Erışr, Tülay; Mumcu, Gökhan; Kiziltuğ, Sezai̇; Akar, Funda

Our article focuses on the study of quaternions topic introduced by Hamilton. Quaternions are a generalization of complex numbers and have multiple applications in mathematical physics. Another application of quaternions is robotics because what generalizes the imaginary axis is the family i, j, k modeling Euler angles and rotations in space. The first part of the article we recall the different definitions of how the algebra of quaternions is well constructed. The main results are given in the third part and concern: spatial quaternionics rectifying-direction (sqRD) curves and and spatial quaternionic rectifying-donor (sqRDnr) curves. We study a new tip of unit speed associated curves in E3, which is also used in robotic systems and kinematics, like a spatial quaternionic rectifying-direction curve and spatial quaternionic rectifying-donor curve. Then, we achieve qualification for the curves. Moreover, we present applications of spatial quaternionic rectifying-direction to some specific curves like helix, slant helix, Salkowski and anti-Salkowski curves or rectifying curves. In addition, we establish different theorems which generalize the results obtained on the quaternionic curves in Q. Then, we give some examples are finally discussed. Consequently, Our paper is centered around theoretical analysis in geometry rather than experimental investigations.

Ozkan, Engin; Akkus, Hakan

- In this study, we define a Dickson k-Fibonacci polynomial inspired by Dickson polynomials and give some terms of these polynomials. Then we present the relations between the terms of Dickson k-Fibonacci polynomials. We find Binet formulas and generating functions for these polynomials. In addition, we give some important identities like Catalan identity, Melham’s identity, and Gelin-Cesaro’s identity. Moreover, Catalan transformation is applied to these polynomials, and their terms are found. Finally, the Hankel transform is applied to the Catalan transform of these polynomials, and the results obtained are associated with known Fibonacci numbers.

Erışır, Tülay; Yildirim, Emrah

In this study, the real quaternions and spinors are studied. The motivation of this study is to express the Hamilton matrices of real quaternions more shortly and elegantly, namely spinors. Therefore, firstly, two transformations between real quaternions and spinors are defined. These transformations are defined for two different spinor matrices corresponding to the left and right Hamilton matrices since the quaternion product is not commutative. Thus, the fundamental spinor matrix corresponding to the fundamental matrix of real quaternions is obtained and some properties are given for these spinor matrices. Finally, the eigenvalues and eigenvectors of the fundamental spinor matrix are obtained.

Kuloǧlu, Bahar; Eser, Engin; Özkan, Engin

In this study, the main goal is to investigate the r-circulant matrices of k-Fermat and k-Mersenne numbers, then to find eigenvalues, determinants of these matrices, to evaluate their different norms (Spectral and Euclidean) and finally to find the right and skew-right circulant matrices.

Ozkan, Engin; Akkus, Hakan

In this study, firstly the k-Chebsyhev sequence is defined, and some terms of this sequence are given. Then, the relations between the terms of the k-Chebsyhev sequence are presented and the generating function of this sequence is obtained. In addition, the Catalan transformation of the sequence is given and the generating function of the Catalan k-Chebsyhev sequence is obtained. Finally, the Hankel transform is applied to the Catalan k-Chebsyhev transform.

Kuloğlu, Bahar; Özkan, Engin

In this study, we define d-Tribonacci polynomials. Some combinatorial properties of the d-Tribonacci polynomials with matrix representations are obtained with the help of Riordan arrays. In addition, d-Tribonacci number sequence, a new generalization of this number sequence, is obtained by considering the Pascal matrix. With the help of the Pascal matrix, two kinds of factors of d-Tribonacci polynomials are found. Also, infinite d-Tribonacci polynomials matrix and the inverses of these polynomials are found.

Kumari, Munesh; Prasad, Kalika; Kuloǧlu, Bahar; Özkan, Engin

In this paper, we have introduced a new infinite cyclic group called the k-Fibonacci group and studied its algebraic properties. Further, we have obtained periods for k-step Fibonacci sequences in the 2-generator groups such as S3, D3, A3, Q8 and in the 3-generator group, Q8 × Z2m.