On Generating Matrices of the K-pell, K-pell-lucas and Modified K-pell Sequences

In this paper we define some tridiagonal matrices depending of a parameter from which we will find the k-Pell, k-Pell-Lucas and Modified k-Pell numbers.


Introduction
The sequences of Pell, Pell-Lucas and Modified Pell numbers are sequences of numbers that are defined by the recursive recurrences.For n a non-negative integer, the Pell sequence {P n } n , Pell-Lucas sequence {Q n } n and Modified Pell sequence {q n } n are given for , respectively, by the following recurrence relations with the respective initial conditions: , , ; , ; , .More detail can be found in the extensive literature dedicated to these sequences.Still, we refer some examples of papers about some their properties: [2], [3], [1], [6], [13], [7], among others.More recently, P. Catarino [8], [11] and P. Catarino and P. Vasco [9], [10] did some research about the sequences of numbers The purpose of this paper is to find the k-Pell, k-Pell-Lucas and Modified k-Pell numbers using some tridiagonal matrices We follow closely some part of what S. Falcon did in the paper [12] for k-Fibonacci numbers.

The determinant of a special kind of tridiagonal matrices
In this section we use the matrices defined by A. Feng in [4] and applied to the three types of numbers referred before and find the k-Pell, k-Pell-Lucas and Modified k-Pell numbers.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 , where , , , e are real numbers.From some properties, the determinant of a tridiagonal matrix of order can be computed from a recurrence relation.In this case, for each , if we compute the several determinants | |, we obtain that 3 Some tridiagonal matrices and the k-Pell numbers • If , and , the matrix above are transformed in the tridiagonal matrices, In this case, and taking into account Table 1, the above formulas are transformed in and ( 1) is given by, for .Then we have the following result that gives us the k-Pell number of order in terms of the determinant of a tridiagonal matrix: • Also, using the tridiagonal matrix (2.3) considered in [5] for any second order linear recurrence sequence such that with From the recurrence relation that define the k-Pell sequence and consider also the respective initial conditions, we consider and the correspondent tridiagonal -by-matrix is, in this case, and again taking into account Table 1, we obtain Taking into account Table 1, we have • Again using the tridiagonal matrix (2.3) considered in [5] from the recurrence relation that define the k-Pell-Lucas sequence and consider also the respective initial conditions, we consider and the correspondent square tridiagonal -by-matrix is, in this case, and we obtain

and then
Proposition 4: Let be the -by-tridiagonal matrix considered in (5), then the th k-Pell-Lucas number is given by| | .∎

∎ 4
in (3), then the th k-Pell number is given by| | .Some tridiagonal matrices and the k-Pell-Lucas numbers • If , , , and , the matrix above are transformed in the tridiagonal matrices, ( ) 1) is given by, If is the -by-tridiagonal matrix considered in (4), then the th k-Pell-Lucas number is given by| | .∎

Paula Catarino that
arising from these sequences: for any positive real number k, the k-Pell sequence {P k,n } n , k-Pell-Lucas sequence {Q k,n } n and Modified k-Pell sequence {q k,n } n , that are also defined by recursive recurrences. of the characteristic equation of the sequences {P k,n } n , {Q k,n } n and {q k,n } n , respectively.As a curiosity, for , we obtain that is the silver ratio which is related with the Pell number sequence.Easily, from their Binet's formula, we have that , for all , one well-know relation between the terms of the k-Pell-Lucas and Modified k-Pell sequences.From the definition of the k-Pell, k-Pell-Lucas and Modified k-Pell numbers, we present the first few values of the sequences in the following table:

Table 1 :
The first eight k-Pell, k-Pell-Lucas and Modified k-Pell numbers.