On Dual Quaternions with $k-$Generalized Leonardo Components

Abstract

In this paper, we define a one-parameter generalization of Leonardo dual quaternions, namely $k-$generalized Leonardo-like dual quaternions. We introduce the properties of $k$-generalized Leonardo-like dual quaternions, including relations with Leonardo, Fibonacci, and Lucas dual quaternions. We investigate their characteristic relations, involving the Binet-like formula, the generating function, the summation formula, Catalan-like, Cassini-like, d'Ocagne-like, Tagiuri-like, and Hornsberger-like identities. The crucial part of the present paper is that one can reduce the calculations of Leonardo-like dual quaternions by considering $k$. For $k=1$, these results are generalizations of the ones for ordered Leonardo quadruple numbers. Finally, we discuss the need for further research.

Keywords

• Leonardo sequence
• recurrence relations
• dual quaternions

References

1. T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley \& Sons, New York, 2001.
2. P. Catarino, A. Borges, On Leonardo Numbers, Acta Mathematica Universitatis Comenianae 89 (1) (2020) 75--86.
3. E. W. Dijkstra, Fibonacci Numbers and Leonardo Numbers (1981), https://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD797.html, Accessed 10 July 2023.
4. K. Kuhapatanakul, J. Chobsorn, On the Generalized Leonardo Numbers, Integers (22) (2022) Article ID A48 7 pages.
5. P. Catarino, A. Borges, A Note on Incomplete Leonardo Numbers, Integers (20) (2020) Article ID A43 7 pages.
6. Y. Alp, E. G. Koçer, Hybrid Leonardo Numbers, Chaos, Solitons \& Fractals (150) (2021) Article ID 111128 5 pages.
7. Y. Alp, E. G. Koçer, Some Properties of Leonardo Numbers, Konuralp Journal of Mathematics 9 (1) (2021) 183--189.
8. A. Shannon, Ö. Deveci, A Note on Generalized and Extended Leonardo Sequences, Notes on Number Theory and Discrete Mathematics 28 (1) (2022) 109--114.

9. A. Karataş, On Complex Leonardo Numbers, Notes on Number Theory and Discrete Mathematics 28 (3) (2022) 458--465.
10. S. Ö. Karakuş, S. K. Nurkan, M. Turan, Hyper-Dual Leonardo Numbers, Konuralp Journal of Mathematics 10 (2) (2022) 269--275.
11. M. Shattuck, Combinatorial Proofs of Identities for the Generalized Leonardo Numbers, Notes on Number Theory and Discrete Mathematics 28 (4) (2022) 778--790.
12. Y. Soykan, Special Cases of Generalized Leonardo Numbers: Modified $p$-Leonardo, $p$-Leonardo-Lucas and $p$-Leonardo Numbers, Earthline Journal of Mathematical Sciences 11 (2) (2023) 317--342.
13. S. K. Nurkan, İ. A. Güven, Ordered Leonardo Quadruple Numbers, Symmetry 15 (1) (2023) Article ID 149 15 pages.
14. E. Tan, H. H. Leung, On Leonardo $p$-Numbers, Integers (23) (2023) Article ID A7 11 pages.
15. O. Dişkaya, H. Menken, P. M. M. C. Catarino, On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions, Journal of New Theory (42) (2023) 74--85.
16. A. F. Horadam, Complex Fibonacci Numbers and Fibonacci Quaternions, American Mathematical Monthly (70) (1963) 289--291.
17. W. R. Hamilton, Elements of Quaternions, Chelsea Publishing Company, New York, 1969.
18. W. R. Hamilton, Lectures on Quaternions, Hodges and Smith, Dublin, 1853.
19. W. R. Hamilton, On Quaternions; or On a New System of Imaginaries in Algebra, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd Series), xxv-xxxvi, (1844–1850), https://www.emis.de/classics/Hamilton/OnQuat.pdf, Accessed 10 July 2023.
20. S. Yüce, F. T. Aydın, A New Aspect of Dual Fibonacci Quaternions, Advances in Applied Clifford Algebras (26) (2016) 873--884.
21. W. K. Clifford, Preliminary Sketch of Bi-Quaternions, Proceedings of the London Mathematical Society s1–4 (1) (1873) 381--395.
22. J. D. Jr. Edmonds, Relativistic Reality: A Modern View, World Scientific, Singapore, 1997.
23. Z. Ercan, S. Yüce, On Properties of the Dual Quaternions, European Journal of Pure and Applied Mathematics 4 (2) (2011) 142--146.
24. V. Majernik, Quaternion Formulation of the Galilean Space-Time Transformation, Acta Physica Slovaca 56 (1) (2006) 9--14.
25. V. Majernik, M. Nagy, Quaternionic Form of Maxwell's Equations with Sources, Lettere al Nuovo Cimento (16) (1976) 165--169.
26. V. Majernik, Galilean Transformation Expressed by the Dual Four-Component Numbers, Acta Physica Polonica A (87) (1995) 919--923.
27. Y. Yaylı, E. E. Tutuncu, Generalized Galilean Transformations and Dual Quaternions, Scientia Magna 5 (1) (2009) 94--100.