Özkan, Engin; Şen, Bayram; Akkuş, Hakan; Uysal, Mine

In this study, we examine an application of k−Mersenne and k−Mersenne-Lucas sequences. We present the Catalan transforms of these sequences and give the properties of these Catalan transforms. Catalan transforms of k−Mersenne and k−Mersenne-Lucas sequences have terms according to different values of k, and some of them are associated with the sequences in OEIS. We obtain the generating functions of the Catalan transforms of k−Mersenne and k−Mersenne-Lucas sequences. Moreover, we apply the Hankel transform to the Catalan transforms of these sequences. Finally, the determinants of the created matrices are calculated by applying the Hankel transform to the terms of the Catalan transforms of these sequences.

Erişir, Tülay; Köse Öztaş, Hilal

The aim of this study is to give spinor representation of successor curves in three-dimensional Euclidean space. In this study, the spinor representations of a curve with arc-length parameter and a successor curve of this curve with some arc-length parameter in three dimensional Euclidean space have been studied. For this, first of all, the curve with unit speed and its successor curve have been corresponded to two different spinors. Then, using the relationships between these curves, the relationships between the spinors corresponding to these curves have been given. Therefore, geometric interpretations of these curves and corresponding spinors have been made. In addition, different spinor equations of the mates and derivatives of these spinors have been examined and geometric interpretations of these spinor equations have been given. Then, spinor equations have been obtained in case the successor curves are helices. Consequently, two examples have been given.