The Third Order Jacobsthal Octonions: Some Combinatorial Properties

TL;DR

This paper introduces new third order Jacobsthal and Jacobsthal-Lucas octonions, exploring their properties and relationships, expanding the family of octonion number sequences with novel algebraic identities.

Contribution

It establishes the first definitions and properties of third order Jacobsthal and Jacobsthal-Lucas octonions, linking them to existing octonion sequences.

Findings

Defined third order Jacobsthal octonions and Jacobsthal-Lucas octonions.

Derived algebraic relations between these new octonion sequences.

Extended the family of octonion number sequences with new identities.

Abstract

Various families of octonion number sequences (such as Fibonacci octonion, Pell octonion and Jacobsthal octonion) have been established by a number of authors in many different ways. In addition, formulas and identities involving these number sequences have been presented. In this paper, we aim at establishing new classes of octonion numbers associated with the third order Jacobsthal and third order Jacobsthal-Lucas numbers. We introduce the third order Jacobsthal octonions and the third order Jacobsthal-Lucas octonions and give some of their properties. We derive the relations between third order Jacobsthal octonions and third order Jacobsthal-Lucas octonions.