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On the zeros of R-Bonacci polynomials and their derivatives

Yıl 2022, Cilt: 71 Sayı: 4, 978 - 992, 30.12.2022

Öznur Öztunç Kaymak Nihal Özgür

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

Öz

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.

Anahtar Kelimeler

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

Destekleyen Kurum

Balıkesir Üniversitesi

Proje Numarası

Mat.BAP.2013.0001

Teşekkür

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

Kaynakça

  • 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.

Yıl 2022, Cilt: 71 Sayı: 4, 978 - 992, 30.12.2022

Öznur Öztunç Kaymak Nihal Özgür

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

Öz

Proje Numarası

Mat.BAP.2013.0001

Kaynakça

  • 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.
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Research Article
Yazarlar

Öznur Öztunç Kaymak 0000-0003-3832-9947

Nihal Özgür 0000-0002-8152-1830

Proje Numarası Mat.BAP.2013.0001
Yayımlanma Tarihi 30 Aralık 2022
Gönderilme Tarihi 15 Aralık 2021
Kabul Tarihi 8 Mayıs 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 71 Sayı: 4

Kaynak Göster

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. Aralık 2022;71(4):978-992. doi:10.31801/cfsuasmas.1037229
Chicago Öztunç Kaymak, Öznur, ve 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, sy. 4 (Aralık 2022): 978-92. https://doi.org/10.31801/cfsuasmas.1037229.
EndNote Öztunç Kaymak Ö, Özgür N (01 Aralık 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 ve N. Özgür, “On the zeros of R-Bonacci polynomials and their derivatives”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 71, sy. 4, ss. 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 (Aralık 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 ve Nihal Özgür. “On the Zeros of R-Bonacci Polynomials and Their Derivatives”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 71, sy. 4, 2022, ss. 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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