Dual-complex Jacobsthal Quaternions

Year 2020, , 145 - 156, 15.10.2020

Fügen Torunbalcı Aydın

https://doi.org/10.36753/mathenot.621169

Abstract

In this paper, dual-complex Jacobsthal quaternions are defined. Also, some algebraic properties of dual-complex 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.

...................................................

..................................................

.......................................

Keywords

Dual number, dual-complex number, Jacobsthal number, dual-complex Jacobsthal number, Jacobsthal quaternion

References

• [1] William Rowan Hamilton. Elements of Quaternions. Longmans, Green & Company,London, 1866.
• [2] NJA Sloane. A handbook of integer sequences. 1973.
• [3] A. F. Horadam. Jacobsthal representation numbers. Fibonacci Quarterly, 34:40–54, 1996.
• [4] A. F. Horadam. Jacobsthal representation polynomials. Fibonacci Quarterly, 35:137– 148, 1997.
• [5] F. Koken and D. Bozkurt. On the jacobsthal numbers by matrix methods. Int. J. Contemp. Math. Sciences, 3(13):605–614, 2008.
• [6] F. Koken and D. Bozkurt. On the jacobsthal-lucas numbers by matrix methods. Int. J. Contemp. Math. Sciences, 3(13):1629–1633, 2008.
• [7] A. Dasdemir. A study on the jacobsthal and jacobsthal-lucas numbers by matrix method. DUFED Journal of Sciences, 3(1):13–18, 2014.
• [8] A. Dasdemir. On the jacobsthal numbers by matrix method. Fen Derg, 7(1):69–76, 2012.
• [9] G. B. Djordjevic. Derivative sequences of generalized jacobsthal and jacobsthal-lucaspolynomials. Fibonacci Quarterly, 38(4):334–338, 2000.
• [10] G. B. Djordjevic. Generalized jacobsthal polynomials. Fibonacci Quarterly, 38(3):239– 243, 2009.
• [11] Z. Cerin. Sums of squares and products of jacobsthal numbers. Journal of Integer Se- quences, 10(07.2):5, 2007.
• [12] Z. Cerin. Formulae for sums of jacobsthal lucas numbers. In Int. Math. Forum, volume 2, pages 1969–1984, 2007.
• [13] Anetta Szynal-Liana and Iwona Wloch. A note on jacobsthal quaternions. Advances in Applied Clifford Algebras, 26(1):441–447, 2016.
• [14] Fugen Torunbalcı Aydın and Salim Yuce. A new approach to jacobsthal quaternions.Filomat, 31(18):5567–5579, 2017.
• [15] Dursun Tasci. On k-jacobsthal and k-jacobsthal-lucas quaternions. JOURNAL OF SCI- ENCE AND ARTS, (3):469–476, 2017.
• [16] Fugen Torunbalcı Aydın. Dual jacobsthal quaternions. submitted, 2018.
• [17] William Kingdon Clifford. A preliminary sketch of biquaternions. 1873.
• [18] AP Kotelnikov. Screw calculus and some of its applications to geometry and mechanics.Annals of Imperial University of Kazan, 1895.
• [19] E. Study. Geometrie der dynamen. Leipzig, 1903.
• [20] V. Majernik. Multicomponent number systems. Acta Pyhsica Polonica A, 90(3):491– 498, 1996.
• [21] Farid Messelmi. DUAL-COMPLEX NUMBERS AND THEIR HOLOMORPHICFUNCTIONS. working paper or preprint, January 2015.
• [22] Mehmet Ali Gungor and Ayse Zeynep Azak. Investigation of dual-complex fibonacci, dual-complex lucas numbers and their properties. Advances in Applied Clifford Algebras,27(4):3083–3096, 2017.