Merve Taştan, Engin Özkan and Anthony G. Shannon

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 2, Pages 148—158

DOI: 10.7546/nntdm.2021.27.2.148-158

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## Details

### Authors and affiliations

Merve Taştan

*Graduate School of Natural and Applied Sciences, Erzincan Binali Yildirim University
Erzincan, Turkey
*

Engin Özkan

*Department of Mathematics, Erzincan Binali Yildirim University
Erzincan, Turkey
*

Anthony G. Shannon

*Warrane College, the University of New South Wales
Kensington, NSW 2033, Australia
*

### Abstract

In this paper, we define new families of Generalized Fibonacci polynomials and Generalized Lucas polynomials and develop some elegant properties of these families. We also find the relationships between the family of the generalized *k*-Fibonacci polynomials and the known generalized Fibonacci polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we establish Cassini’s Identities for the families and their polynomials. Finally, we suggest avenues for further research.

### Keywords

- Generalized Fibonacci polynomials
*k*-Fibonacci numbers- Generalized Lucas polynomials
*k*-Lucas numbers

### 2020 Mathematics Subject Classification

- 11B39
- 11B83
- 11C20

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## Cite this paper

Taştan, M., Özkan, E., & Shannon, A. G. (2021). The generalized *k*-Fibonacci polynomials and generalized *k*-Lucas polynomials. Notes on Number Theory and Discrete Mathematics, 27(2), 148-158, doi: 10.7546/nntdm.2021.27.2.148-158.