Some basic properties and a two-by-two matrix involving the k-Pell numbers

Abstract

In this paper we consider the k-Pell numbers sequence and present some properties involving the k-Pell numbers. The theoretical basis of using generating matrices for deriving the explicit formula for the term of order n of the k-Pell numbers sequence and also to get the well-known Cassini's identity using linear algebra.

A note involving two-by-two matrices of the k-Pell and k-Pell-Lucas sequences

“…Sometimes, in the literature, are considered other sequences namely, Pell-Lucas and Modified Pell sequences (see, for example, [17]) and also Dasdemir, in [1], consider new matrices which are based on these sequences as well as that they have the generating matrices The Pell-Lucas sequence is defined by ܳ ൌ 2ܳ ିଵ ܳ ିଶ , ݊ 2, with the initial conditions ܳ ൌ ܳ ଵ ൌ 2. Other sequences, the k-Pell and the k-Pell-Lucas are considered in Catarino [13], Catarino and Vasco [15] and [14]. For any positive real number k, the k-Pell sequence say ሼܲ , ሽ ∈ Գ is defined recurrently by ܲ ,ାଵ ൌ 2ܲ , + kܲ ,ିଵ , for ݊ 1,…”

“…with the initial conditions given by ܲ , ൌ 0, ܲ ,ଵ ൌ 1 (see [13] and [15]) and the k-Pell-Lucas ሼܳ , ሽ ∈ Գ sequence satisfies the recursive recurrence given by ܳ ,ାଵ ൌ 2ܳ , + kܳ ,ିଵ , for ݊ 1,…”

On some identities for k-Fibonacci sequence

“…So, for the ݇-Fibonacci sequence, using (8) and then (10) we know that she can be written as a power series with radius of convergence equal to…”

“…In these cases, for ݊ 1, we have, respectively, the following: ܲ ,ାଵ ൌ 2ܲ , + kܲ ,ିଵ , ܲ , ൌ 0, ܲ ,ଵ ൌ 1; ܳ ,ାଵ ൌ 2ܳ , + kܳ ,ିଵ , ܳ , ൌ ܳ ,ଵ ൌ 2; ‫ݍ‬ ,ାଵ ൌ ‫ݍ2‬ , + k‫ݍ‬ ,ିଵ , ‫ݍ‬ , = ‫ݍ‬ ,ଵ ൌ 1. For more detail about these sequences, see the works of Catarino [7], [11] and Catarino and Vasco [8], [9] and [10].…”

On generating matrices of the k-Pell, k-Pell-Lucas and modified k-Pell 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 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. In these cases, for , we have, respectively, the following recurrence relations with the respective initial conditions:…”