Hyper-number generalized quaternions

The main aim of this paper is to introduce generalized quaternions with hyper-number coefficients. For this, firstly, a new number system is defined, which is the generalization of bicomplex numbers, hyper-double numbers and hyper-dual numbers. And any element of this generalization is called a hyper-number. Then, real matrix representation and vector representation of a hyper-number are given. Secondly, hyper-number generalized quaternions and their algebraic properties are introduced. For a hyper-number generalized quaternion, $4\times4$ real generalized quaternion matrix representation is presented. Next, because of lack of commutativity, for a hyper-number generalized quaternion, two different hyper-number matrix representations are calculated. Moreover, real matrix representations of a hyper-number generalized quaternion is expressed by matrix representation of a hyper-number. Finally, vector representations of a hyper-number generalized quaternion are given and properties of this representations are investigated.