**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

**Full paper (PDF, 802 Kb)**

## 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

### References

- Bolat, C., & Köse, H. (2010). On the Properties of
*k*-Fibonacci Numbers. International Journal of Contemporary Mathematical Sciences, 22(5), 1097–1105. - Borel, É. (1899). Mémoire sur les séries divergentes. Annales scientifiques de l’École Normale Supérieure. Serie 3, 16(1), 9–131.
- Dikici, R., & Özkan, E. (2003). An application of Fibonacci sequences in groups. Applied Mathematics and Computation, 136 (2–3), 323–331.
- Falcon, S., & Plaza, A. (2007). On the Fibonacci
*k*-numbers. Chaos, Solitons & Fractals, 32(5), 1615–1624. - Hardy, G. H., & Ramanujan, S. (1917). Asymptotic formulae in combinatory analysis. Proceedings of the London Mathematical Society. Series 2, 16(1), 75–115.
- Hoggatt, V. E. Jr. (1969). Fibonacci and Lucas Numbers. Boston, MA: Houghton Mifflin.
- Hoggatt, V. E, Jr., & Bicknell, M. (1973). Roots of Fibonacci polynomials. The Fibonacci Quarterly, 11(3), 271–274.
- Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3(3), 161–176.
- Lucas, E. (1878). Théorie des Fonctions Numériques Simplement Périodiques. American Journal of Mathematics, 1, 184–240.
- Mikkawy, M., & Sogabe, T. (2010). A new family of
*k*-Fibonacci numbers. Applied Mathematics and Computation. 215, 4456–4461. - Özkan, E., & Altun, İ. (2019). Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials. Communications in Algebra, 47(10), 4020–4030.
- Özkan, E., Altun, İ., & Göçer, A. (2017). On Relationship among A New Family of
*k*-Fibonacci,*k*-Lucas Numbers, Fibonacci and Lucas Numbers. Chiang Mai Journal of Science, 44, 1744–1750. - Özkan, E., & Taştan, M. (2020). On Gauss Fibonacci polynomials, on Gauss Lucas polynomials and their applications. Communications in Algebra, 48(3), 952–960.
- Özkan, E., Taştan, M., & Aydoğdu, A. (2018). 2-Fibonacci polynomials in the family of Fibonacci numbers.
*Notes on Number Theory and Discrete Mathematics*, 24 (3), 47–55. - Rota, G.-C., Kahaner, D., & Oslyzko, A. (1975). Finite Operator Calculus. New York: Academic Press.
- Shannon, A. G. (1975). Fibonacci analogs of the classical polynomials. Mathematics Magazine, 48(3), 123–130.
- Shannon, A. G., & Deveci, O. (2020). A note on the coefficient array of a generalized Fibonacci polynomial.
*Notes on Number Theory and Discrete Mathematics*, 26(4), 206–212. - Singh, D. (1952). The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers. Mathematics Student, 20(1), 66–70.
- Sloane, N. J. A. (1973). A Handbook of Integer Sequences. New York: Academic Press.
- Taştan, M., & Özkan, E. (2021). On The Gauss
*K*−Fibonacci Polynomials. Electronic Journal of Mathematical Analysis and Applications, 9(1), 124–130. - Yılmaz, N., Aydoğdu, A., & Özkan, E. (2021). Some properties of
*k*-generalized Fibonacci Numbers. Mathematica Montisnigri, 50(7), 73–79.

## Related papers

- Akkuş, H., Deveci, Ö, Özkan, E., & Shannon, A. G. (2024). Discatenated and lacunary recurrences.
*Notes on Number Theory and Discrete Mathematics*, 30(1), 8-19.

## 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.