.svg){kind=logo}
$Q_1$ AND $Q_2$ MATRIX REPRESENTATIONS OF GENERALIZED $k$-FIBONACCI SEQUENCE
Keywords:
generalized $k$-Fibonacci sequence, Cassini’s identity, Gelin-Cesàro identity, Catalan’s identity, d’Ocagne’s identity, $Q_1$ and $Q_2$ matrices, Binet formulaDOI:
https://doi.org/10.17654/0972555522013Abstract
In this paper, we show some $Q_1$ and $Q_2$ matrix representations of generalized $k$-Fibonacci sequence by $2 \times 2$ matrix method. Also, by using Binet’s formula for generalized $k$-Fibonacci sequence, Cassini’s identity, d’Ocagne’s identity, Catalan’s identity, Gelin-Cesàro identity, and generalized identity for $k$-Fibonacci sequence by using Binet’s formula.
Received: January 5, 2022
Revised: March 8, 2022
Accepted: March 9, 2022
References
T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2011.
G. Cerda-Morales, On generalized Fibonacci and Lucas numbers by matrix methods, Hacet. J. Math. Stat. 42(2) (2013), 173-179.
A. D. Godase and M. B. Dhakne, On the properties of generalized k-Fibonacci-like sequence using matrices, Indian Journal in Number Theory 1 (2014), 35-42.
D. Kalman, Generalized Fibonacci numbers by matrix methods, Fibonacci Quart. 20(1) (1982), 73-76.
Z. Akyuz and S. Halici, Some identities deriving from the nth power of a special matrix, Advances in Difference Equations 2012(1) (2012), 223.
A. Borges, P. Catarino, A. P. Aires, P. Vasco and H. Campos, Two-by-two matrices involving k-Fibonacci and k-Lucas sequences, Appl. Math. Sci. 8(34) (2014), 1659-1666.
Henry W. Gould, A history of the Fibonacci Q-matrix and a higher-dimensional problem, Fibonacci Quart. 19(3) (1981), 250-257.
Douglas A. Lind, The Q matrix as a counterexample in group theory, Fibonacci Quart. 5 (1967), 44.
Charles H. King, Some properties of the Fibonacci numbers, Master’s Thesis, San Jose State College, June 1960.
S. L. Basin and V. E. Hoggatt, Jr., A primer for the Fibonacci sequence, Part II, Fibonacci Quart. 1(2) (1963), 61-68.
J. R. Silvester, Fibonacci properties by matrix methods, The Mathematical Gazette 63(425) (1979), 188-191.
R. A. Rosenbaum, An application of matrices to linear recursion relations, Amer. Math. Monthly 66 (1959), 792-793.
E. P. Miles, Jr., Generalized Fibonacci numbers and associated matrices, Amer. Math. Monthly 67 (1960), 745-752.
J. Ivie, A general Q-matrix, Fibonacci Quart. 10(3) (1972), 255-264.
J. L. Brenner, Lucas matrix, Amer. Math. Monthly 58 (1951), 220-221.
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Pushpa Publishing House, Prayagraj, India
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
