Generalized k-Order Fibonacci Hybrid Quaternions

Year 2022, , 670 - 683, 31.08.2022

Kübra Gül

https://doi.org/10.18185/erzifbed.1132164

Abstract

In this study, generalized k-order Fibonacci hybrid quaternion is defined. We give recurrence relation, generating function, the summation formula and some properties for these quaternions. Furthermore, the matrix representation for generalized k-order Fibonacci hybrid quaternions is determined. The Q_k matrix defined for k-order Fibonacci numbers is given for the generalized k-order Fibonacci hybrid quaternions. By the means of this matrix and another defined matrices, several identities of these quaternions are also obtained.

Keywords

Fibonacci sequence, k-order Fibonacci sequence, hybrid numbers, quaternions.

References

• Asci M., Aydinyuz S., (2021) Generalized k-order Fibonacci and Lucas hybrid numbers, Journal of Information and Optimization Sciences, 42(8), 1765-1782.
• Asci M., Aydinyuz S., (2021) k-order Fibonacci quaternions, Journal of Science and Arts; Targoviste, 21(1), 29-38.
• Dağdeviren A., Kürüz F., (2020) On the Horadam hybrid quaternions, https://arXiv.2012.08277.
• Deveci Ö., Shannon A., (2018) The quaternion-Pell sequence, Communications in Algebra, 46(12), 5403-5409.
• Gül K., (2019) On bi-periodic Jacobsthal and Jacobsthal-Lucas quaternions, Journal of Mathematics Research ,11(2), 44-52.
• Gül K., (2020) Dual bicomplex Horadam quaternions, Notes on Number Theory and Discrete Mathematics, 26(4), 187–205.
• Halici S., (2012) On Fibonacci quaternions, Adv. Appl. Clifford Algebras, 22(2), 321-327.
• Hamilton W. R., (1866) Elements of quaternions. Longmans, Green and Co., London.
• Horadam A. F., (1963) Complex Fibonacci numbers and Fibonacci quaternions, Math. Mag., 70(3), 289-291.
• Kızılateş C., (2020) A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos, Solitons & Fractals, 130.
• Szynal-Liana A., Wloch I. (2019). The Fibonacci hybrid numbers, Utilitas Math., 110, 3–10.
• Özdemir M., (2018) Introducion to hybrid numbers, Adv. Appl. Clifford Algebra, 28(1), 11-32.
• Petroudi S. H. J., Pirouz M., . Karaca E and Yılmaz F., (2021) A combined approach to Perrin and Padovan hybrid sequences, Heliyon, 7 (8).
• Soykan Y., (2019) Simson Identity of Generalized m-step Fibonacci Numbers, Int. J. Adv. Appl. Math. and Mech., 7(2), 45 – 56.
• Şentürk T. D., Bilgici G., Daşdemir A. and Ünal Z., (2020) A Study on Horadam Hybrid Numbers, Turkish Journal of Mathematics, 44, 1212-1221.
• Er M.C., (1984) Sums of Fibonacci numbers by matrix methods, Fibonacci Quart, 22(3), 204-207.
• Uysal M., Özkan E., (2022) Padovan Hybrid Quaternions and some properties, Journal of Science and Arts, 22(1), 121-132.
• Kilic E., Dursun T., (2006) On the generalized order-k Fibonacci and Lucas numbers, Rocky Mountain J. Math., 36(6), 1915-1926.
• Lee G.Y., Lee S.G., Kim J.S., Shin H.K., (2001) The Binet Formula and Representations of k-Generalized Fibonacci Numbers, The Fibonacci Quart. 158-164.
• Lee G.Y., Lee S.G., (1995) A note on generalized Fibonacci numbers, Fibonacci Quart., 33, 273-278.