2-Fibonacci polynomials in the family of Fibonacci numbers

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

  1. Berg, C. (2011) Fibonacci numbers and orthogonal polynomials, Arab Journal of Mathematical Sciences, 17, 75–88.
  2. Falcon S. & Plaza, A. (2009) On 𝑘-Fibonacci sequences and polynomials and their derivatives, Chaos, Solutions and Fractals, 39, 1005–1019.
  3. Hoggatt, Jr. V. E. & Lind, D. A. (1968) Symbolic Substitutions in to Fibonacci Polynomials, The Fibonacci Quarterly, 6(5), 55–74.
  4. Hoggatt, Jr. V. E., Leonard, H. T. Jr. & Philips, J. W. (1971) Twenty-four Master Identities, The Fibonacci Quarterly, 9(1), 1–17.
  5. Hoggatt, Jr. V. E. & Bicknell, M. (1973) Generalized Fibonacci polynomials, Fibonacci Quarterly, 11(5), 457–465.
  6. Ivie, J. (1972) A General Q-Matrix, Fibonacci Quarterly, 10(3), 255–261.
  7. Koshy, T. (2001) Fibonacci and Lucas Numbers with Applications, JohnWiley & Sons, Inc., Canada.
  8. 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.
  9. Mikkawy M. & Sogabe, T. (2010) A new family of 𝑘-Fibonacci numbers, Applied Mathematics and Computation, 215, 4456–4461.
  10. Nalli, A. & Haukkanen, P. (2009) On generalized Fibonacci and Lucas polynomials, Chaos, Solitons and Fractals, 42(5), 3179–3186.
  11. Panwar, Y. K., Singh, B. & Gupta, V. K. (2013) Generalized Fibonacci Polynomials, Turkish Journal of Analysis and Number Theory, 1(1), 43–47.
  12. Ramirez, J. (2014) On convelved generalized Fibonacci and Lucas polynomials, Applied Mathematics and Computation, 229, 208–213.
  13. Swamy, M. N. S. (1965) Problem B-74, The Fibonacci Quarterly, 3(3), 236.
  14. Tuğlu, N., Koçer, E. G. & Stakhov, A. (2011) Bivariate fibonacci like 𝑝-polynomials, Applied Mathematics and Computation, 217, 10239–10246.
  15. 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

Cite this paper

APA

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.

Chicago

Özkan, Engin, Merve Taştan and Ali Aydoğdu. “2-Fibonacci Polynomials in the Family of Fibonacci Numbers.” Notes on Number Theory and Discrete Mathematics 24, no. 3 (2018): 47-55, doi: 10.7546/nntdm.2018.24.3.47-55.

MLA

Özkan, Engin, Merve Taştan and Ali Aydoğdu. “2-Fibonacci Polynomials in the Family of Fibonacci Numbers.” Notes on Number Theory and Discrete Mathematics 24.3 (2018): 47-55. Print, doi: 10.7546/nntdm.2018.24.3.47-55.

Comments are closed.