The Modified Pell and the Modified k-Pell Quaternions and Octonions

The generalized bi-periodic Fibonacci quaternions and octonions

“…In this section, we introduce the generalized bi-periodic Fibonacci quaternions and give some basic properties of them. These results can be seen as a generalization of the results in [4,22,23].…”

“…By analogously, we defined the generalized bi-periodic Fibonacci octonions and gave some basic properties of them. Our results not only gave a generalization of the papers in [4,22,23,27,28], but also included new results. The main contribution of this research is one can get a great number of distinct quaternion and octonion sequences by providing the initial values in the generalized bi-periodic Fibonacci sequence {w n } .…”

Horadam Octonions

“…More recently octonions have been studied by many authors. For example, the Fibonacci octonions, Pell octonions and Modified Pell octonions appeared in [3,17,22]. There are many studies about Fibonacci numbers over dual octonions and generalized octonions [11,21,23].…”

Third-Order Jacobsthal Generalized Quaternions

“…In this sense, Horadam [Ho1] defined the quaternions with the classic Fibonacci and Lucas number components as QF n = F n + F n+1 i + F n+2 j + F n+3 k (F n 1 = F n ) and QL n = L n + L n+1 i + L n+2 j + L n+3 k (L n 1 = L n ), respectively, where F n and L n are the n-th classic Fibonacci and Lucas numbers, respectively, and the author studied the properties of these quaternions. Several interesting and useful extensions of many of the familiar quaternion numbers (such as the Fibonacci and Lucas quaternions [Ak,Ha1,Ho1], Pell quaternion [Ca,Ci1] and Jacobsthal quaternions [Szy-Wl] have been considered by several authors.…”