Horadam Octonions

Abstract

In this paper, first we define Horadam octonions by Horadam sequence which is a generalization of second order recurrence relations. Also, we give some fundamental properties involving the elements of that sequence. Then, we obtain their Binet-like formula, ordinary generating function and Cassini identity.Mathematics Subject Classification (2010). 11B39, 17A20.

On Generalized Tribonacci Octonions

“…Note that, the second order linear recurence octonion sequences for example in [8], they defined modified Pell and Modified −Pell octonions and in [9], authors studied Pell octonions. Moreover in [10], ( , ) −Fibonacci octonions are obtained which is the same results in [3] but the initial conditions are ( , ) = 0 and ( , ) = 1 , also in [11]. Another octonionic sequence was devoted to studying Jacobsthal and Jacobsthal-Lucas octonions in [12].…”

“…In our work, we introduce for finding special identities of generalized tribonacci octonions. We motivate by their results in [2], [3] and [4]. In [3], they studied Horadam octonions.…”

“…We motivate by their results in [2], [3] and [4]. In [3], they studied Horadam octonions. In [5], they considered Padovan and Pell-Padovan quater-nions which is the third order quaternions.…”

The generalized bi-periodic Fibonacci quaternions and octonions

“…If we take the initial conditions w 0 = 2 and w 1 = b, we get the bi-periodic Lucas octonions in [28]. Also, if we take a = b = 1 in {w n }, we get the Horadam octonion numbers in [15] with the case of q = 1.…”

“…For related studies on different types of sequences over octonion algebra, we refer to [4,15,16,20,21].…”