Puangjumpa, Piyanut Some \(k\)-Fibonacci and \(k\)-Lucas identities by a matrix approach with applications. (English) Zbl 1511.11021 Thai J. Math. 20, No. 1, 417-423 (2022). Summary: In this research, we study and find some identities involving \(k\)-Fibonacci and \(k\)-Lucas numbers by using a matrix approach. As an application of these identities we then obtain the solutions of some Diophantine equations. MSC: 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11D09 Quadratic and bilinear Diophantine equations Keywords:\(k\)-Fibonacci numbers; \(k\)-Lucas numbers; Diophantine equations; Binet’s formula Software:OEIS × Cite Format Result Cite Review PDF Full Text: Link References: [1] S. Falcon, A. Plaza, On thek-Fibonacci numbers, Chaos, Solitions & Fractals 32 (5) (2007) 1615-1624. · Zbl 1158.11306 [2] S. Falcon, A. Plaza, Thek-Fibonacci sequence and Pascal 2-triangle, Chaos, Solitions & Fractals 33 (1) (2007) 38-49. · Zbl 1152.11308 [3] N. Taskara, K. Uslu, H.H. Gulec, On the propeties of Lucas numbers with binomial coefficients, Applied Mathematics Letters 23 (1) (2010) 68-72. · Zbl 1213.11040 [4] T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley and Sons Inc., New York, 2001. · Zbl 0984.11010 [5] S. Srisawat, W. Sriprad, Some Pell and Pell-Lucas identities by matrix methods and their applications, Sciecnce and Technol. RMUTT J. 6 (1) (2016) 170-174. · Zbl 1374.11025 [6] A.F. Horadam, Basic properties of a certain generalized sequence of numbers, The Fibonacci Quarterly 3 (3) (1965) 61-76. [7] P. Filipponi, A.F. Horadam, A matrix approach to certain identities, The Fibonacci Quarterly 26 (2) (1988) 115-126. · Zbl 0643.15004 [8] N. Sloane, The On-Line Encyclopedia of Integer Sequences (OEIS). Web site:https: //oeis.org/. accessed 1 March 2018. · Zbl 1439.11001 [9] A. D. Godase, M. B. Dhakne, On the properties ofk-Fibonacci andk-Lucas numbers, International Journal of Advances in Applied Mathematics and Mechanics 2 (2014) 100-106 · Zbl 1359.11016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.