Generalized k-Order Fibonacci and Lucas Quaternions

Suleyman Aydinyuz^1, and Mustafa Asci¹

¹Department of Mathematics, Pamukkale University, Denizli, Turkey

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

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 d_i 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 d_i 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

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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.