Hybrid hyper-Fibonacci and hyper-Lucas numbers

Yasemin Alp
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 1, Pages 154–170
DOI: 10.7546/nntdm.2023.29.1.154-170
Full paper (PDF, 311 Kb)

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Authors and affiliations

Yasemin Alp
Department of Education of Mathematics and Science, Selcuk University
Konya, Turkey

Abstract

Different number systems have been studied lately. Recently, many researchers have considered the hybrid numbers which are generalization of the complex, hyperbolic and dual number systems. In this paper, we define the hybrid hyper-Fibonacci and hyper-Lucas numbers. Furthermore, we obtain some algebraic properties of these numbers such as the recurrence relations, the generating functions, the Binet’s formulas, the summation formulas, the Catalan’s identity, the Cassini’s identity and the d’Ocagne’s identity.

Keywords

• Hybrid numbers
• Hyper-Fibonacci numbers
• Hyper-Lucas numbers

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 11R52

References

1. Alp, Y., & Kocer, E.G. (2021). Hybrid Leonardo numbers. Chaos, Solitons and Fractals, 150, Article 111128.
2. Ait-Amrane, L., & Behloul, D. (2020). Cassini determinant involving the
   (𝑎, 𝑏)-hyper-Fibonacci numbers. Malaya Journal of Matematik, 8(3), 939–944.
3. Ait-Amrane, L., & Behloul, D. (2022). Generalized hyper-Lucas numbers and applications. Indian Journal of Pure and Applied Mathematics, 53, 62–75.
4. Bahsi, M., Mező, I., & Solak, S. (2014). A symmetric algorithm for hyper-Fibonacci and hyper-Lucas numbers. Annales Mathematicae et Informaticae, 43, 19–27.
5. Bahsi, M., & Solak, S. (2020). On the norms of another form of 𝑟−circulant matrices with the hyper-Fibonacci and Lucas numbers. Turkish Journal of Mathematics and Computer Science, 12(2), 76–85.
6. Cao, N. N., & Zhao, F. Z. (2010). Some properties of hyper-Fibonacci and hyper-Lucas numbers. Journal of Integer Sequences, 13, Article 10.8.8.
7. Cerda-Moreles, G. (2021). Introduction to third-order Jacobsthal and modified third-order Jacobsthal hybrinomials. Discussiones Mathematicae General Algebra and Applications, 41, 139–152.
8. Cerda-Moreles, G. (2021). Investigation of generalized hybrid Fibonacci numbers and their properties. Applied Mathematics E-Notes, 21, 110–118.
9. Cristea, L. L., Martinjak, I., & Urbiha, I. (2016). Hyperfibonacci sequences and polytopic numbers. Journal of Integer Sequences, 19(7), Article 16.7.6.
10. Dil, A., & Mező, I.(2008). A symmetric algorithm for hyperharmonic and Fibonacci numbers. Applied Mathematics and Computation, 206, 942–951.
11. Kızılateş, C.(2020). A new generalization of Fibonacci hybrid and Lucas hybrid numbers. Chaos, Solitons and Fractals, 130, Article 109449.
12. Kızılates, C. (2022). A note on Horadam hybrinomials. Fundamental Journal of Mathematics and Applications, 5(1), 1–9.
13. Komatsu, T., & Szalay, L. (2017). A new formula for hyper-Fibonacci numbers and the number of occurrences. Turkish Journal of Mathematics, 43, 993–1004.
14. Koshy, T. (2018). Fibonacci and Lucas Numbers with Applications. Volume 1. John Wiley & Sons.
15. Liana, M., Szynal-Liana A., & Wloch I. (2019). On Pell hybrinomials. Miskolc Mathematical Notes, 20, 1051–1062.
16. Mangueira, M., Vieira, R., Alves, F., & Catarino, P. (2020). The hybrid numbers of Padovan and some identities. Annales Mathematicae Silesianae, 34(2), 256–267.
17. Özdemir M. (2018). Introduction to hybrid numbers. Advances in Applied Clifford Algebras, 28, Article 11.
18. Szynal-Liana, A. (2018). The Horadam hybrid numbers. Discussiones Mathematicae – General Algebra and Applications, 38(1), 91–98.
19. Szynal-Liana, A., & Wloch I. (2018). On Pell and Pell–Lucas Hybrid Numbers.
    Commentationes Mathematicae, 58(1), 11–17.
20. Szynal-Liana, A., & Wloch I. (2019). Introduction to Fibonacci and Lucas hybrinomials. Complex Variables and Elliptic Equations, 65(10), 1736–1747.
21. Szynal-Liana, A., & Wloch I. (2019). On Jacosthal and Jacosthal–Lucas hybrid numbers. Annales Mathematicae Silesianae, 33(1), 276–283.
22. Szynal-Liana, A., & Wloch I. (2019). The Fibonacci hybrid numbers. Utilitas Mathematica, 110, 3–10.
23. Szynal-Liana, A., & Wloch, I. (2020). On generalized Mersenne hybrid numbers. Annales Universitatis Mariae Curie-Sklodowska Lublin-Polonia, LXXIV(1), 77–84.
24. Szynal-Liana, A., & Wloch, I. (2020). On special spacelike hybrid numbers. Mathematics, 8, Article 1671.
25. Tan, E., & Ait-Amrane, N.R. (2022). On a new generalization of Fibonacci hybrid numbers, Indian Journal of Pure and Applied Mathematics, https://doi.org/10.1007/s13226-022-00264-3.
26. Tascı, D., & Sevgi, E. (2021). Some properties between Mersenne, Jacobsthal and
    Jacobsthal–Lucas hybrid numbers. Chaos, Solitons and Fractals, 146, Article 110862.
27. Vajda, S. (1989). Fibonacci and Lucas numbers, and the Golden Section: Theory and Applications. Halsted Press.

Manuscript history

• Received: 15 August 2022
• Revised: 4 February 2023
• Accepted: 21 March 2023
• Online First: 27 March 2023

Copyright information

Ⓒ 2023 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).