On Characterization of Being a Matrix Q (k) g(a2,b2) of Linear Combinations of a Matrix Q (n) g(a1,b1) and a Matrix Q m

It is given a characterization of all solution of the matrix equation $c_{1}Q_{g(a_{1}, b_{1})}^{(n)}+c_{2}Q^{m}=Q_{g(a_{2}, b_{2})}^{(k)}$ with unknowns $c_{1}, c_{2} \in \mathbb{C}^{*}$. Here the matrix $Q_{g(a, b)}^{(l)}$, called an $l$-generalized Fibonacci $Q$-matrix, is defined by means of the Fibonacci $Q$-matrix, where $l$ is an integer, and $a, b \in \mathbb{R}^{*}$.

Keywords:

Fibonacci Numbers, Fibonacci $Q$-matrix Generalized Fibonacci Numbers, Linear Combination, Matrix Equations,

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[1] A. Horadam, A Generalized Fibonacci Sequence, The American Mathematical Monthly, 68(5) (1961), pp. 455-459.
[2] B. Demirtürk, Fibonacci and Lucas Sums with Matrix Method, International Mathematical Forum, 5(3) (2010), pp. 99-107.
[3] C.H. King, Some Properties of the Fibonacci Numbers, Master’s Thesis, San Jose State College, June, (1960).
[4] G. Cerda-Morales, On Generalized Fibonacci and Lucas Numbers by Matrix Methods, Hacettepe Journal of Mathematics and Statistics, 42(2) (2013), pp. 173-179.
[5] H.W. Gould, A History of the Fibonacci Q-Matrix and a Higher-Dimensional Problem, The Fibonacci Quarterly, 19(3) (1981), pp. 250-257.
[6] H. Özdemir, S. Karakaya, and T. Petik, On Characterization of Some Linear Combinations Involving the Matrices Q and R, Honam Mathematical Journal, 42(2) (2020), pp. 235-249.
[7] J.R. Silvester, Fibonacci properties by matrix methods, Mathematical Gazette, 63(425) (1979), pp. 188-191.
[8] A.M. Meinke, Fibonacci Numbers and Associated Matrices, Master’s Thesis, Kent States University, (2011).
[9] R. Euler and J. Sadek, Elementary Problems and Solutions, Fibonacci Quarterly, 41(2) (2003), pp. 181-186.
[10] R. P. Grimaldi, Fibonacci and Catalan Numbers an Introduction, John Wiley and Sons, (2012).
[11] R. Keskin, B. Demirtürk, Some new Fibonacci and Lucas identities by matrix methods, Internat. J. Math. Ed. Sci. Tech. 41(3) (2010), pp. 379-387.
[12] S. Sinha, The Fibonacci Numbers and Its Amazing Applications, International Journal of Engineering Science Invention, 6(9) (2017), pp. 07-14.
[13] S.L. Basin, V.E. Hoggatt JR., A Primer on the Fibonacci Sequence Part I, The Fibonacci Quarterly, 1(1) (1963), pp. 65-72.
[14] S. Karakaya, H. Özdemir, and T. Petik, $3\times3$ Dimensional Special Matrices associated with Fibonacci and Lucas numbers, Sakarya University Journal of Science 22(6) (2018), pp. 1917-1922.
[15] S. Karakaya, On the Linear Combinations of Matrices associated with Fibonacci and Lucas numbers, Master’s Thesis, (2018).
[16] S. Uçar, N. Taş, and N. Yılmaz Özgür, A New Application to Coding Theory via Fibonacci and Lucas Numbers, Mathematical Sciences and Applications E-Notes, 7(1) (2019), pp. 62-70.
[17] T. Koshy, Fibonacci and Lucas Numbers With Applications, John Wiley and Sons, (2001).
[18] T. Koshy, Fibonacci and Lucas Identities, Fibonacci and Lucas Numbers With Applications, John Wiley & Sons, (2018).
[19] V.E. Hoggatt JR., I.D. Ruggles, A Primer for the Fibonacci numbers-Part IV, Fibonacci Quarterly, 1(4) (1963), pp. 65-71.
[20] V.E. Hoggatt, Fibonacci and Lucas Numbers, The Fibonacci Association, University of Santa Clara, (1969).
[21] V. Gupta, Y.K. Panwar, and O. Sikhwal, Generalized Fibonacci Sequences, Theoretical Mathematics & Applications, 2(2), (2012), pp. 115-124.
[22] Y.K. Panwar, B. Singh, V.K. Gupta, Generalized Fibonacci Sequences and its Properties, Palestine Journal of Mathematics, 3(1) (2014), pp. 141-147.
[23] Z. Akyüz, S. Halıcı, On Some Combinatorial Identities Involving the terms of Generalized Fibonacci and Lucas Sequences, Hacettepe Journal of Mathematics and Statistics, 42(4) (2013), pp. 431-435.
[24] Z. Şiar, R. Keskin, Some New Identities Concerning Generalized Fibonacci and Lucas Numbers, Hacettepe Journal of Mathematics and Statistics, 42(3) (2013), pp. 211-222.

Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2013
Yayıncı: Mehmet Zeki SARIKAYA

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