On some 3 × 3 dimensional matrices associated with generalized Fibonacci numbers

Halim Özdemir, Sinan Karakaya and Tuğba Petik
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 3, Pages 63–72
DOI: 10.7546/nntdm.2021.27.3.63-72
Full paper (PDF, 209 Kb)

Details

Authors and affiliations

Halim Özdemir
Department of Mathematics, University of Sakarya,
54187, Serdivan, Sakarya, Turkey

Sinan Karakaya
Department of Mathematics, University of Sakarya,
54187, Serdivan, Sakarya, Turkey

Tuğba Petik
Department of Mathematics, University of Sakarya,
54187, Serdivan, Sakarya, Turkey

Abstract

In this work, it is presented a procedure to find some 3 × 3 dimensional matrices whose integer powers can be characterized by generalized Fibonacci numbers. Moreover, some numerical examples are given to exemplify the procedure established.

Keywords

• Fibonacci numbers
• Generalized Fibonacci numbers
• Fibonacci Q-matrix
• Eigenvalue
• Eigenvector
• Matrix equation

2020 Mathematics Subject Classification

• 11B39
• 15A24

References

1. Cerda-Morales, G. (2013). On generalized Fibonacci and Lucas numbers by matrix  methods. Hacettepe Journal of Mathematics and Statistics, 42(2), 173–179.
2. Demirtürk, B. (2010). Fibonacci and Lucas sums by matrix methods. International  Mathematical Forum, 5(3), 99–107.
3. Gould, H. W. (1981). A history of the Fibonacci Q-matrix and a
   higher-dimensional problem. Fibonacci Quarterly, 19(3), 250–257.
4. Gupta, V. K., Panwar, Y. K., & Sikhwal, O. (2012). Generalized Fibonacci
   sequences, Theoretical Mathematics & Applications, 2(2), 115–124.
5. Kalman, D., & Mena, R. (2003). The Fibonacci numbers – exposed. Mathematics Magazine, 76, 167–181.
6. Karakaya, S., Özdemir, H., & Petik, T. (2018). 3 × 3 Dimensional special
   matrices associated with Fibonacci and Lucas numbers. Sakarya University Journal of Science, 22(6), 1917–1922.
7. Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications, Wiley Interscience.
8. Omotheinwa, T. O., & Ramon, S. O. (2013). Fibonacci numbers and golden ratio in  mathematics and science. International Journal of Computer and Information Technology, 2(4), 630–638.
9. Ribenboim, P. (2000). My Numbers, My Friends. Springer-Verlag New York, Inc.
10. Şiar, Z., & Keskin, R. (2013). Some new identities concerning generalized Fibonacci and Lucas numbers. Hacettepe Journal of Mathematics and Statistics, 42(3), 211–222.
11. Vajda, S. (1989). Fibonacci and Lucas Numbers, and the Golden Section. Theory and Applications. Ellis Horwood Limited.
12. Vorobiev, N. N. (2002). Fibonacci Numbers, Birkhäuser, Basel, (Russian original 1950).
13. Yordzhev, K. (2014). Factor-set of binary matrices and Fibonacci numbers. Applied Mathematics and Computation, 236, 235–238.