On the zeros of R-Bonacci polynomials and their derivatives

Year 2022, Volume: 71 Issue: 4, 978 - 992, 30.12.2022

Öznur Öztunç Kaymak , Nihal Özgür

https://doi.org/10.31801/cfsuasmas.1037229

https://izlik.org/JA87JK34KX

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

Supporting Institution

Balıkesir Üniversitesi

Project Number

Mat.BAP.2013.0001

Thanks

This work is supported by the Scientific Research Projects Unit of Balıkesir University under the project number Mat.BAP.2013.0001.

References

• Brousseau, A., Fibonacci statistics in conifers, Fibonacci Quart., 7(4) (1969), 525–532.
• Carson, J., Fibonacci numbers and pineapple phyllotaxy, The Two-Year College Mathematics Journal, 9(3) (1978), 132–136. https://doi.org/10.2307/3026682
• 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
• 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.
• Filipponi, P., Horadam, A., Second derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Quart., 31(3) (1993), 194–204.
• 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
• He, M. X., Simon, D., Ricci, P. E., Dynamics of the zeros of Fibonacci polynomials, Fibonacci Quart., 35(2) (1997), 160–168.
• He, M. X., Ricci, P. E., Simon, D., Numerical results on the zeros of generalized Fibonacci polynomials, Calcolo, 34 (1-4) (1997), 25–40.
• Hoggatt, V. E., Bicknell, M., Generalized Fibonacci polynomials, Fibonacci Quart., 11(5) (1973), 457–465.
• Hoggatt, V. E., Bicknell, M., Roots of Fibonacci polynomials, Fibonacci Quart., 11(3) (1973), 271–274.
• Öztunç Kaymak, Ö., R-Bonacci polynomials and Their Derivatives, Ph. D. Thesis, Balıkesir University, 2014.
• 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
• Koshy, T., Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics, Wiley-Interscience, New York, 2001.
• 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.
• Matyas, F., Szalay, L., A note on Tribonacci-coefficient polynomials, Ann. Math. Inform. 38 (2011), 95–98.
• Mitchson, G. J., Phyllotaxis and the Fibonacci series, Science, 196 (1977), 270–275.
• 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.
• 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
• 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
• 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.
• Vieira, R. S., Polynomials with Symmetric Zeros, In: Polynomials – Theory and Application, IntechOpen, 2019. https://doi.org/10.5772/intechopen.82728
• 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
• Wang, J., On the k-th derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Quart., 33(2) (1995), 174–178.
• Web, W. A., Parberry, E. A., Divisibility properties of Fibonacci polynomials, Fibonacci Quart., 7(5) (1969), 457–463.
• Yuan, Y., Zhang, W., Some identities involving the Fibonacci polynomials, Fibonacci Quart., 40(4) (2002), 314–318.