On higher order Fibonacci hyper complex numbers

Abstract

• The aim of this paper is to study hyper complex numbers with higher order Fibonacci numbers coefficient. • Given a simple and practical way to obtain many properties of higher order Fibonacci 2 m -ions such as Binet type formula, generating functions, exponential generating function, Vajda’s identity, etc. • We develop a matrix identity involving higher order Fibonacci 2 m -ions which allow us to obtain some properties of these higher order hyper complex numbers. This paper deals with developing a new class of quaternions , octonions and sedenions called higher order Fibonacci 2 m -ions (or-higher order Fibonacci hyper complex numbers) whose components are higher order Fibonacci numbers. We give recurrence relation, Binet formula, generating function and exponential generating function of higher order Fibonacci 2 m -ions. We also derive some identities such as Vajda’s identity, Catalan’s identity, Cassini’s identity, and d’Ocagne’s identity with the aid of the Binet formula. Finally, we develop some matrix identities involving higher order Fibonacci 2 m -ions which allow us to obtain some properties of these higher order hyper complex numbers.