On the generating matrices of thee Κ-Fibonacci numbers

 

Sergio Falcon
Universidad de las Palmas, España

---

ABSTRACT

In this paper we define some tridiagonal matrices depending of a parameter from which we will find the k-Fibonacci numbers. And from the cofactor matrix of one of these matrices we will prove some formulas for the k-Fibonacci numbers differently to the traditional form. Finally, we will study the eigenvalues of these tridiagonal matrices.

Keyword : k-Fibonacci numbers, Cofactor matrix, Eigenvalues.

AMS : 11B39, 34L16.

---

 

REFERENCES

1 Falcon S. and Plaza A., On the Fibonacci k-numbers, Chaos, Solit. & Fract. 32 (5), pp. 1615—1624, (2007).         [ Links ]

2 Falcon S. and Plaza A., The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solit. & Fract. 33 (1), pp. 38—49, (2007).         [ Links ]

3 Falcon S. and Plaza A., The k-Fibonacci hyperbolic functions, Chaos, Solit.& Fract. 38 (2), pp. 409—420, (2008).         [ Links ]

4 Feng A., Fibonacci identities via determinant of tridiagonal matrix, Applied Mathematics and Computation, 217, pp. 5978—5981, (2011).         [ Links ]

5 Horn R. A. and Johnson C. R., Matrix Analysis, p. 506, Cambridge University Press (1991)        [ Links ]

6 Hoggat V. E. Fibonacci and Lucas numbers, Houghton—Miffin, (1969).         [ Links ]

7 Horadam A. F. A generalized Fibonacci sequence, Mathematics Magazine, 68, pp. 455—459, (1961).         [ Links ]

8 Usmani R., Inversion of a tridiagonal Jacobi matrix, Linear Algebra Appl. 212/213, pp. 413—414, (1994).         [ Links ]

9 Wikipedia, http://en.wikipedia.org/wiki/Cofactor_matrix        [ Links ]

 

Sergio Falcon
University of Las Palmas de Gran Canaria
Department of Mathematics
35017-Las Palmas de Gran Canaria
España
e-mail : sfalcon@dma.ulpgc.es

This work has been supported in part by CICYT Project number MTM2008-05866-C03-02/MTM from Ministerio de Educación y Ciencia of Spain.

Received : February 2013. Accepted : September 2013.