On Higher Order Jacobsthal Hyper Complex Numbers

Year 2024, , 35 - 44, 30.06.2024

Hayrullah Özimamoğlu

https://doi.org/10.47000/tjmcs.1195463

Abstract

In this work, we define a new class of hyper complex numbers whose components are higher order Jacobsthal numbers, and call such numbers as the higher order Jacobsthal $ 2^{s} $-ions. We obtain some algebraic properties of the higher order Jacobsthal $ 2^{s} $-ions such as recurrence relation, Binet-like formula, generating function, exponential generating function, Vajda's identity, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. Morever we derive the matrix representation of the higher order Jacobsthal $ 2^{s} $-ions, and so prove Cassini's identity as a further type.

Keywords

Hyper complex numbers, higher order Jacobsthal numbers, higher order Jacobsthal $ 2^{s} $-ions, Binet-like formula, recurrence relation

References

• Baez, J.C., The octonions, Bulletin of the American Mathematical Society, 39(2)(2002), 145–205.
• Cariow, A., Cariowa, G., Algorithm for multiplying two octonions, Radioelectronics and Communications Systems, 55(10)(2012), 464–473.
• Cariow, A., Cariowa, G., An algorithm for fast multiplication of sedenions, Information Proccessing Letters, 113(9)(2013), 324–331.
• Carmody, K., Circular and hyperbolic quaternions, octonions and sedenions, Applied Mathematics and Computation, 28(1)(1988), 47–72.
• Cawagas, R., On the structure and zero divisors of the Cayley-Dickson sedenion algebra, Discussiones Mathematicae-General Algebra and Applications, 24(2)(2004), 251–265.
• Çimen, C.B., İpek, A., On Jacobsthal and Jacobsthal–Lucas octonions, Mediterranean Journal of Mathematics, 14(2)(2017), 1–13.
• Çimen, C.B., İpek, A. On Jacobsthal and the Jacobsthal-Lucas sedenions and several identities involving these numbers,Mathematica Aeterna, 7(4)(2017), 447–454.
• Göcen, M., Soykan, Y., Horadam 2k-ions, Konuralp Journal of Mathematics, 7(2)(2019), 492–501.
• Hamilton, W.R., Elements of quaternions, Green & Company, London: Longman, 1866.
• Horadam, A.F., Jacobsthal representation numbers, The Fibonacci Quarterly, 34(1)(1996), 40–54.
• Imaeda, K., Imaeda, M., Sedenions: algebra and analysis, Applied Mathematics and Computation, 115(2000), 77–88.
• Mironov, V.L., Mironov, S.V., Associative space-time sedenions and their application in relativistic quantum mechanics and field theory, Applied Mathematics, 6(1)(2015), 46–56.
• Özimamoğlu, H., On hyper complex numbers with higher order Pell numbers components, The Journal of Analysis, 31(4)(2023), 2443–2457
• Özkan, E., Uysal, M., On quaternions with higher order Jacobsthal numbers components, Gazi University Journal of Science, 36(1)(2022), 336–347.
• Özvatan, M., Generalized Golden-Fibonacci calculus and applications, Master of Science Thesis,İzmir Institute of Technology, 2018.
• Szynal-Liana, A., Wloch, I., A note on Jacobsthal quaternions, Advances in Applied Clifford Algebras, 26(1)(2016), 441–447.
• Torunbalcı-Aydın, F., Yüce, S., A new approach to Jacobsthal quaternions, Advances in Applied Clifford Algebras, Filomat, 31(18)(2017), 5567–5579.
• Uysal, M., Özkan, E., Higher order Jacobsthal–Lucas quaternions, Axioms, 11(12)(2022), 671.