Fibonacci Generalized Quaternions

In this paper, the Fibonacci generalized quaternions are introduced. We use the well-known identities related to the Fibonacci and Lucas numbers to obtain the relations regarding these quaternions. Furthermore, the Fibonacci generalized quaternions are classified by considering the special cases of quaternionic units.     

On Fibonacci Quaternions

In this paper, we investigate the Fibonacci and Lucas quaternions. We give the generating functions and Binet formulas for these quaternions. Moreover, we derive some sums formulas for them.     

On Generalized Fibonacci Quaternions and Fibonacci-Narayana Quaternions

In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions in a generalized quaternion algebra.     

On Complex Fibonacci Quaternions

Horadam defined the Fibonacci quaternions and established a few relations for the Fibonacci quaternions. In this paper, we investigate the complex Fibonacci quaternions and give the generating function and Binet formula for these quaternions. Moreover, we also give the matrix representations of them.     

Involutions of Complexified Quaternions and Split Quaternions

An involution or anti-involution is a self-inverse linear mapping. Involutions and anti-involutions of real quaternions were studied by Ell and Sangwine [15]. In this paper we present involutions and antiinvolutions of biquaternions (complexified quaternions) and split quaternions. In addition, while only quaternion conjugate can be defined for a real quaternion and split quaternion, also complex conjugate can be defined for a biquaternion. Therefore, complex conjugate of a biquaternion is used in some transformations beside quaternion conjugate in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion, biquaternion and split quaternion involutions and anti-involutions are given.     

Split Quaternions and Integer-valued Polynomials

The integer split quaternions form a noncommutative algebra over ?. We describe the prime and maximal spectrum of the integer split quaternions and investigate integer-valued polynomials over this ring. We prove that the set of such polynomials forms a ring, and proceed to study its prime and maximal ideals. In particular we completely classify the primes above 0, we obtain partial characterizations of primes above odd prime integers, and we give sufficient conditions for building maximal ideals above 2.     

A Generalization of Fibonacci and Lucas Quaternions

In this paper, we give a generalization of the Fibonacci and Lucas quaternions. We obtain the Binet formulas, generating functions, and some certain identities for these quaternions which include generalizations of some results of Halici.     

On the Semisimilarity and Consemisimilarity of Split Quaternions

In this study, we introduce the concept of semisimilarity and consemisimilarity of split quaternions. Moreover, we examine the solvability conditions and general solutions of systems \({xay=b,ybx=a {\rm and} \widetilde{x}ay=b,\widetilde{y}bx=a}\) in split quaternions. If there exist x and y that satisfy first equations system, then a and b are said to be semisimilar, if there exist x and y that satisfy second equations system, then a and b are said to be consemisimilar.     

Dual Split Quaternions and Screw Motion in 3-Dimensional Lorentzian Space

In this paper we obtain screw axis of a displacement in \mathbb L³{{\mathbb L}^3}. Then by using the L-screw axis, L-Rodrigues equation for a spatial displacement is obtained in the space \mathbb L³{{\mathbb L}^3}. Moreover, the components of a dual split quaternion are obtained by replacing the L-Euler parameters with their split dual versions.     

On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions

In this paper, we examine eigenvalue problem of a rotation matrix in Minkowski 3 space by using split quaternions. We express the eigenvalues and the eigenvectors of a rotation matrix in term of the coefficients of the corresponding unit timelike split quaternion. We give the characterizations of eigenvalues (complex or real) of a rotation matrix in Minkowski 3 space according to only first component of the corresponding quaternion. Moreover, we find that the casual characters of rotation axis depend only on first component of the corresponding quaternion. Finally, we give the way to generate an orthogonal basis for ${\mathbb{E}^{3}_{1}}$ by using eigenvectors of a rotation matrix.     

On Pell Quaternions and Pell-Lucas Quaternions

The main object of this paper is to present a systematic investigation of new classes of quaternion numbers associated with the familiar Pell and Pell-Lucas numbers. The various results obtained here for these classes of quaternion numbers include recurrence relations, summation formulas and Binet’s formulas.     

Quaternions and applications

This note gives an overview of two novel applications of Quaternions which appeared in [1]–[3]: First, the evaluation of certain three dimensional real integrals without integrating with respect to the real variables. This is the generalisation of the well-known Cauchy Residue Theorem from the case of two dimensions to the case of four dimensions. Second, the solution of boundary value problems for linear elliptic PDEs in four dimensions. This is the extension of some of the results of [4] from two to four dimensions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Canal Surfaces with Quaternions

Quaternions are more usable than three Euler angles in the three dimensional Euclidean space. Thus, many laws in different fields can be given by the quaternions. In this study, we show that canal surfaces and tube surfaces can be obtained by the quaternion product and by the matrix representation. Also, we show that the equation of canal surface given by the different frames of its spine curve can be obtained by the same unit quaternion. In addition, these surfaces are obtained by the homothetic motion. Then, we give some results.     

Wiener Algebra for the Quaternions

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener–Lévy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener–Hopf operators.