On fourth-order jacobsthal quaternions
Abstract
In this paper, we present for the first time a sequence of quaternions of order 4 that we will call the fourth-order Jacobsthal and the fourth-order Jacobsthal-Lucas quaternions. In particular, we are interested in the generating function, Binet formula, explicit formula and some interesting results for fourth-order Jacobsthal quaternions and fourth-order Jacobsthal-Lucas quaternions. This generalizes some previous results given by Szynal-Liana and Wloch in [13], Torunbalci Aydin and Yüce in [14] and Cerda-Morales in [2].
Keywords
Fourth-order Jacobsthal number,Jacobsthal number,quaternion,Recurrence relation
References
- [1] P. Barry, Triangle geometry and Jacobsthal numbers, Irish Math. Soc. Bulletin 51 (2003), 45–57.
- [2] G. Cerda-Morales, Identities for Third Order Jacobsthal Quaternions, Adv. Appl. Clifford Algebras 27(2) (2017), 1043–1053.
- [3] C. K. Cook and M.R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae 41 (2013), 27–39.
- [4] O. Deveci, E. Karaduman and G. Saˇglam, The Jacobsthal sequences in finite groups, Bull. Iranian Math. Soc. 42(1) (2016), 79–89.
- [5] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebras 22 (2012), 321–327.
- [6] S. Halici, On complex Fibonacci quaternions, Adv. Appl. Clifford Algebras 23 (2013), 105–112.
- [7] A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Month. 70 (1963), 289–291.
- [8] A. F. Horadam, Quaternion recurrence relations, Ulam Quarterly 2 (1993), 23–33.
- [9] A. F. Horadam, Jacobsthal representation numbers, Fibonacci Quarterly 34 (1996), 40–54.
- [10] M. R. Iyer, A note on Fibonacci quaternions, Fibonacci Quaterly 7(3) (1969), 225–229.
