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On Higher Order Jacobsthal Hyper Complex Numbers

Year 2024, Volume: 16 Issue: 1, 35 - 44, 30.06.2024
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.

References

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  • Cariow, A., Cariowa, G., Algorithm for multiplying two octonions, Radioelectronics and Communications Systems, 55(10)(2012), 464–473.
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  • 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.
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  • 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.
Year 2024, Volume: 16 Issue: 1, 35 - 44, 30.06.2024
https://doi.org/10.47000/tjmcs.1195463

Abstract

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.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Hayrullah Özimamoğlu 0000-0001-7844-1840

Publication Date June 30, 2024
Published in Issue Year 2024 Volume: 16 Issue: 1

Cite

APA Özimamoğlu, H. (2024). On Higher Order Jacobsthal Hyper Complex Numbers. Turkish Journal of Mathematics and Computer Science, 16(1), 35-44. https://doi.org/10.47000/tjmcs.1195463
AMA Özimamoğlu H. On Higher Order Jacobsthal Hyper Complex Numbers. TJMCS. June 2024;16(1):35-44. doi:10.47000/tjmcs.1195463
Chicago Özimamoğlu, Hayrullah. “On Higher Order Jacobsthal Hyper Complex Numbers”. Turkish Journal of Mathematics and Computer Science 16, no. 1 (June 2024): 35-44. https://doi.org/10.47000/tjmcs.1195463.
EndNote Özimamoğlu H (June 1, 2024) On Higher Order Jacobsthal Hyper Complex Numbers. Turkish Journal of Mathematics and Computer Science 16 1 35–44.
IEEE H. Özimamoğlu, “On Higher Order Jacobsthal Hyper Complex Numbers”, TJMCS, vol. 16, no. 1, pp. 35–44, 2024, doi: 10.47000/tjmcs.1195463.
ISNAD Özimamoğlu, Hayrullah. “On Higher Order Jacobsthal Hyper Complex Numbers”. Turkish Journal of Mathematics and Computer Science 16/1 (June 2024), 35-44. https://doi.org/10.47000/tjmcs.1195463.
JAMA Özimamoğlu H. On Higher Order Jacobsthal Hyper Complex Numbers. TJMCS. 2024;16:35–44.
MLA Özimamoğlu, Hayrullah. “On Higher Order Jacobsthal Hyper Complex Numbers”. Turkish Journal of Mathematics and Computer Science, vol. 16, no. 1, 2024, pp. 35-44, doi:10.47000/tjmcs.1195463.
Vancouver Özimamoğlu H. On Higher Order Jacobsthal Hyper Complex Numbers. TJMCS. 2024;16(1):35-44.