Some Properties of Dual Fibonacci and Dual Lucas Octonions

Abstract

Halici (Adv Appl Clifford Algebr 25(4):905–914, 2015) defined dual Fibonacci and dual Lucas octonions by the relations ${\widetilde{Q}_{n}={Q}_n+\varepsilon Q_{n+1}}$ and ${\widetilde{P}_n=P_n+\varepsilon P_{n+1}}$ for every integer n where ${Q_n}$ and ${P_n}$ are the Fibonacci and Lucas octonions respectively, and ${\varepsilon}$ is the dual unit. The aim of this paper is to investigate properties of dual Fibonacci and dual Lucas octonions. After obtaining the Binet formulas for the sequences ${\{\widetilde{Q}_n \}_{n=0}^\infty}$ and ${\{\widetilde{P}_n \}_{n=0}^\infty}$, we derive some identities for these sequences such as Catalan’s, Cassini’s and d’Ocagne’s identities.