On Some New Generalized Gaussian Oresme Numbers

Year 2024, Volume: 14 Issue: 2, 241 - 248, 31.12.2024

Elifcan Sayın , Serpil Halıcı

https://doi.org/10.54370/ordubtd.1459920

Abstract

In this study, we defined and examined a new generalization to a special sequence of rational numbers. We gave the similarities and relationships of these newly defined Gaussian Oresme numbers with the generalized Fibonacci numbers and Lucas numbers existing in the literature. Moreover, we have obtained some important identities provided by these numbers we have discussed. We also obtained the relationship of these numbers with Gaussian Fibonacci numbers.

Keywords

fibonacci sequences, gaussian numbers, recurrence relation, oresme numbers

Ethical Statement

There are no ethical issues with the publication of this article.

Supporting Institution

This paper was suported Pamukkale University with Project No: 2023FEBE002.

Project Number

2023FEBE002

Bazı Yeni Genelleştirilmiş Gaussian Oresme Sayıları

Abstract

Bu çalışmada rasyonel sayıların özel bir dizisine yönelik yeni bir genelleme tanımladık ve inceledik. Yeni tanımlanan bu Gaussian Oresme sayılarının literatürde var olan genelleştirilmiş Fibonacci sayıları ve Lucas sayıları ile benzerliklerini ve ilişkilerini verdik. Üstelik ele aldığımız bu sayıların sağladığı bazı önemli kimlikleri de elde ettik. Bu sayıların Gaussian Fibonacci sayılarıyla ilişkisini de elde ettik.

Keywords

fibonacci dizileri, gaussian sayıları, tekrarlama bağıntısı, oresme sayıları

Project Number

2023FEBE002

References

• Akyuz, Z. (2013). Some identities and Fibonacci type polynomials containing terms of generalized Fibonacci and Lucas sequences [Doctoral dissertation, Sakarya University].
• Berzsenyi, G. (1977). Gaussian Fibonacci numbers. Fibonacci Quarterly, 15(3), 233-236. https://doi.org/10.1080/00150517.1977.12430445
• Cagman, A. (2021). Explicit solutions of powers of three as sums of three Pell numbers based on Baker’s type inequalities. Turkish J. Ineq, 5(1), 93-103. . https://www.tjinequality.com/volume-5/issue-1/
• Cagman, A. (2021). An approach to Pillai's problem with the Pell sequence and the powers of 3. Miskolc Mathematical Notes, 22(2), 599-610. https://doi.org/10.18514/MMN.2021.3659
• Cagman, A. (2023). Repdigits as sums of three Half-companion Pell numbers. Miskolc Mathematical Notes, 24(2), 687-697. https://doi.org/10.18514/MMN.2023.4143
• Cook, C. K. (2004). 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, 87-99. https://doi.org/10.1007/978-0-306-48517-6_10
• Halici, S., & Sayin, E. (2023). On Some Gaussian Oresme Numbers, In International Conference on Mathematics and its Applications in Science and Engineering, 66. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-031-49218-1_25
• Halici, S., & Oz, S. (2016). On some Gaussian Pell and Pell-Lucas numbers. Ordu Üniversitesi Bilim ve Teknoloji Dergisi, 6(1), 8-18. https://dergipark.org.tr/tr/pub/ordubtd/issue/24680/261001
• Halici, S., & Curuk, S. (2020). On Dual k- Pell Bicomplex Numbers and Some Identities Including Them. Fundamental Journal of Mathematics and Applications, 3(1), 86-93. https://doi.org/10.33401/fujma.718298
• Horadam, A. F. (1963). Complex Fibonacci Numbers and Fibonacci Quaternions, Amer. Math. Monthly, 70, 289-291. https://doi.org/10.2307/2313129
• Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. Fibonacci Quarterly, 3(3), 161–176. https://doi.org/10.1080/00150517.1965.12431416
• Horadam, A. F. (1974). Oresme numbers. Fibonacci Quarterly ,12(3), 267–270. https://doi.org/10.1080/00150517.1974.12430733
• Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. A.Wiley- Interscience Publication. https://doi.org/10.1002/9781118033067
• Ozkan, E., & Tastan, M. (2020). On Gauss Fibonacci polynomials, on Gauss Lucas polynomials and their applications. Communications in Algebra, 48(3), 952960. https://doi.org/10.1080/00927872.2019.1670193
• Pethe, S., & Horadam, A. F. (1986). Generalised Gaussian Fibonacci numbers. Bulletin of the Australian Mathematical Society, 33(1), 37-48. https://doi.org/10.1017/S0004972700002847
• Yilmaz, F., & Ertas, A. (2023). On Quaternions with Gaussian Oresme Coefficients. Turkish Journal of Mathematics and Computer Science, 15(1), 192-202. https://doi.org/10.47000/tjmcs.1133973