Hyperbolic k-Fibonacci and k-Lucas octonions

Abstract

In this paper, we introduce the hyperbolic k-Fibonacci and k-Lucas octonions. We present Binet's formulas, Catalan's identity, Cassini's identity, d'Ocagne's identity and generating functions for the k-Fibonacci and k-Lucas hyperbolic octonions.

On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

“…The hyperbolic Fibonacci and hyperbolic Lucas quaternions and some of their generalizations are given in [19][20][21][22][23].…”

On hyperbolic k-Jacobsthal and k-Jacobsthal–Lucas octonions

“…where ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 are real components and 𝑖 1 , 𝑖 2 , 𝑖 3 , 𝑖 4 are hyperbolic quaternion units, which have the rules as given in Table 1 In [10,20], the hyperbolic k-Fibonacci and k-Fibonacci-Lucas, hyperbolic k-Jacobsthal and k-Jacobsthal-Lucas quaternions were defined and given some of their properties.…”

“…A hyperbolic octonion 𝑂 has the form In [9], A. Godase defined the hyperbolic k-Fibonacci and k-Fibonacci-Lucas octonions and gave some of their properties.…”

On $k$-Mersenne and k-Mersenne-Lucas Octonions

“…Octonions with Fibonacci and Lucas components were introduced by Akkus and Kec ¸ilioglu [1] and they studied their properties like Binet formula, generating function and some well-known identities. Recently A.D. Godse [12] studied the hyperbolic Octonions involving k-Fibonacci & k-Lucas and Özkan et.al. [16] studied the hyperbolic Octonions with k-Jacobsthal & k-Jacobsthal Lucas sequences.…”