On the generating matrices of the Κ-Fibonacci numbers

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.

On Generalized Jacobsthal and Jacobsthal-Lucas polynomials

“…Horadam in [11] present the ordinary generating function for sequences (2) and (3). Djordjević in [6] present the ordinary generating function for polynomials (6) and (9). In [13] we can find the ordinary generating function for sequence (4) and in [2] we have the ordinary generating function for sequence (5).…”

“…Following the ideas of [9], we know that the determinant of a special kind of tridiagonal matrices is related to a special nth order polynomial. If we consider the (n × n) tridiagonal matrices M n , defined as:…”

“…Note that in the particular case where h(x) = 2x, (7) and (8) reduces to (6), (10) and (11) reduces to (9). The paper is organized as follows: in Section 2 we present the generating functions for polynomials (7) and (8) and also (10) and (11).…”

On generating matrices of the k-Pell, k-Pell-Lucas and modified k-Pell sequences

“…We consider tridiagonal matrices in a similar way that Falcon did in [12]. In linear algebra a tridiagonal matrix is a matrix that has nonzero elements only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal.…”

“…A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. Let us consider the square matrix of order , denoted by , and defined (as in Falcon [12]) by…”