A new perspective on bicomplex numbers with Leonardo number components

Year 2023, Volume: 72 Issue: 2, 340 - 351, 23.06.2023

Murat Turan , Sıddıka Özkaldı Karakuş , Semra Kaya Nurkan

https://doi.org/10.31801/cfsuasmas.1181930

Cited By: 2

https://izlik.org/JA49BT33BH

Abstract

In the present paper, the bicomplex Leonardo numbers will be introduced with the use of Leonardo numbers and some important algebraic properties including recurrence relation, generating function, Catalan’s and Cassini’s identities, Binet’s formula, sum formulas will also be obtained.

Keywords

Fibonacci numbers , bicomplex numbers , Leonardo numbers

References

• Alp, Y., Koçer, E. G., Some properties of Leonardo numbers, Konuralp J. Math., 9(1) (2021), 183–189.
• Alp, Y., Koçer, E. G., Hybrid Leonardo numbers, Chaos, Solitons and Fractals, 150 (2021), 111–128. https://doi.org/10.1016/j.chaos.2021.111128
• Alves, F. R. V., Catarino, P. M. M. C., A forma matricial dos n´umeros de Leonardo, Ciencia e natura, 42 (2020), 1–6. https://doi.org/10.5902/2179460X41839
• Catarino, P., Borges, A., On Leonardo numbers, Acta Mathematica Universitatis Comenianae, 89(1) (2019), 75–86.
• Catarino, P., Borges, A., A note on incomplete Leonardo numbers, Integers, 20(7) (2020).
• Halıcı, S., On bicomplex Fibonacci numbers and their generalization, In Models and Theories in Social Systems, (2019), 509–524. https://doi.org/10.1007/978-3-030-00084-426
• Hamilton, W. R., Lectures on Quaternions, Hodges and Smith, Dublin, 1853.
• Hoggatt, V. E., Fibonacci and Lucas Numbers, A publication of the Fibonacci Association University of Santa Clara, Santa Clara, Houghton Mifflin Company, 1969.
• Horadam, A. F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quarterly 3 (1965), 161–176.
• Kızılates C, Kone T. On higher order Fibonacci hyper complex numbers, Chaos Solitons Fractals, 148 (2021), 111044. https://doi.org/10.1016/j.chaos.2021.111044
• Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, Hoboken, NJ, USA, 2019.
• Kuruz, F., Dagdeviren, A., Catarino, P., On Leonardo Pisano hybrinomials, Mathematics, 9(22) (2021), 2923. https:/doi.org/10.3390/math9222923
• Luna-Elizarraras, M. E., Shapiro, M., Struppa, D. C., Bicomplex numbers and their elementary functions, Cubo 14 (2012), 61–80.
• Nurkan, S. K., Guven, I. A., A Note on bicomplex Fibonacci and Lucas numbers, International Journal of Pure and Applied Mathematics, 120(3) (2018), 365–377. https:/doi.org/10.12732/ijpam.v120i3.7
• Price, G. B., An Introduction to Multicomplex Spaces and Functions, Monographs and Textbooks in Pure and Applied Mathematics, M. Dekker, New York, NY, USA, 1991.
• Rochon, D., Shapiro, M., On algebraic properties of bicomplex and hyperbolic numbers, Anal. Univ. Oradea Fasc. Math., 11 (2004), 71–110.
• Segre, C., The real representation of complex elements and hyperalgebraic entities (Italian), Math. Ann., 40 (1892), 413–467.
• Shannon, A. G., A note on generalized Leonardo numbers, Notes Number Theory Discrete Math., 25(3) (2019), 97–101. https:/doi.org/10.7546/nntdm.2019.25.3.97-101.
• Sloane, N. J. A., The On-line Encyclopedia of Integers Sequences. 1964.
• Tan, E., Leung H. H., On Leonardo p-numbers, Integers, 23 (2023), 1-11. DOI: 10.5281/zenodo.7569221
• Torunbalcı, A., Bicomplex Fibonacci quaternions, Chaos, Solitons and Fractals, 106 (2018), 147–153. https://doi.org/10.1016/j.chaos.2017.11.026
• Vajda, S., Fibonacci and Lucas Numbers and The Golden Section, Ellis Horwood Limited Publ., England, 1989.