On a generalization of dual-generalized complex Fibonacci quaternions

Elif Tan and Umut Öcal
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 4, Pages 635–646
DOI: 10.7546/nntdm.2023.29.4.635-646
Full paper (PDF, 199 Kb)

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Authors and affiliations

Elif Tan
Department of Mathematics, Ankara University
06100 Tandogan Ankara, Turkey

Umut Öcal
Department of Mathematics, Ankara University
06100 Tandogan Ankara, Turkey

Abstract

In this study, we introduce a new class of generalized quaternions whose components are dual-generalized complex Horadam numbers. We investigate some algebraic properties of them.

Keywords

• Dual-generalized complex numbers
• Quaternions, Fibonacci numbers
• Horadam numbers
• Fibonacci quaternions

2020 Mathematics Subject Classification

• 11B39
• 11R52

References

1. Ait-Amrane, N. R., Gok, I., & Tan, E. (2021). Hyper-dual Horadam quaternions. Miskolc Mathematical Notes. 22(2), 903–913.
2. Akyigit, M., Kosal, H. H., & Tosun, M. (2014). Fibonacci generalized quaternions.
   Advances in Applied Clifford Algebras, 24, 631–641.
3. Alfsmann, D. (2006). On families of 2N-dimensional hypercomplex algebras suitable for digital signal processing. 14th European Signal Processing Conference, Florence, Italy.
4. Aslan, S., Bekar, M., & Yayli, Y. (2020). Hyper-dual split quaternions and rigid body motion. Journal of Geometry and Physics, 158, Article ID 103876.
5. Catoni, F., Cannata, R., Catoni, V., & Zampetti, P. (2004). Two-dimensional hypercomplex numbers and related trigonometries and geometries. Advances in Applied Clifford Algebras, 14, 47–68.
6. Cihan, A., Azak, A.Z., Gungor, M.A., & Tosun, M. (2019). A Study on Dual hyperbolic Fibonacci and Lucas numbers. Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 27(1), 35–48.
7. Clifford, W. K. (1873). Preliminary sketch of biquaternions. Proceedings of London Mathematical Society, 4, 361–395.
8. Cockle, J. (1849). On a new imaginary in algebra. Philosophical Magazine, London–Dublin–Edinburgh, 3(34), 37–47.
9. Cohen, A., & Shoham, M. (2018). Principle of transference–An extension to hyper-dual numbers. Mechanism and Machine Theory, 125, 101–110.
10. Dickson, L. E. (1924). On the theory of numbers and generalized quaternions. American Journal of Mathematics, 46(1), 1–16.
11. Fike, J. A. (2009). Numerically exact derivative calculations using hyper-dual numbers. 3rd Annual Student Joint Workshop in Simulation-Based Engineering and Design.
12. Fike, J. A., & Alonso, J. J. (2011). The development of hyper-dual numbers for exact second-derivative calculations. 49th AIAA Aerospace Sciences Meeting 28 including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 4–7.
13. Fjelstad, P., & Gal, S. G. (2001). Two-dimensional geometries, topologies, trigonometries, and physics generated by complex-type numbers. Advances in Applied Clifford Algebras, 11(1), 81–107.
14. Griffiths, L. W. (1928). Generalized quaternion algebras and the theory of numbers. American Journal of Mathematics, 50(2), 303–314.
15. Gungor, M. A., & Azak, A. Z. (2017). Investigation of dual-complex Fibonacci,
    dual-complex Lucas numbers and their properties. Advances in Applied Clifford Algebras, 27, 3083–3096.
16. Gurses, N., Senturk, G. Y., & Yuce, S. (2021). A study on dual-generalized complex and hyperbolic-generalized complex numbers. Gazi University Journal of Science, 34(1), 180–194.
17. Halici, S., & Karatas, A. (2017). On a generalization for quaternion sequences. Chaos, Solitons and Fractals, 98, 178–182.
18. Hamilton, W. R. (1853). Lectures on Quaternions. Hodges and Smith, Dublin.
19. Harkin, A. A., & Harkin, J. B. (2004). Geometry of generalized complex numbers.
    Mathematics Magazine, 77(2), 118–129.
20. Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70, 289–291.
21. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3(3), 161-176.
22. Jafari, M., & Yaylı, Y. (2015). Generalized quaternions and their algebraic properties. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 64(1), 5–27.
23. Kantor, I., & Solodovnikov, A. (1989). Hypercomplex Numbers. Springer-Verlag, New York.
24. Lie, S., & Scheffers G. (1893). Vorlesungen uber continuerliche Gruppen, Kap. 21, Taubner, Leipzig.
25. Majernik, V. (1996). Multicomponent number systems. Acta Physica Polonica Series A, 90, 491–498.
26. Messelmi, F. (2015). Dual-complex numbers and their holomorphic functions. https://hal.archives-ouvertes.fr/hal-01114178.
27. Motter, A. E., & Rosa, A. F. (1998). Hyperbolic calculus. Advances in Applied Clifford Algebras, 8(1), 109–128.
28. Nurkan, S. K., & Guven, I. A. (2015). Dual Fibonacci quaternions. Advances in Applied Clifford Algebras, 25(2), 403–414.
29. Pottman, H., & Wallner, J. (2000). Computational Line Geometry. Springer-Verlag Berlin Heidelberg New York.
30. Prasad, B. (2021). Dual complex Fibonacci p-numbers. Chaos, Solitons and Fractals, 145, Article ID 109922.
31. Senturk, G. Y., Gurses, N., & Yuce, S. (2022). Construction of dual-generalized complex Fibonacci and Lucas quaternions. Carpathian Mathematical Publications, 14(2), 406–418.
32. Senturk, T. D., Dasdemir, A., Bilgici, G., & Unal, Z. (2019). On unrestricted Horadam generalized quaternions. Utilitas Mathematica, 110, 89-98.
33. Tan, E. (2017). Some properties of the bi-periodic Horadam sequences. Notes on Number Theory and Discrete Mathematics, 23(4), 56–65.
34. Tan, E., Dagli, M., & Belkhir, A. (2023). Bi-periodic incomplete Horadam numbers. Turkish Journal of Mathematics, 47(2), 554–564.
35. Tan, E., & Leung, H. H. (2020). Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences. Advances in Difference Equations, 2020(26).
36. Tan, E., & Leung, H. H. (2020). Some results on Horadam quaternions. Chaos, Solitons and Fractals, 138, Article ID 109961.
37. Yaglom, I. M. (1968). Complex Numbers in Geometry. Academic Press, New York.

Manuscript history

• Received: 3 February 2023
• Revised: 20 August 2023
• Accepted: 12 September 2023
• Online First: 13 September 2023

Copyright information

Ⓒ 2023 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).