Quaternions. Adv. Appl. Clifford Algebras, 22, 321-327.
[8]   | Çimen, B. C., & Ipek, A. (2016). On Pell Quaternions and Pell-Lucas Quaternions. Adv. Appl. Clifford Algebras, 26 (1), 39-51.
[9]   | Szynal-Liana, & A., Wloch, I. (2016). A Note on Jacobsthal Quaternions. Adv. Appl. Clifford Algebras, 26, 441-447.
[10]   | Polatli, E., Kizilates, & C., Kesim, S. (2016). On Split k-Fibonacci and k-Lucas quaternions. Adv. Appl. Clifford Algebras, 26, 353-362.
[11]   | Cerda-Morales, & G. (2017). On a generalization for Tribonacci Quaternions. Mediterr. J. Math., 14: 239-251.
[12]   | Tasci, D., & Yalcin, F. (2015). Fibonacci p-quaternions. Adv. Appl. Clifford Algebras, 25, 245-254.
[13]   | Tasci, D. (2018). Padovan and Pell-Padovan Quaternions. Journal of Science and Arts, No. I(42), 125-132.
[14]   | Asci, M., & Aydinyuz, S. (2021). k-order Fibonacci Quaternions. Journal of Science and Arts, No. 1(54), 29-38.
[15]   | Asci, M., & Aydinyuz, S. (2021). Generalized k-order Fibonacci and Lucas Hybrid Numbers. Journal of Information and Optimization Sciences, 42:8, 1765-1782.