On Higher Order Lucas Hybrid Quaternions

Abstract

In this article, we introduced higher order Lucas hybrid quaternions with the help of higher order Lucas numbers. We also examined some algebraic properties of these quaternions. By obtaining the recurrence relation, we found the Binet formula, the generating function and the exponential generating function. Finally, we calculated the Vajda identity for the higher order Lucas hybrid quaternions and obtained the Catalan, Cassini and d'Ocagne identities with the help of this identity.

Keywords

• Fibonacci number
• Lucas number
• Hybrid number
• Higher order Lucas number
• Binet formula

References

1. [1] Hamilton, W. R.: Elements of Quaternions. Longmans, Green and Co., London, (1866)
2. [2] Horadam, A. F.: Complex Fibonacci Numbers and Fibonacci Quaternions. American Math. Monthly. 70 (3), 289-291 (1963)
3. [3] Halıcı S.: On Fibonacci Quaternions. Adv. Appl. Clifford Algebras. 22 (2), 321-327 (2013)
4. [4] E. Polatli, S. Kesim, On quaternions with generalized fibonacci and lucas number components, Advances in Difference Equations 2015 (1) (2015) 1–8.
5. [5] E. Polatlı, A generalization of fibonacci and lucas quaternions, Advances in Applied Clifford Algebras 26 (2016) 719–730.
6. [6] E. Tan, S. Yilmaz, M. Sahin, On a new generalization of fibonacci quaternions, Chaos, Solitons & Fractals 82 (2016) 1–4.
7. [7] M. Akyigit, H. H¨uda K¨osal, M. Tosun, Fibonacci generalized quaternions, Advances in Applied Clifford Algebras 24 (2014) 631–641.
8. [8] S. Halici, A. Karata¸s, On a generalization for fibonacci quaternions, Chaos, Solitons & Fractals 98 (2017) 178–182.

9. [9] A. Horadam, Quaternion recurrence relations, Ulam Quarterly 2 (2) (1993) 23–33.
10. [10] M. R. Iyer, A note on fibonacci quaternions, Fibonacci Quart 7 (3) (1969) 225–229.
11. [11] M. Ozvatan, Generalized golden-fibonacci calculus and applications, Ph.D. thesis, Izmir Institute of Technology (Turkey) (2018).
12. [12] C. Kizilates¸, T. Kone, On higher order fibonacci quaternions, The Journal of Analysis (2021) 1–12.
13. [13] M. Uysal, E. Ozkan, Higher-order jacobsthal–lucas quaternions, Axioms 11 (12) (2022) 671.
14. [14] E. O¨ zkan, M. Uysal, On quaternions with higher order jacobsthal numbers components, Gazi University Journal of Science (2023) 1–1.
15. [15] M. Ozdemir, Introduction to hybrid numbers, Advances in applied Clifford algebras 28 (2018) 1–32.
16. [16] A. Da˘gdeviren, F. K¨ur¨uz, On the horadam hybrid quaternions, arXiv preprint arXiv:2012.08277.
17. [17] M. d. S. MAngueria, F. Alves, P. Catarino, Hybrid quaternions of leonardo, Trends in Computational and Applied Mathematics 23 (2022) 51–62.