The quaternion-type cyclic-Fibonacci sequences in groups

Nazmiye Yilmaz, Esra Kırmızı Çetinalp and Ömür Deveci
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 2, Pages 226–240
DOI: 10.7546/nntdm.2023.29.2.226-240
Full paper (PDF, 267 Kb)

Details

Authors and affiliations

Nazmiye Yilmaz
University of Karamanoğlu Mehmetbey, Kamil Özdağ Science Faculty,
Department of Mathematics, Turkey

Esra Kırmızı Çetinalp
University of Karamanoğlu Mehmetbey, Kamil Özdağ Science Faculty,
Department of Mathematics, Turkey

Ömür Deveci
Kafkas University, Faculty of Science and Letter, Department of Mathematics,
36100 Kars, Turkey

Abstract

In this paper, we define the six different quaternion-type cyclic-Fibonacci sequences and present some properties, such as, the Cassini formula and generating function. Then, we study quaternion-type cyclic-Fibonacci sequences modulo . Also we present the relationships between the lengths of periods of the quaternion-type cyclic-Fibonacci sequences of the first, second, third, fourth, fifth and sixth kinds modulo m and the generating matrices of these sequences. Finally, we introduce the quaternion-type cyclic-Fibonacci sequences in finite groups. We calculate the lengths of periods for these sequences of the generalized quaternion groups and obtain quaternion-type cyclic-Fibonacci orbits of the quaternion groups Q₈ and Q₁₆ as applications of the results.

Keywords

• Group
• Period
• Presentation
• Quaternion Fibonacci sequence

2020 Mathematics Subject Classification

• 11C20
• 11B39
• 11B50
• 20F05
• 20G20

References

1. Akuzum, Y. (2021). The complex-type Pell 𝑝-numbers in finite groups. Turkish Journal of Science, 6(3), 142–147.
2. Atanassov, K. (1989). On a generalization of the Fibonacci sequence in the case of three sequences. The Fibonacci Quarterly, 27(1), 7–10.
3. Aydin, H., & Dikici, R. (1998). General Fibonacci sequences in finite groups. The Fibonacci Quarterly, 36(3), 216–221.
4. Bernardini, M., Marques, D., & Trojovský, P. (2021). On two-generator Fibonacci numerical semigroups with a prescribed genus. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas, 115(3), 1–18.
5. Campbell, C. M., Campbell, P. P., Doostie, H., & Robertson, E. F. (2004). On the Fibonacci length of powers of dihedral groups. In: Howard, F. T. (Ed.). Applications of Fibonacci Numbers. Vol. 9, 69–85. Dordrecht: Kluwer Academic Publisher.
6. Campbell, C. M., Doostie, H., & Robertson, E. F. (1990). Fibonacci length of generating pairs in groups. In: Bergum, G. E. (Ed.). Applications of Fibonacci Numbers, Vol. 3, 27–35.  Dordrecht: Kluwer Academic Publishers.
7. Deveci, Ö. (2018). The Padovan-circulant sequences and their applications. Mathematical Reports, 20.70(4), 401–416.
8. Deveci, Ö., Erdag, O., & Gungoz, U. (2023). The complex-type cyclic-Fibonacci sequence and its applications. Journal of Mahani Mathematical Research, 12(2), 235–246.
9. Deveci, Ö., Karaduman, E., & Campbell, C. M. (2011). On the 𝑘-nacci sequences in finite binary polyhedral groups. Algebra Colloquium, 18(1), 945–954.
10. Deveci, Ö., Karaduman, E., & Campbell, C. M. (2017). The Fibonacci-circulant sequences and their applications. Iranian Journal of Science and Technology, Transaction A: Science, 41(4), 1033–1038.
11. Deveci, Ö., & Shannon, A. G. (2018). The quaternion-Pell sequence. Communications in Algebra, 46(12), 5403–5409.
12. Dikici, R., & Özkan, E., (2003). An application of Fibonacci sequences in groups. Applied Mathematics and Computation, 136(2), 323–331.
13. Doostie, H., & Hashemi, M. (2006). Fibonacci lengths involving the Wall number 𝐾(𝑛). Journal of Applied Mathematics and Computing, 20(1–2), 171–180.
14. Falcón, S., & Plaza, A. (2009). 𝑘-Fibonacci sequences modulo 𝑚. Chaos, Solitons & Fractals, 41(1), 497–504.
15. Halici, S. (2012). On Fibonacci quaternions. Advances in Applied Clifford Algebras, 22(2), 321–327.
16. Hamilton, W. R. (1844). On quaternions or on a new system of imaginaries in algebra. Philosophical Magazine, 25(3), 489–495.
17. Horadam, A. F. (1961). A generalized Fibonacci sequence. The American Mathematical Monthly, 68(5), 455–459.
18. Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70(3), 289–291.
19. Iyer, M. R. (1969). Some results on Fibonacci quaternions. The Fibonacci Quarterly, 7(2), 201–210.
20. Kalman, D. (1982). Generalized Fibonacci numbers by matrix methods. The Fibonacci Quarterly, 20(1), 73–76.
21. Knox, S. W. (1992). Fibonacci sequences in finite groups. The Fibonacci Quarterly, 30(2), 116–120.
22. Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. New York: John Wiley & Sons.
23. Lancaster, P., & Tismenetsky, M. (1985). The Theory of Matrices: With Applications. Elsevier.
24. Lu, K., & Wang, J. (2006). 𝑘-step Fibonacci sequence modulo 𝑚. Utilitas Mathematica, 71, 169–177.
25. Özkan, E. (2003). On general Fibonacci sequences in groups. Turkish Journal of
    Mathematics, 27(4), 525–537.
26. Stakhov, A., & Rozin, B. (2006). Theory of Binet formulas for Fibonacci and Lucas 𝑝-numbers. Chaos, Solitons & Fractals, 27(5), 1162–1177.
27. Van der Waerden, B. L. (1976). Hamilton’s discovery of quaternions. Mathematics Magazine, 49(5): 227-234.
28. Wall, D. D. (1960). Fibonacci series modulo 𝑚. The American Mathematical Monthly, 67(6), 525–532.
29. Wilcox, H. J. (1986). Fibonacci sequences of period 𝑛 in groups. The Fibonacci Quarterly, 24(4), 356–361.

Manuscript history

• Received: 7 January 2023
• Revised: 4 April 2023
• Accepted: 24 April 2023
• Online First: 26 April 2023

Copyright information

Ⓒ 2023 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).