On the Pell p-circulant sequences

Yeşim Aküzüm, Ömür Deveci, and A. G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 2, Pages 91–103
Full paper (PDF, 197 Kb)

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Authors and affiliations

Yeşim Aküzüm
Dr., Faculty of Science and Letters,
Kafkas University 36100, Turkey

Ömür Deveci
Associate Professor, Faculty of Science and Letters,
Kafkas University 36100, Turkey

A. G. Shannon
Emeritus Professor, Faculty of Engineering & IT,
University of Technology Sydney, 2007, Australia

Abstract

In this paper, we define the generalized Pell p-circulant sequence and the Pell pcirculant sequence by using the circulant matrices which are obtained from the characteristic polynomial of the generalized Pell (p,i)-sequence and then, we obtain miscellaneous properties of these sequences. Also, we consider the cyclic groups which are generated by the generating matrices and the auxiliary equations of the defined recurrence sequences and then, we study the orders of these groups. Furthermore, we extend the Pell p-circulant sequence to groups. Finally, we obtain the lengths of the periods of Pell p-circulant sequences in the semidihedral group SD_2^m for m ≥ 4 as applications of the results obtained.

Keywords

• Circulant Matrix
• Sequence
• Group
• Period

AMS Classification

• 11B50
• 20F05
• 15A36
• 20D60

References

1. Aydın, H.& Smith, G.C. (1994) Finite p-quotients of some cyclically presented groups, J. London Math. Soc., 49, 83-92.
2. Bozkurt, D. & Tin-Yau, T. (2012) Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers, Applied Mathematics and Computation, 219 (2), 544-551.
3. Campbell, C.M., P.P. Campbell, The Fibonacci lengths of binary polyhedral groups and related groups, Congr. Numer., 2009, 194: 95-102.
4. Davis, Philip J. (1979) Circulant Matrices, John Wiley, New York.
5. Deveci, O. (2015) The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in finite groups, Util. Math., 98, 257-270.
6. Deveci, O. & Akuzum, Y., (2014) The cyclic groups and the semigroups via MacWilliams and Chebyshev matrices, Journal of Math. Research, 6(2), 55-58.
7. Deveci, O. & Akuzum, Y., The recurrence sequences via Hurwitz matrices, The Scientific Annals of “Al. I. Cuza” University of Iasi, to appear.
8. Deveci, O. & Karaduman, E. (2012) The cyclic groups via the Pascal matrices and the generalized Pascal matrices, Linear Algebra and its Appl., 437, 2538-2545.
9. Deveci, O. & Karaduman, E. (2015) The Pell sequences in finite groups, Util. Math., 96, 263-276.
10. Doostie, H. & Hashemi, M. (2006) Fibonacci lengths involving the Wall number k(n), J. Appl. Math. Comput., 20, 171-180.
11. Dikici, R. & Smith, G.C. (1997) Fibonacci sequences in finite nilpotent groups, Turkish J. Math., 21, 133-142.
12. Gorenstein, D. (1980) Finite Groups, Chelsea, New York, 188-195.
13. Gray, Robert M. Toeplitz and Circulant Matrices: A Review. https://www.ee.stanford.edu./~gray/toeplitz.pdf.
14. Huppert, B. (1967) Endliche Gruppen, Springer, Heidelberg, 90-93.
15. Ingleton, A.W. (1956) The rank of circulant matrices, J. London Math. Soc., 1-31 (4), 445-460.
16. Kalman, D. (1982) Generalized Fibonacci numbers by matrix methods, The Fibonacci Quart., 20(1), 73-76.
17. Kilic, E. (2009) The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial representations, sums, Chaos, Solitons and Fractals, 40(4), 2047-2063.
18. Knox, S.W. (1992) Fibonacci sequences in finite groups, The Fibonacci Quart., 30(2), 116-120.
19. Lü, K., & Wang, J. (2007) k-step Fibonacci sequence modulo m, Util. Math., 71, 169-178.
20. Muir, T. (1911) The Theory of Determinants in Historical Order of Development, Volume 4, Macmillan, London.
21. Ozkan, E. (2014) Truncated Lucas Sequences and its period, Appl. Math. and Compt., 232, 285-291.
22. Pan, Hongyan & Jiang, Zhaolin (2015) VanderLaan circulant type matrices, Abstract and Applied Analysis, Vol.2015, Article ID 329329, http://dx.doi.org/10.1155/2015/329329, 11 pp.
23. Shannon, A.G. & Bernstein, L. (1973) The Jacobi-Perron Algorithm and the Algebra of Recursive Sequences, Bulletin of the Australian Mathematical Society, 8 (4), 261-277.
24. Stephen, B. (1990) Matrices Methods and Applications, Oxford University Press, New York.
25. Tas, S., E. Karaduman, (2014) The Padovan sequences in finite groups, Chaing Mai J. Sci., 41(2), 456-462.
26. Wall, D.D. (1960), Fibonacci series modulo m, Amer. Math. Monthly, 67, 525-532.