Research Article
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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

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.

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

Öznur Öztunç Kaymak Nihal Özgür

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

Abstract

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.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Öznur Öztunç Kaymak 0000-0003-3832-9947

Nihal Özgür 0000-0002-8152-1830

Project Number Mat.BAP.2013.0001
Publication Date December 30, 2022
Submission Date December 15, 2021
Acceptance Date May 8, 2022
Published in Issue Year 2022 Volume: 71 Issue: 4

Cite

APA Öztunç Kaymak, Ö., & Özgür, N. (2022). On the zeros of R-Bonacci polynomials and their derivatives. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4), 978-992. https://doi.org/10.31801/cfsuasmas.1037229
AMA Öztunç Kaymak Ö, Özgür N. On the zeros of R-Bonacci polynomials and their derivatives. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2022;71(4):978-992. doi:10.31801/cfsuasmas.1037229
Chicago Öztunç Kaymak, Öznur, and Nihal Özgür. “On the Zeros of R-Bonacci Polynomials and Their Derivatives”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 4 (December 2022): 978-92. https://doi.org/10.31801/cfsuasmas.1037229.
EndNote Öztunç Kaymak Ö, Özgür N (December 1, 2022) On the zeros of R-Bonacci polynomials and their derivatives. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 4 978–992.
IEEE Ö. Öztunç Kaymak and N. Özgür, “On the zeros of R-Bonacci polynomials and their derivatives”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 4, pp. 978–992, 2022, doi: 10.31801/cfsuasmas.1037229.
ISNAD Öztunç Kaymak, Öznur - Özgür, Nihal. “On the Zeros of R-Bonacci Polynomials and Their Derivatives”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/4 (December 2022), 978-992. https://doi.org/10.31801/cfsuasmas.1037229.
JAMA Öztunç Kaymak Ö, Özgür N. On the zeros of R-Bonacci polynomials and their derivatives. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:978–992.
MLA Öztunç Kaymak, Öznur and Nihal Özgür. “On the Zeros of R-Bonacci Polynomials and Their Derivatives”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 4, 2022, pp. 978-92, doi:10.31801/cfsuasmas.1037229.
Vancouver Öztunç Kaymak Ö, Özgür N. On the zeros of R-Bonacci polynomials and their derivatives. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(4):978-92.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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