On Some identities for Generalized Fibonacci and Lucas Sequences with Rational Subscript

Saadet Arslan, Fikri Köken, Youness El Khatabi

Abstract


In this paper, we exploit general techniques from matrix theory to establish some identities for generalized Fibonacci and Lucas sequences with rationalsubscripts of the forms \(\frac{n}{2}\) and \(\frac{r}{s}\). For this purpose, we consider matrix functions \(X\mapsto X^{n/2}\) (resp. \(X\mapsto X^{r/s}\)) of two special matrices, and discuss whether the \(\frac{n}{2}\) (resp. \(\frac{r}{s}\)) are integers or irreducible fractions.

Keywords


Horadam Sequences; Generalized Fibonacci Sequences; Generalized Lucas Sequences; Matrix Functions

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References


S. Arslan and F. Köken, The Jacobsthal and Jacobsthal-Lucas numbers via square roots of matrices, Int. Math. Forum 11(11) (2016), 513 – 520.

S. Arslan and F. Köken, The Pell and Pell-Lucas numbers via square roots of matrices, Journal of Informatics and Mathematical Sciences 8(3) (2016), 159 – 166, DOI: 10.26713/jims.v8i3.393.

M. Bicknell, Fibonacci fantasy: The square root of the Q matrix, The Fib. Quart. 3(1) (1965), 67 – 71.

J. Ercolano, Matrix generators of Pell sequences, Fibonacci Quart. 17(1) (1979), 71 – 77.

F. R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea, New York (1960), https://books. google.co.in/books/about/The_theory_of_matrices.html?id=GOdQAAAAMAAJ&redir_esc=y.

H. W. Gould, A history of the Fibonacci Q-matrix and a higher-dimensional problem, Fibonacci Quart. 19 (1981), 250 – 257.

S. Halici and Z. Akyuz, Fibonacci and Lucas sequences at negative indices, Konuralp J. Math. 4(1) (2016), 172 – 178.

N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2008).

A. F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly 68 (1961), 455 – 459.

A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart. 3 (1965), 161 – 176.

A. F. Horadam, Jacobshthal and Pell curves, Fibonacci Quart. 26 (1988), 77 – 88.

E. Lucas, Theorie des fonctions numeriques simplement periodiques, Amer. J. Math. (1978), 189 – 240.

J. J. McDonald, P. Paparella and M. J. Tsatsomeros, Matrix roots of eventually positive matrices, Linear Algebra and Its Applications 456 (2014), 122 – 137, DOI: 10.1016/j.laa.2013.10.052.

R. S. Melham and A. G. Shannon, Some summation identities using generalized Q-matrices, The Fib. Quart. 33 (1995), 64 – 73, http://hdl.handle.net/10453/17915.

G. Udrea, A note on the sequence {Wn}n¸0 of A. F. Horadam, Port. Math. 53 (1996), 143 – 155.

Z. Zhang, Generalized Fibonacci sequences and generalization of the Q-matrix, Fibonacci Quart. 37 (1999), 203 – 207.




DOI: http://dx.doi.org/10.26713%2Fjims.v11i1.648

eISSN 0975-5748; pISSN 0974-875X