q-Leonardo Bicomplex Numbers

Abstract

Abstract: In the paper, we define the $q$-Leonardo bicomplex numbers by using the $q$-integers. Also, we give some algebraic properties of $q$-Leonardo bicomplex numbers such as recurrence relation, generating function, Binet's formula, D'Ocagne's identity, Cassini's identity, Catalan's identity and Honsberger identity.

Keywords

• Bicomplex number
• q-integer
• Fibonacci number
• Bicomplex Fibonacci number
• Leonardo number
• q-Leonardo number
• q-Leonardo bicomplex number

References

1. [1] Adler S. L., and others. Quaternionic quantum mechanics and quantum fields, Oxford University Press on Demand, 88, (1995).
2. [2] Y. Alp and E.G. Koc¸er, Some properties of Leonardo numbers., Konuralp Journal of Mathematics (KJM), 9(1), (2021), 183-189.
3. [3] I. Akkus, G. Kizilaslan, Quaternions: Quantum calculus approach with applications., Kuwait Journal of Science, 46 (4), (2019).
4. [4] Andrews G. E., Askey R., and Roy R. Special functions, Cambridge University Press, 71, (1999).
5. [5] Arfken G. B., and Weber H. J. Mathematical methods for physicists, American Association of Physics Teachers, (1999).
6. [6] F. Torunbalcı Aydın, Bicomplex fibonacci quaternions., Chaos Solitons Fractals, Vol. 106, (2018), 147-153.
7. [7] F. Torunbalcı Aydın, q-Fibonacci bicomplex and q-Lucas bicomplex numbers., Notes on Number Theory and Discrete Mathematics, 28(2),(2022), 261-275.
8. [8] M. Turan, S. O¨ zkaldı Karakus¸ and S. Nurkan, A New Perspective on Bicomplex Numbers: Leonardo Component Extension., preprint, (2022).

9. [9] P. Catarino, and Anabela Borges, On leonardo numbers., Acta Mathematica Universitatis Comenianae, 89(1), (2019), 75-76.
10. [10] P. Catarino and A. Borges, A note on incomplete Leonardo numbers., Integers, Vol. 20(7), (2020).
11. [11] S. Halıcı, On bicomplex Fibonacci numbers and their generalization., Models and Theories in Social Systems, (2019), 509-524.
12. [12] Kac V. G., and Cheung P. Quantum calculus, Springer 113, (2002).
13. [13] C. Kızılates¸, A new generalization of Fibonacci hybrid and Lucas hybrid numbers., Chaos, Solitons and Fractals, 130, 109449, (2020).
14. [14] M. Luna-Elizarraras, M. Shapiro, D. C. Struppa, A. Vajiac, Bicomplex numbers and their elementary functions., Cubo (Temuco) 14 (2), (2012), 61-80.
15. [15] M. Luna-Elizarraras, M. Shapiro, D. C. Struppa, A. Vajiac, Bicomplex holomorphic functions: the algebra, geometry and analysis of bicomplex numbers., Birkhauser, (2015)
16. [16] S. Kaya Nurkan and A. I. G¨uven, A note on bicomplex Fibonacci and Lucas numbers., arXiv preprint arXiv:1508.03972, (2015).
17. [17] D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers., Anal. Univ. Oradea, fasc. math, 11, (71), (2004), 110.
18. [18] Segre C. Le rappresentazioni reali delle forme complesse e gli ente iper-algebrici, Mathematische Annalen, Vol. 40(3), 413-467 (1892).
19. [19] A. G. Shannon, A note on generalized Leonardo numbers., Notes on Number Theory and Discrete Mathematics, 25(3), (2019), 97-101.
20. [20] R. P. M. Vieira, F. R. V. Alves, and P. M. M. C. Catarino, Relac¸ ˜oes bidimensionais e identidades da sequˆencia de Leonardo., Revista Sergipana de Matem´atica e Educac¸ ˜ao Matem´atica, 4(2), (2019), 156-173 .