More identities on Fibonacci and Lucas hybrid numbers

Nazmiye Yilmaz
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 2, Pages 159—167
DOI: 10.7546/nntdm.2021.27.2.159-167
Download full paper: PDF, 159 Kb


Authors and affiliations

Nazmiye Yilmaz
Department of Mathematics, University of Karamanoğlu Mehmetbey
Kamil Özdağ Science Faculty, Turkey


We give several identities about Fibonacci and Lucas hybrid numbers. We introduce the Fibonacci and Lucas hybrid numbers with negative subscripts. We obtain different Cassini identities for the conjugate of the Fibonacci and Lucas hybrid numbers by two different determinant definitions of a hybrid square matrix (whose entries are hybrid numbers).


  • Binomial
  • Conjugate
  • Fibonacci hybrid number
  • Lucas hybrid number

2020 Mathematics Subject Classification

  • 11B39
  • 97F50


  1. Catarino, P. (2019). On k-Pell hybrid numbers. Journal of Discrete Mathematical Sciences and Cryptography, 22(1), 83–89.
  2. Cerda-Morales, G. (2018). Investigation of Generalized Hybrid Fibonacci Numbers and Their Properties. arXiv preprint, arXiv:1806.02231.
  3. Irmak, N. (2020). More identities for Fibonacci and Lucas quaternions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(1), 369–375.
  4. Kızılates, C. (2020). A new generalization of Fibonacci hybrid and Lucas hybrid numbers. Chaos, Solitons and Fractals, 130, 109449.
  5. Kızılates, C. (2020). A Note on Horadam Hybrinomials. Preprint, doi:10.20944/
  6. Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. John Wiley and Sons Inc., NY.
  7. Liana, M., Szynal-Liana, A., & Włoch, I. (2019). On Pell Hybrinomials. Miskolc
    Mathematical Notes, 20(2), 1051–1062.
  8. Morais, J. P., Georgiev, S., & Spröbig, W. (2014). Real Quaternionic Calculus Handbook. Birkhauser, London.
  9. Özdemir, M. (2018). Introduction to Hybrid Numbers. Advances in Applied Clifford Algebras, 28(11), 1–32.
  10. Szynal-Liana, A. (2018). The Horadam Hybrid Numbers. Discussiones Mathematicae – General Algebra and Applications, 38(1), 91–98.
  11. Szynal-Liana, A., & Włoch, I. (2018). On Pell and Pell–Lucas Hybrid Numbers.
    Commentationes Mathematicae, 58(1–2), 11–17.
  12. Szynal-Liana, A., & Włoch, I. (2019). On Jacobsthal and Jacobsthal–Lucas Hybrid Numbers. Annales Mathematicae Silesianae, 33(1), 276–283.
  13. Szynal-Liana, A., &Włoch, I. (2019). The Fibonacci hybrid numbers. Utilitas Mathematica, 110, 3–10.
  14. Szynal-Liana, A., & Włoch, I. (2020). Introduction to Fibonacci and Lucas hybrinomials. Complex Variables and Elliptic Equations, 65(10), 1736–1747.
  15. Sentürk, T.D., Bilgici, G., Dasdemir, A., & Ünal, Z. (2020). A Study on Horadam Hybrid Numbers. Turkish Journal of Mathematics, 44(4), 1212–1221.
  16. Tasyurdu, Y. (2019). Tribonacci and Tribonacci–Lucas Hybrid Numbers. International Journal of Contemporary Mathematical Sciences, 14(4), 245–254.

Related papers

Cite this paper

Yilmaz, N. (2021). More identities on Fibonacci and Lucas hybrid numbers. Notes on Number Theory and Discrete Mathematics, 27(2), 159-167, doi: 10.7546/nntdm.2021.27.2.159-167.

Comments are closed.