Modified k-Pell sequence: some identities and ordinary generating function

Abstract

Some identities of the k-Pell-Lucas sequence and their relationship to the Modified k-Pell sequence allow us to obtain some identities for Modified k-Pell numbers. Also the ordinary generating function for the Modified k-Pell sequence and another expression for the general term of the sequence, using the ordinary generating function, is provided.

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].…”

Incomplete k-Pell, k-Pell-Lucas and Modified k-Pell Numbers

“…Some identities related with these sequences involving sums formulae for products involving its terms have been studied by several authors (see, for example, [17], among others). Catarino in [5] studied a generalization of Pell sequence, as well as in [6] and in [7], where Catarino and Vasco studied generalizations of Pell-Lucas and Modied Pell sequences, respectively. Also in [30], the authors studied some identities and norms of Hankel matrices whose entries are elements of these sequences, as well as in [31], we can nd some properties involving sums of products with its terms.…”

On some identities of k-Jacobsthal-Lucas numbers

“…More recently, some of these sequences were generalized for any positive real number k: the study of the k-Fibonacci sequence, the k-Lucas sequence, the k-Pell sequence, the k-Pell-Lucas sequence,the Modified k-Pell sequence and the k-Jacobhstal sequence appeared (see [1], [11], [2], [4], [5], [6] and [3]).…”