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

Abstract

We use a diagonal matrix for getting the Binet's formula for k-Pell sequence Also the n th power of the generating matrix for k-Pell-Lucas sequence is established and basic properties involving the determinant allow us to obtain its Cassini's identity.Mathematics Subject Classification: 11B37, 05A15, 11B83.

k-Pell, k-Pell-Lucas and modified k-Pell numbers: some identities and norms of Hankel matrices

“…Now using Proposition 4 of [2] we have exactly the same as in (15). Now we will consider the Hadamard product, denoted by •, of the previously defined matrices, that is the entrywise product, see [8].…”

“…Proposition 3 of [2] is required to obtain vi) and Proposition 4 of the same article to obtain vii).…”

On generalized K-Fibonacci sequence by two-cross-two matrix

“…where c 0 , c 1 · · · c p−1 are real constants and for detailed illustration about the generating matrix one can see [11]. If we put p = 2 in (3.1) we get a n+2 = c 1 a n+1 + c 0 a n and after that if we recall recurrence (2.1) and take c 0 = 1 and c 1 = k then the matrix associated called generating matrix is given by…”

“…In [11] author used the same concept as in [1] and studied the k-PellLucas sequences by matrix methods.…”

On some identities for k-Fibonacci sequence

“…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].…”