On Quaternions with Higher Order Jacobsthal Numbers Components

Abstract

In this study, we present higher order Jacobsthal numbers. Then we define higher order Jacobsthal quaternions by using higher order Jacobsthal numbers. We give the concept of the norm and conjugate for these quaternions. We express and prove some propositions related to these quaternions. Also, we find the recurrence relation, the Binet formula and the generating function for these quaternions. Finally, we calculate Cassini, Catalan, Vajda and d’Ocagne identities for higher order Jacobsthal quaternions.

Keywords

• Exponential generating functions
• Higher order Jacobsthal numbers
• Higher order Jacobsthal quaternions
• Binet formula

References

1. [1] Hamilton, W.R. Elements of quaternions, Green & Company, London: Longman, (1866).
2. [2] Horadam, A. F., ‘‘Complex Fibonacci numbers and Fibonacci quaternions”, The American Mathematical Monthly, 70(3): 289-291, (1963).
3. [3] Halici, S. ‘‘On Fibonacci quaternions”, Advances in applied Clifford algebras, 22(2): 321-327, (2012).
4. [4] Iyer, M. R., ‘‘Some results on Fibonacci quaternions”, The Fibonacci Quarterly, 7(2): 201-210, (1969).
5. [5] Kizilates, C., Kone, T., ‘‘On higher order Fibonacci quaternions”, J. Anal., 29(4): 1071-1082, (2021).
6. [6] Kizilates, C., ‘‘On quaternions with incomplete Fibonacci and Lucas numbers components”, Util. Math., 110: 263–269, (2019).
7. [7] Kızılates, C., Polatlı, E., ‘’New families of Fibonacci and Lucas octonions with q-integer components’’, Indian Journal of Pure and Applied Mathematics, 52(1): 231–240, (2021).
8. [8] Polatlı, E., Kesim, S., ‘’On quaternions with generalized Fibonacci and Lucas number components’’, Advances in Difference Equations, 2015(1): 1-8, (2015).

9. [9] Polatlı, E., ‘’A generalization of Fibonacci and Lucas Quaternions’’, Advances in Applied Clifford Algebras, 26 (2): 719–730, (2016).
10. [10] Kızılateş, C., Catarino, P., Tuğlu, N., ‘’On the bicomplex generalized Tribonacci quaternions’’, Mathematics, 7(1): 1-8, (2019).
11. [11] Koshy, T., Fibonacci and Lucas numbers with applications, Wiley-Interscience publishing, Canada, (2001).
12. [12] Özkan, E., Taştan, M., Aydoğdu, A., “2-Fibonacci polynomials in the family of Fibonacci numbers”, Notes on Number Theory and Discrete Mathematics, 24(3): 47- 55, (2018).
13. [13] Yılmaz, N., Aydoğdu, A., Özkan, E., ‘‘Some properties of k-generalized fibonacci numbers”, Mathematica Montisnigri, 50(7): 73-79, (2021).
14. [14] Özkan, E., “3-step fibonacci sequences in nilpotent groups”, Applied Mathematics and Computation, 144(2-3): 517-527, (2003).
15. [15] Özkan, E., Altun, İ., and Göçer, A., “On relationship among a new family of k-fibonacci, k-lucas numbers, fibonacci and lucas numbers”, Chiang Mai Journal of Science, 44(3): 1744-1750, (2017).
16. [16] Kizilates, C., Kone, T., ‘‘On higher order Fibonacci hyper complex numbers’’, Chaos Solitons Fractals, 148, 111044, (2021).
17. [17] Çelik, S., Durukan, İ., Özkan, E., New recurrences on pell numbers, Pell-Lucas numbers, Jacobsthal numbers, and Jacobsthal-Lucas numbers, Chaos, Solitons and Fractals, 150, 111173, (2021).
18. [18] Deveci, Ö., Shannon, A. G., ‘‘The complex-type k-Fibonacci sequences and their applications’’, Communications in Algebra, 49(3): 1352-1367, (2021).
19. [19] Özkan, E., Taştan, M., ‘‘A New Families of Gauss k-Jacobsthal Numbers and Gauss k-Jacobsthal-Lucas Numbers and Their Polynomials’’, Journal of Science and Arts, 4(53): 893-908, (2020).
20. [20] Uygun, S., ‘‘A New Generalization for Jacobsthal and Jacobsthal Lucas Sequences”, Asian Journal of Mathematics and Physics, 2(1): 14-21, (2018).
21. [21] Jhala, D., Sisodiya, K., Rathore, G. P. S., ‘‘On some identities for k-Jacobsthal numbers”, International Journal of Mathematical Analysis, (Ruse), 7(9-12): 551-556, (2013).
22. [22] Cook, C. K., Bacon, M. R., ‘‘Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations”, In Annales mathematicae et informaticae, 41, 27-39, (2013).
23. [23] Horadam A.F., ‘‘Jacobsthal representation numbers’’, Fibonacci Quart., 34(1): 40-53, (1996).
24. [24] Szynal-Liana, A., Włoch, I., ‘‘A note on Jacobsthal quaternions”, Advances in Applied Clifford Algebras, 26(1): 441-447, (2016).
25. [25] Torunbalcı-Aydın, F., Yüce, S., ‘‘A new approach to Jacobsthal quaternions”, Filomat, 31(18): 5567-5579, (2017).
26. [26] Tasci, D., ‘‘On k-Jacobsthal and k-Jacobsthal-Lucas quaternions”, Journal of Science and Arts, 17(3): 469-476, (2017).
27. [27] Polatlı E., ‘’On certain properties of Quadrapell quaternions’’, Karaelmas Science & Engineering Journal, 8 (1): 305–308, (2018).
28. [28] Özkan, E., Uysal, M., Godase, A. D., ‘‘Hyperbolic k-Jacobshtal and k-Jacobshtal-Lucas Quaternions, Indian Journal of Pure and Applied Mathematics, 52(3): 1-12, 2021.
29. [29] Cerda-Morales, G., ‘‘Identities for third order Jacobsthal quaternions”, Advances in Applied Clifford Algebras, 27(2): 1043-1053, (2017).