On generalized K-Fibonacci sequence by two-cross-two matrix

  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract

    In this study we define a new generalized k-Fibonacci sequence associated with its two cross two matrix called generating matrix. After use the matrix representation we find many interesting properties such as nth power of the matrix, Cassini's Identity of generalized k-Fibonacci sequence as well as Binet's formula for generalized k-Fibonacci sequence by the method of matrix diagonalization.

  • Keywords

    k-Fibonacci Sequence; Generalized k-Fibonacci Sequence; Binet’s Formula; Diagonalization of a Matrix.

  • References

      [1] A. Borges, P. Catarino, A. P. Aires, P. Vasco and H. Campos. “Two-by-Two Matrices Involving k-Fibonacci and k-Lucas Sequences", Applied Mathematical Sciences, 8(34):1659-1666, 2014.

      [2] A. Dasdemir. “On the Pell, Pell-Lucas and Modified Pell Numbers By Matrix Method", Applied Mathematical Sciences, 5(64):3173-3181, 2011.

      [3] A. F. Horadam. Generalized Fibonacci Sequences”, The American Mathematical Monthly, 68(5):455-459, 1982.

      [4] A. Wloch. “Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers", Applied Mathematics and Computation, 219:5564-5568, 2013.

      [5] C. Bolat. “On the Properties of k-Fibonacci numbers", Int. J. contemp. Math. Sciences, 5(22):1097-1105, 2010.

      [6] C. K. Ho. and C. Y. Chong. “Odd and Even Sums of Generalized Fibonacci Numbers byMatrix Methods", AIP Conference Proceedings, 1602, 1026 (2014); doi: 10.1063/1.4882610

      [7] D. Kalman. “Generalized Fibonacci numbers by Matrix Methods”, The Fibonacci Quarterly, 20(1):73-76, 1982.

      [8] G. C. Morales.“On Generalized Fibonacci and lucas Numbers by Matrix Methods", Hacettepe Journal of Mathematics and Statistics, 42(2):173-179, 2013.

      [9] http://mathworld.wolfram.com/CramersRule.html.

      [10] N. N. Vorobyov. “The Fibonacci Numbers", D. C. Health and company Boston, 1963.

      [11] P. Catarino. “A Note Involving Two-by-Two Matrices of the k-Pell and k-Pell-Lucas Sequences", International Mathematical Forum}, 8(32):1561-1568, 2013.

      [12] P. Catarino. “On Some Identities for k-Fibonacci Sequence", Int. J. contemp. Math. Sciences, 9(1):37-42, 2014.

      [13] S. Falcon and A. Plaza. “On k-Fibonacci numbers of arithmetic indexes", Applied Mathematics and Computation, 208:180--185, 2009.

      [14] S. Falcon. “On the k-Lucas Numbers", Int. J. contemp. Math. Sciences, 6(21):1039--1050, 2011.

      [15] S. Falcon. “Generalized (k, r) Fibonacci Numbers", Gen. Math. Notes, 25(2):148--158, 2014.

      [16] S. Falcon and A. Plaza. “On the Fibonacci k-numbers", Chaos, Solitons and Fractals, 32:1615-1624, 2007.

      [17] S. Vajda. “Fibonacci and Lucas Numbers and the Golden Section. Theory and Applications", Ellis Horwood Limited, 1989.

      [18] T. Koshy. “Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, New york, 2001.

      [19] V. E. Hoggatt. “Fibonacci and Lucas Numbers", Houghton-Mifflin, Co. Boston, 1969.

      [20] Z. Akyuz and S. Halici. “Some identities deriving from the nth power of a special matrix", Advances in Difference Equations, DOI: 10.1186/1687-1847-2012-223.




Article ID: 6949
DOI: 10.14419/gjma.v5i1.6949

Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.