On the zeros of R-Bonacci polynomials and their derivatives

Abstract

The purpose of the present paper is to examine the zeros of R-Bonacci polynomials and their derivatives. We obtain new characterizations for the zeros of these polynomials. Our results generalize the ones obtained for the special case r=2. Furthermore, we find explicit formulas of the roots of derivatives of R-Bonacci polynomials in some special cases. Our formulas are substantially simple and useful.

Keywords

• $R$-Bonacci polynomial
• symmetric polynomial
• complex polynomial
• complex zeros

References

1. Brousseau, A., Fibonacci statistics in conifers, Fibonacci Quart., 7(4) (1969), 525–532.
2. Carson, J., Fibonacci numbers and pineapple phyllotaxy, The Two-Year College Mathematics Journal, 9(3) (1978), 132–136. https://doi.org/10.2307/3026682
3. Falcon, S., Plaza, A., On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals, 30(3) (2009), 1005-1019. https://doi.org/10.1016/j.chaos.2007.03.007
4. Filipponi, P., Horadam, A. F., Derivative Sequences of Fibonacci and Lucas Polynomials, Applications of Fibonacci Numbers, Vol. 4 (Winston-Salem, NC, 1990), 99–108, Kluwer Acad. Publ., Dordrecht, 1991.
5. Filipponi, P., Horadam, A., Second derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Quart., 31(3) (1993), 194–204.
6. Goh, W., He, M. X., Ricci, P. E., On the universal zero attractor of the Tribonacci-related polynomials, Calcolo, 46(2) (2009), 95–129. https://doi.org/10.1007/s10092-009-0002-0
7. He, M. X., Simon, D., Ricci, P. E., Dynamics of the zeros of Fibonacci polynomials, Fibonacci Quart., 35(2) (1997), 160–168.
8. He, M. X., Ricci, P. E., Simon, D., Numerical results on the zeros of generalized Fibonacci polynomials, Calcolo, 34 (1-4) (1997), 25–40.

9. Hoggatt, V. E., Bicknell, M., Generalized Fibonacci polynomials, Fibonacci Quart., 11(5) (1973), 457–465.
10. Hoggatt, V. E., Bicknell, M., Roots of Fibonacci polynomials, Fibonacci Quart., 11(3) (1973), 271–274.
11. Öztunç Kaymak, Ö., R-Bonacci polynomials and Their Derivatives, Ph. D. Thesis, Balıkesir University, 2014.
12. Öztunç Kaymak, Ö., Some remarks on the zeros of tribonacci polynomials, Int. J. Anal. Appl., 16(3) (2018), 368-373. https://doi.org/10.28924/2291-8639-16-2018-368
13. Koshy, T., Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics, Wiley-Interscience, New York, 2001.
14. Marden, M., Geometry of Polynomials, Second edition, Mathematical Surveys, No. 3 American Mathematical Society, Providence, R.I. 1966. Matyas, F., Szalay, L., A note on Tribonacci coefficient polynomials, Ann. Math. Inform. 38 (2011), 95–98.
15. Matyas, F., Szalay, L., A note on Tribonacci-coefficient polynomials, Ann. Math. Inform. 38 (2011), 95–98.
16. Mitchson, G. J., Phyllotaxis and the Fibonacci series, Science, 196 (1977), 270–275.
17. Özgür, N. Y., Öztunç Kaymak, Ö., On the zeros of the derivatives of Fibonacci and Lucas polynomials, Journal of New Theory, 7 (2015), 22-28.
18. Taş, N., Uçar, S., Özgür, N., Öztunç Kaymak, Ö., A new coding/decoding algorithm using Finonacci numbers, Discrete Math. Algorithms Appl., 10(2) (2018), 1850028. https://doi.org/10.1142/S1793830918500283
19. Taş, N., Uçar, S., Özgür, N., Pell coding and Pell decoding methods with some applications, Contrib. Discrete Math. 15(1) (2020), 52-66. https://doi.org/10.11575/cdm.v15i1.62606
20. Uçar, S., Taş, N., Özgür, N. Y., A new application to coding theory via Fibonacci and Lucas numbers, Mathematical Sciences and Applications E-Notes, 7(1) (2019), 62–70.
21. Vieira, R. S., Polynomials with Symmetric Zeros, In: Polynomials – Theory and Application, IntechOpen, 2019. https://doi.org/10.5772/intechopen.82728
22. Vieira, R. S., How to count the number of zeros that a polynomial has on the unit circle?, J Comp. Appl. Math., 384 (2021), Paper No. 113169, 11 pp. https://doi.org/10.1016/j.cam.2020.113169
23. Wang, J., On the k-th derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Quart., 33(2) (1995), 174–178.
24. Web, W. A., Parberry, E. A., Divisibility properties of Fibonacci polynomials, Fibonacci Quart., 7(5) (1969), 457–463.
25. Yuan, Y., Zhang, W., Some identities involving the Fibonacci polynomials, Fibonacci Quart., 40(4) (2002), 314–318.