The Complex-type Cyclic-Pell Sequence and its Applications

Abstract

In this paper, we define the complex-type cyclic-Pell sequence and then, we give miscellaneous properties of this sequence by using matrix method. Also, we study the complex-type cyclic-Pell sequence modulo m. In addition, we describe the complex-type cyclic-Pell sequence in a 2-generator group and we investigate that in finite groups in detail. Finally, we obtain the lengths of the periods of the complex-type cyclic-Pell sequences in dihedral groups D2, D3, D4, D6, D8, D16 and D32 with respect to the generating pair (x, y).

Keywords

• The complex-type cyclic-Pell sequence
• Matrix
• Group
• Period

Kaynakça

1. Akuzum Y, Deveci O. The Hadamard-type k-step Fibonacci sequences in groups. Communications in Algebra. 48(7), 2020, 2844−2856.
2. Akuzum Y, Deveci O, Rashedi ME. The Hadamard-type k-step Pell sequences in groups. Caspian Journal of Mathematical Sciences. 11(1), 2022, 304−312.
3. Akuzum Y, Deveci O, Shannon AG. On the Pell p-circulant sequences. Notes on Number Theory and Discrete Mathematics. 23(2), 2017, 91-103.
4. Aydin H, Dikici R. General Fibonacci sequences in finite groups. Fibonacci Quarterly. 36(3), 1998, 216−221.
5. Berzsenyi G. Sums of products of generalized Fibonacci numbers. Fibonacci Quarterly. 13(4), 1975),343−344.
6. Campbell CM, Campbell PP, Doostie H, Robertson EF. On the Fibonacci length of powers of dihedral groups. In Applications of Fibonacci numbers. F. T. Howard, Ed., vol. 9, 2004, pp. 69−85, Kluwer Academic Publisher, Dordrecht, The Netherlands.
7. Campbell CM, Campbell PP. The Fibonacci lengths of binary polyhedral groups and related groups. Congressus Numerantium. 194, 2009, 95−102.
8. Campbell CM, Doostie H, Robertson EF. Fibonacci length of generating pairs in groups. In: Bergum, G. E., ed. Applications of Fibonacci Numbers. Vol. 3, 1990, pp. 27−35, Springer, Dordrecht: Kluwer Academic Publishers.

9. Deveci O, Akdeniz M, Akuzum Y. The Periods of The Pell p-Orbits of Polyhedral and Centro-Polyhedral Groups. Jordanian Journal of Mathematics and Statistics. 10(1), 2017, 1−9.
10. Deveci O, Akuzum Y, Karaduman E. The Pell-Padovan p-sequences and its applications. Utilitas Mathematica. 98, 2015, 327−347.
11. Deveci O, Karaduman E, Campbell CM. On the k-nacci sequences in finite binary polyhedral groups. Algebra Colloquium. 18(1), 2011, 945−954.
12. Deveci O, Shannon AG. The complex-type k-Fibonacci sequences and their applications. Communications in Algebra. 49(3), 2021, 1352−1367.
13. Deveci O, Shannon AG. The quaternion-Pell sequence. Communications in Algebra. 46(12), 2018, 5403−5409.
14. Doostie H, Hashemi M. Fibonacci lengths involving the Wall number K(n). Journal of Applied Mathematics and Computing. 20(1), 2006, 171−180.
15. Falcon S, Plaza A. k-Fibonacci sequences modulo m. Chaos Solitons & Fractals 41(1), 2009, 497−504.
16. Horadam AF. A generalized Fibonacci sequence. American Mathematical Monthly. 68(5), 1961, 455−459.
17. Horadam AF. Complex Fibonacci numbers and Fibonacci quaternions. American Mathematical Monthly. 70(3), 1963, 289−291.
18. Kalman D. Generalized Fibonacci numbers by matrix methods. Fibonacci Quarterly. 20(1), 1982, 73−76.
19. Karaduman E, Aydın H. k-nacci sequences in some special groups of finite order. Mathematical and Computer Modelling of Dynamical Systems. 50(1-2), 2009, 53−58.
20. Knox SW. Fibonacci sequences in finite groups. Fibonacci Quarterly. 30(2), 1992, 116−120.
21. Lu K, Wang J. k-step Fibonacci sequence modulo m. Utilitas Mathematica. 71, 2006, 169−177.
22. Ozkan, E. Truncated Lucas sequences and its period. Applied Mathematics and Computation. 232, 2014, 285−291.
23. Ozkan E, Alp T. Bigaussian Pell and Pell-Lucas Polynomials. Mathematica Montisnigri. 53(3), 2022, 17−25.
24. Tastan M, Ozkan E. On the Gauss k-Fibonacci Polynomials. Electronic Journal of Mathematical Analysis and Applications. 9(1), 2021, 124−130.
25. Wall DD. Fibonacci series modulo m, American Mathematical Monthly. 67(6), 1960, 525−532.
26. Wilcox HJ. Fibonacci sequences of period n in Groups. Fibonacci Quarterly. 24(4), 1986, 356−361.