Dual-Complex Jacobsthal Quaternions
Dual-Complex Jacobsthal Quaternions
In this paper, dual-complex Jacobsthal quaternions are defined. Also, some algebraic properties of dualcomplex Jacobsthal quaternions which are connected with dual-complex numbers and Lucas numbers are investigated. Furthermore, the Honsberger identity, the d’Ocagne’s identity, Binet’s formula, Cassini’s identity, Catalan’s identity for these quaternions and their real representations are given.
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- [1] Hamilton W. R.: Elements of Quaternions. Longmans, Green and Co., London, (1866).
- [2] Sloane N. J. A.: A Handbook of Integer Sequences. New York, Press, (1973).
- [3] Horadam A. F.: Jacobsthal Representation Numbers. The Fibonacci Quarterly. 34, 40-54 (1996).
- [4] Horadam A. F.: Jacobsthal Representation Polynomials. The Fibonacci Quarterly. 35, 137-148 (1997).
- [5] Clifford W. K.: A preliminary sketch of biquaternions, (1873).
- [6] Majernik, V.: Quaternion formulation of the Galilean space-time transformation. Acta Phy. Slovaca. 56 (1), 9-14 (2006).
- [7] Kotelnikov, A. P.: Screw calculus and some of its applications to geometry and mechanics. Annals of Imperial University of Kazan (1895).
- [8] Study, E.: Geometrie der Dynamen. Leipzig. (1903).
- [9] Majernik, V.: Multicomponent number systems. Acta Pyhsica Polonica A. 90 (3), 491-498 (1996).
- [10] Messelmi, F.: Dual -complex numbers and their holomorphic functions. https://hal.archives-ouvertes.fr/hal01114178, (2015).
- [11] Gungor, M. A., Azak, A. Z.: Investigation of Dual-Complex Fibonacci, Dual-Complex Lucas Numbers and Their Properties. Advances in Applied Clifford Algebras. 27 (4), 3083-3096 (2017).
- [12] Djordjevid G. B.: Generalized Jacobsthal Polynomials. The Fibonacci Quarterly, 38 (3), 239-243 (2009).
- [13] Djordjevid G. B.: Derivative sequences of generalized Jacobsthal and Jacobsthal-Lucas polynomials. Fibonacci Quarterly. 38 (4), 334-338 (2000).
- [14] Cerin Z.: Sums of Squares and Products of Jacobsthal Numbers. Journal of Integer Sequence. 10, Article 07.2.5, (2007).
- [15] Cerin Z.: Formulae for Sums of Jacobsthal-Lucas Numbers. International Mathematical Forum. 2 (40), 1969-1984 (2007).
- [16] Köken F., Bozkurt D.: On the Jacobsthal Numbers by Matrix Methods. International Journal of Contemporary Mathematical Sciences. 3 (13), 605-614 (2008).
- [17] Köken F., Bozkurt D.: On the Jacobsthal –Lucas Numbers by Matrix Methods. International Journal of Contemporary Mathematical Sciences. 3 (13), 1629-1633 (2008).
- [18] Da¸sdemir A.: On the Jacobsthal Numbers by Matrix Method. SDU Journal of Science (E-Journal). 71, 69-76 (2012).
- [19] Da¸sdemir A.: A study on the Jacobsthal and Jacobsthal -Lucas numbers by matrix method. DUFED Journal of Sciences 3, (1), 13-18 (2014).
- [20] Szynal-Liana A., Włoch I.: A Note on Jacobsthal Quaternions. Advances in Applied Clifford Algebras 26 (1), 441-447 (2016).
- [21] Aydın Torunbalcı F., Yüce, S.: A New Approach to Jacobsthal Quaternions. Filomat 31 (18), 5567-5579 (2017).
- [22] Ta¸sçı D.: On k-jacobsthal and k-jacobsthal-Lucas quaternions. Journal of Science and Arts 3 469-476 (2017).
- [23] Aydın, Torunbalcı F.: Dual Jacobstthal quaternions. Communication in advanced Mathamatical Sciences. 3 (3), 130-142 (2020).