On higher order Fibonacci hyper complex numbers

This thesis deals with the study of the higher order Fibonacci quaternions and the higher order Fibonacci hyper complex numbers. The thesis consists of five chapters. Chapter 1 is a concise overview of what this thesis is about and also is a literature summary for Fibonacci numbers, Lucas numbers, quaternions, octonions, sedenions, and 2m-ions. In Chapter 2, we shall define the fundamental concepts, the definition of Fibonacci, Lucas, and higher order Fibonacci numbers. Their basic identities and the relations between these identities will be given. We will also define quaternions, octonions, and sedenions whose components Fibonacci numbers, and give some properties about these algebras. In Chapter 3, we will define the higher order Fibonacci quaternions and study some properties of these quaternions. In the Chapter 4, we will define and work on the higher order Fibonacci hyper complex numbers. Then many algebraic properties of the higher order Fibonacci 2m -ions will be given. We will 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. Moreover, we develop some matrix identities involving higher order Fibonacci 2m -ions which allow us to obtain some properties of these higher order hyper complex numbers.