Engin Özkan, Merve Taştan and Ali Aydoğdu

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 3, Pages 47—55

DOI: 10.7546/nntdm.2018.24.3.47-55

**Download full paper: PDF, 206 Kb**

## Details

### Authors and affiliations

Engin Özkan

*Department of Mathematics, University of Erzincan Binali Yıldırım
Faculty of Arts and Sciences, Yalnızbag Campus, 24100, Erzincan, Turkey*

Merve Taştan

*Department of Mathematics, University of Erzincan Binali Yıldırım
Faculty of Arts and Sciences, Yalnızbag Campus, 24100, Erzincan, Turkey*

Ali Aydoğdu

*Department of Mathematics, University of Beykent
Ayazaga Campus, Ayazaga-Maslak, Sarıyer, 34485, Istanbul, Turkey*

### Abstract

In the present study, we define new 2-Fibonacci polynomials by using terms of a new family of Fibonacci numbers given in [4]. We show that there is a relationship between the coefficient of the 2-Fibonacci polynomials and Pascal’s triangle. We give some identities of the 2-Fibonacci polynomials. Afterwards, we compare the polynomials with known Fibonacci polynomials. We also express 2-Fibonacci polynomials by the Fibonacci polynomials. Furthermore, we prove some theorems related to the polynomials. Also, we introduce the derivative of the 2-Fibonacci polynomials.

### Keywords

- Fibonacci numbers
- Fibonacci polynomials
- Generalized Fibonacci polynomials

### 2010 Mathematics Subject Classification

- 11B39

### References

- Berg, C. (2011) Fibonacci numbers and orthogonal polynomials, Arab Journal of Mathematical Sciences, 17, 75–88.
- Falcon S. & Plaza, A. (2009) On 𝑘-Fibonacci sequences and polynomials and their derivatives, Chaos, Solutions and Fractals, 39, 1005–1019.
- Hoggatt, Jr. V. E. & Lind, D. A. (1968) Symbolic Substitutions in to Fibonacci Polynomials, The Fibonacci Quarterly, 6(5), 55–74.
- Hoggatt, Jr. V. E., Leonard, H. T. Jr. & Philips, J. W. (1971) Twenty-four Master Identities, The Fibonacci Quarterly, 9(1), 1–17.
- Hoggatt, Jr. V. E. & Bicknell, M. (1973) Generalized Fibonacci polynomials, Fibonacci Quarterly, 11(5), 457–465.
- Ivie, J. (1972) A General Q-Matrix, Fibonacci Quarterly, 10(3), 255–261.
- Koshy, T. (2001) Fibonacci and Lucas Numbers with Applications, JohnWiley & Sons, Inc., Canada.
- Lee G. Y. & Asci, M. (2012) Some Properties of the (𝑝, 𝑞)-Fibonacci and (𝑝, 𝑞)-Lucas Polynomials, Journal of Applied Mathematics, 2012, 18 pages.
- Mikkawy M. & Sogabe, T. (2010) A new family of 𝑘-Fibonacci numbers, Applied Mathematics and Computation, 215, 4456–4461.
- Nalli, A. & Haukkanen, P. (2009) On generalized Fibonacci and Lucas polynomials, Chaos, Solitons and Fractals, 42(5), 3179–3186.
- Panwar, Y. K., Singh, B. & Gupta, V. K. (2013) Generalized Fibonacci Polynomials, Turkish Journal of Analysis and Number Theory, 1(1), 43–47.
- Ramirez, J. (2014) On convelved generalized Fibonacci and Lucas polynomials, Applied Mathematics and Computation, 229, 208–213.
- Swamy, M. N. S. (1965) Problem B-74, The Fibonacci Quarterly, 3(3), 236.
- Tuğlu, N., Koçer, E. G. & Stakhov, A. (2011) Bivariate fibonacci like 𝑝-polynomials, Applied Mathematics and Computation, 217, 10239–10246.
- Ye, X. & Zang, Z. (2017) A common generalization of convolved generalized Fibonacci and Lucas polynomials and its applications, Applied Mathematics and Computation, 306, 31–37.

## Related papers

Taştan, M. & Özkan, E. (2021). Catalan transform of the *k*-Pell, *k*-Pell–Lucas and modified *k*-Pell sequence. Notes on Number Theory and Discrete Mathematics, 27(1), 198-207.

## Cite this paper

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, doi: 10.7546/nntdm.2018.24.3.47-55.