On Generalized Hybrid Oresme Numbers with Hybrid k-Oresme Coefficients

Year 2025, Volume: 13 Issue: 1 , 28 - 36 , 30.04.2025

Elifcan Sayın , Serpil Halıcı

https://izlik.org/JA69PK86SF

Abstract

In current paper, we have defined a generalization of Oresme numbers with hybrid $\textit{k- }$ Oresme coefficients. We obtained the generating function of these newly defined numbers. Using generating function, we obtained some fundamentel and important identities containing the elements of the newly defined number sequence. We also gave some special sum formulas of this sequence we are considering.

Keywords

Hybrid numbers , Recurrence relations , Oresme numbers

References

• [1] Cook, C. K., Some sums related to sums of Oresme numbers,In Applications of Fibonacci Numbers: Volume 9: Proceedings of The Tenth International Research Conference on Fibonacci Numbers and Their Applications, Dordrecht: Springer Netherlands, (2004), 87-99.
• [2] Diskaya, O. and Menken, H., On the (s, t)-Padovan and (s, t)-Perrin Quaternions. J. Adv. Math. Stud., 12(2), (2019), 186–192.
• [3] Diskaya, O. and Menken, H., On the bicomplex Padovan and bicomplex Perrin numbers. Acta Universitatis, (2023).
• [4] Halici, S., On Fibonacci quaternions. Adv. Appl. Clifford Algebras, 22(2), (2012), 321-327.
• [5] Halici, S., On complex Fibonacci quaternions. Advances in applied Clifford algebras, 23, (2013), 105-112.
• [6] Halici, S. and Sayin, E., On some k- Oresme hybrid numbers. Utilitas Mathematica, 120, (2023), 1-11.
• [7] Horadam, A. F., Fibonacci Numbers and Fibonacci Quaternions, American Mathematical Monthly, 70, (1963), 289–291.
• [8] Horadam, A. F., Basic properties of a certain generalized sequence of numbers,The Fibonacci Quarterly, 3(3), (1965), 161-176.
• [9] Horadam, A. F., Oresme numbers,The Fibonacci Quarterly, 12(3), (1974), 267-271.
• [10] Iyer, M. R., Some Results on Fibonacci Quaternions, Fibonacci Quarterly, 7(2), (1969), 201–210.
• [11] Kızılates, C., A new generalization of Fibonacci hybrid and Lucas hybrid numbers. Chaos, Solitons and Fractals, (2020), 130, 109449.
• [12] Oresme, N., Quaestiones super geometriam Euclidis (Vol. 3), Brill Archive, (1961).
• [13] Ozdemir, M., Introduction to hybrid numbers, Advances in applied Clifford algebras, 28(1),(2018), 1-32.
• [14] Polatlı E., Hybrid numbers with Fibonacci and Lucas hybrid number coefficients, Universal Journal of Mathematics and Applications, 6(3), (2020), 106-113.
• [15] Polatlı, E., A note on ratios of Fibonacci hybrid and Lucas hybrid numbers. Notes Number Theory Discrete Math, 27(3), (2021), 73-78.
• [16] Szynal-Liana, A., The Horadam hybrid numbers, Discussiones Mathematicae-General Algebra and Applications, 38(1), (2018), 91-98.
• [17] Szynal-Liana, A. and Wloch, I, Oresme hybrid numbers and hybrationals, Kragujevac Journal of Mathematics, 48(5), (2024), 747-753.
• [18] Tan, E. and Ait-Amrane, N. R., On a new generalization of Fibonacci hybrid numbers, Indian Journal of Pure and Applied Mathematics, 54(2), (2023), 428-438.
• [19] Uysal, M. Kumarı, M., Kulog˘lu, B., Prasad, K. and O¨ zkan, E., On the hyperbolic k􀀀 Mersenne and k􀀀 Mersenne-Lucas octonions. Kragujevac Journal of Mathematics, 49(5), (2025).
• [20] Dagdeviren, A. and K¨ur¨uz, F., On the Horadam hybrid quaternions, arXiv preprint arXiv:2012,08277, (2020).