Generalized bivariate conditional Fibonacci and Lucas hybrinomials

Abstract

The Hybrid numbers are generalizations of complex, hyperbolic and dual numbers. In recent years, studies related with hybrid numbers have been increased significantly. In this paper, we introduce the generalized bivariate conditional Fibonacci and Lucas hybrinomials. Also, we present the Binet formula, generating functions, some significant identities, Catalan’s identities and Cassini’s identities of the generalized bivariate conditional Fibonacci and Lucas hybrinomials. Finally, we give more general results compared to the previous works.

Keywords

• Bivariate conditional polynomials
• hybrid numbers
• Binet formula's
• generating function
• Catalan's identities
• Cassini's identities

References

1. Ait-Amrane, N. R., Belbachir, H., Bi-periodic r-Fibonacci sequence and bi-periodic r-Lucas sequence of type s, Hacettepe Journal of Mathematics and Statistics, 51 (3) (2022), 680–699. https://dx.doi.org/10.15672/hujms.825908.
2. Ait-Amrane, N. R., Belbachir, H., Tan, E., On generalized Fibonacci and Lucas hybrid polynomials, Turkish Journal of Mathematics, 46 (6) (2022), 2069–2077. https://dx.doi.org/10.55730/1300-0098.3254.
3. Bala, A., Verma, V., Some properties of bi-variate bi-periodic Lucas polynomials, Annals of the Romanian Society for Cell Biology (2021), 8778–8784.
4. Belbachir, H., Bencherif, F., On some properties on bivariate Fibonacci and Lucas polynomials, arXiv preprint arXiv:0710.1451 (2007). https://dx.doi.org/10.48550/arXiv.0710.1451.
5. Bilgici, G., Two generalizations of Lucas sequence, Applied Mathematics and Computation, 245 (2014), 526–538. https://dx.doi.org/10.1016/j.amc.2014.07.111.
6. Edson, M., Yayenie, O., A new generalization of Fibonacci sequence & extended Binet’s formula, Integers, 9 (6) (2009), 639–654. https://dx.doi.org/10.1515/INTEG.2009.051.
7. Falcon, S., Plaza, ´A., The k−Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals, 33 (1) (2007), 38–49. https://dx.doi.org/10.1016/j.chaos.2006.10.022.
8. Kızılateş, C., A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos, Solitons & Fractals, 130 (2020), 109449. https://dx.doi.org/10.1016/j.chaos.2019.109449.

9. Koshy, T., Fibonacci and Lucas Numbers with Applications, Volume 2, John Wiley & Sons, 2019.
10. Özdemir, M., Introduction to hybrid numbers, Advances in applied Clifford algebras, 28 (2018), 1–32. https://dx.doi.org/10.1007/s00006-018-0833-3.
11. Panwar, Y. K., Singh, M., Generalized bivariate Fibonacci-like polynomials, International Journal of Pure Mathematics, 1 (8) (2014), 13.
12. Sevgi, E., The generalized Lucas hybrinomials with two variables, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70 (2) (2021), 622–630, https://dx.doi.org/10.31801/cfsuasmas.854761.
13. Szynal-Liana, A., The Horadam hybrid numbers., Discussiones Mathematicae: General Algebra & Applications, 38 (1) (2018),.https://dx.doi.org/10.7151/dmgaa.1287.
14. Szynal-Liana, A., W loch, I., Introduction to Fibonacci and Lucas hybrinomials, Variables and Elliptic Equations, 65 (10) (2020), 1736–1747. https://dx.doi.org/10.1080/17476933.2019.1681416.
15. Verma, A. B., Bala, A., On properties of generalized bi-variate bi-periodic Fibonacci polynomials, International journal of Advanced science and Technology, 29 (3) (2020), 8065–8072.
16. Yayenie, O., A note on generalized Fibonacci sequences, Applied Mathematics and Computation, 217 (12) (2011), 5603–5611. https://dx.doi.org/10.1016/j.amc.2010.12.038.
17. Yazlik, Y., Köme, C., Madhusudanan, V., A new generalization of Fibonacci and Lucas p-numbers, Journal of computational analysis and applications, 25 (4) (2018), 657–669.
18. Yilmaz, N., Coskun, A., Taskara, N., On properties of bi-periodic Fibonacci and Lucas polynomials, In AIP Conference Proceedings (2017), vol. 1863, AIP Publishing LLC, p. 310002. https://dx.doi.org/10.1063/1.4992478.