On some identities and generating functions for k-Pell-Lucas sequence

Abstract

We obtain the Binet's formula for k-Pell-Lucas numbers and as a consequence we obtain some properties for k-Pell-Lucas numbers. Also we give the generating function for the k-Pell-Lucas sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided.Mathematics Subject Classification: 11B37, 05A15, 11B83.

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 ܳ , ൌ 2, ܳ ,ଵ ൌ 2 (see the work of Catarino and Vasco, in [14]). The Binet's formula is also well known for several of these sequences.…”

On some identities for k-Fibonacci sequence

“…The d'Ocagne's identity can also obtained using the Binet's formula as it was done in Jhala et al, in [3] for the k-Jacobsthal sequence, Catarino in [7] for the k-Pell numbers and Catarino and Vasco in [9], [10] for k-Pell-Lucas and Modified k-Pell numbers, respectively. Hence we have…”

“…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].…”

Fibonacci-Like sequence associated with k-Pell, k-Pell-Lucas and Modified k-Pell sequences

“…In [5][6][7] the authors delineated Binet's formula, generating and some other identities for Modified k-Pell numbers, k-Pell-Lucas numbers and k-Pell numbers respectively. In [8,9] the authors acquainted identities of generalized Fibonacci sequences with Jacobsthal and Jacobsthal-Lucas sequences.…”