On the k- Pell Quaternions and the k- Pell-Lucas Quaternions

Abstract

The quaternions form a four-dimensional associative and non-commutative algebra over the set of real numbers. In this paper, firstly, we give some relations for k-Pell quaternions and k-Pell-Lucas quaternions. Then, by using Binet's formula, we obtain their sums formulas, their the identities such as Cassini's identity and generating function, also derive relationships between these quaternions.

On Bi-periodic Jacobsthal and Jacobsthal-Lucas Quaternions

“…The study of the quaternions of the sequences began with the earlier work of Horadam (Horadam, 1993) where the Fibonacci quaternion was investigated. Several authors worked on different quaternions and their generalizations (Gül, 2018;Halıcı, 2012;Szynal-Liana & Wloch, 2016;Tan, Yılmaz & Sahin, 2016a;Tan, Yılmaz & Sahin, 2016b;Tascı, 2017).…”

Dual-complex k-Pell quaternions

“…First the idea to consider Pell quaternions was suggested by Horadam in paper [13]. In the literature, the reader can find Pell quaternions and studies on their properties in [1,2,4,8,9,11,18,19].…”

Dual-complex k-Pell quaternions

“…First the idea to consider Pell quaternions it was suggested by Horadam in paper [11]. In the literature the reader can find Pell quaternions and studies on their properties in [12]- [19].…”

“…where P k,n = P k,n + ε P k,n+1 , P k,n = 2 P k,n−1 + k P k,n−2 , n ≥ 2 e 0 = 1, e i 2 = −1, e i e j = −e j e i , i, j = 1, 2, 3, ε = 0, 0 ε = ε 0 = 0, 1 ε = ε 1 = ε, ε 2 = 0. In 2018, Gül introduced k-Pell quaternions and k-Pell-Lucas quaternions [19] as follows:…”