On Some Identities Related to Generalized Fibonacci and Lucas Numbers
Abstract
In this paper, we define some new matrices similar to the classical matrices introduced by Gould in [9]. We calculate the nth powers of the new matrices by diagonalizing them with the help of eigenvalues and eigenvectors. Thus, by making use of Binomial expansions, we obtain new identities containing generalized Fibonacci and Lucas numbers. These new results inform us about the relationships between matrix algebra and sequence theory, especially in the context of generalized Fibonacci and Lucas sequences.
References
Sloane, N. J. (2025). The On-Line Encyclopedia of Integer Sequences. Available online: https://oeis.org/
Lucas, E. (1878). Théorie des fonctions numériques simplement périodiques. American Journal of Mathematics, 1(4), 289-321.
Koshy, T. (2019). Fibonacci and Lucas numbers with applications, Volume 2. John Wiley & Sons. https://doi.org/10.1002/9781118742297
Silvester, J. R. (1979). Fibonacci properties by matrix methods. The Mathematical Gazette, 63(425), 188-191. https://doi.org/10.2307/3617892
Kalman, D., & Mena, R. (2003). The Fibonacci numbers-exposed. Mathematics Magazine, 76(3), 167-181. https://doi.org/10.1080/0025570X.2003.11953176
Siar, Z., & Keskin, R. (2013). Some new identities concerning generalized Fibonacci and Lucas numbers. Hacettepe Journal of Mathematics and Statistics, 42(3), 211-222.
Ribenboim, P. (2000). My numbers, my friends: Popular lectures on number theory. Springer Science & Business Media. https://doi.org/10.1007/b98892
Ballot, C. J.-C., & Williams, H. C. (2023). The Lucas sequences: Theory and applications. Springer.
Gould, H. W. (1981). A history of the Fibonacci Q-matrix and a higher-dimensional problem. Fibonacci Quarterly, 19(3), 250-257. https://doi.org/10.1080/00150517.1981.12430088
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