Algebraic properties of bi−-periodic dual Fibonacci quaternions

The purpose of the paper is to construct a new representation of dual quaternions called bi$-$periodic dual Fibonacci quaternions. These quaternions are originated as a generalization of the known quaternions in literature such as dual Fibonacci quaternions, dual Pell quaternions and dual $k-$Fibonacci quaternions. Furthermore, some of them have not been introduced until this time. Then, we give generating function, Binet formula and Catalan's identity in terms of these quaternions

Identities for third order Jacobsthal quaternions

In this paper we introduce the third order Jacobsthal quaternions and the third order Jacobsthal-Lucas quaternions and give some of their properties. We derive the relations between third order Jacobsthal numbers and third order Jacobsthal quaternions and we give the matrix representation of these quaternions

Hyperbolic k-Fibonacci Quaternions

In this paper, hyperbolic k-Fibonacci quaternions are defined. Also, some algebraic properties of hyperbolic k-Fibonacci quaternions which are connected with hyperbolic numbers and k-Fibonacci numbers are investigated. Furthermore, D'Ocagne's identity, the Honsberger identity, Binet's formula, Cassini's identity and Catalan's identity for these quaternions are given

Bicomplex k-Fibonacci quaternions

In this paper, bicomplex k-Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex k-Fibonacci quaternions which are connected with bicomplex numbers and k-Fibonacci numbers are investigated. Furthermore, the Honsberger identity, the d'Ocagne's identity, Binet's formula, Cassini's identity, Catalan's identity for these quaternions are given

The Moore-Penrose inverses of split quaternions

In this paper, we find the roots of lightlike quaternions. By introducing the concept of the Moore-Penrose inverse in split quaternions, we solve the linear equations $axb=d$, $xa=bx$ and $xa=b\bar{x}$. Also we obtain necessary and sufficient conditions for two split quaternions to be similar or consimilar.Comment: 6 page

Dual complex Pell quaternions

In this paper, dual complex Pell numbers and quaternions are defined. Also, some algebraic properties of dual-complex Pell numbers and quaternions which are connected with dual complex numbers and Pell numbers are investigated. Furthermore, the Honsberger identity, Binet's formula, Cassini's identity, Catalan's identity for these quaternions are given.Comment: arXiv admin note: substantial text overlap with arXiv:1810.0500

De- Moivre's and Euler Formulas for Matrices of Split Quaternions

In this paper, real matrix representations of split quaternions are examined in terms of the casual character of quaternion. Then, we give De-Moivre' s formula for real matrices of timelike and spacelike split quaternions, separately. Finally, we state the Euler theorem for real matrices of pure split quaternions

Third-order Jacobsthal Generalized Quaternions

In this paper, the third-order Jacobsthal generalized quaternions are introduced. We use the well-known identities related to the third-order Jacobsthal and third-order Jacobsthal-Lucas numbers to obtain the relations regarding these quaternions. Furthermore, the third-order Jacobsthal generalized quaternions are classified by considering the special cases of quaternionic units. We derive the relations between third-order Jacobsthal and third-order Jacobsthal-Lucas generalized quaternions.Comment: Submitted to Journal. 16 page

About special elements in quaternion algebras over finite fields

In this paper we study special Fibonacci quaternions and special generalized Fibonacci-Lucas quaternions in quaternion algebras over finite fields.Comment: This is a preliminary form of the pape

Noncommutative multiplicative norm identities for the quaternions and the octonions

We present Capelli type identities associated with the quaternions and the octonions, which are noncommutative versions of multiplicative norm identities for the quaternions and the octonions.Comment: 6 page