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On k-Fibonacci sequences and polynomials and their derivatives

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  • Falcón, Sergio
  • Plaza, Ángel

Abstract

The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven.

Suggested Citation

  • Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:3:p:1005-1019
    DOI: 10.1016/j.chaos.2007.03.007
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    References listed on IDEAS

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    1. Stakhov, Alexey & Rozin, Boris, 2005. "The Golden Shofar," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 677-684.
    2. Falcón, Sergio & Plaza, Ángel, 2008. "The k-Fibonacci hyperbolic functions," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 409-420.
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    7. Stakhov, A. & Rozin, B., 2006. "The “golden” algebraic equations," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1415-1421.
    8. El Naschie, M.S., 2007. "Determining the number of Fermions and the number of Boson separately in an extended standard model," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1241-1243.
    9. El Naschie, M.S., 2005. "Non-Euclidean spacetime structure and the two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 1-6.
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    11. Stakhov, Alexey & Rozin, Boris, 2006. "Theory of Binet formulas for Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1162-1177.
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    2. Chung-Chuan Chen & Lin-Ling Huang, 2021. "Some New Identities for the Generalized Fibonacci Polynomials by the Q(x) Matrix," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 13(2), pages 1-21, April.
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    5. Nalli, Ayse & Haukkanen, Pentti, 2009. "On generalized Fibonacci and Lucas polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3179-3186.
    6. Postavaru, Octavian, 2023. "An efficient numerical method based on Fibonacci polynomials to solve fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 406-422.
    7. Flaut, Cristina & Shpakivskyi, Vitalii & Vlad, Elena, 2017. "Some remarks regarding h(x) – Fibonacci polynomials in an arbitrary algebra," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 32-35.
    8. Flavio Pressacco & Giacomo Plazzotta & Laura Ziani, 2014. "K-Fibonacci sequences and minimal winning quota in Parsimonious game," Working Papers hal-00950090, HAL.
    9. A. D. Godase, 2019. "Properties of k-Fibonacci and k-Lucas octonions," Indian Journal of Pure and Applied Mathematics, Springer, vol. 50(4), pages 979-998, December.
    10. Esmaeili, M. & Gulliver, T.A. & Kakhbod, A., 2009. "The Golden mean, Fibonacci matrices and partial weakly super-increasing sources," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 435-440.
    11. W. M. Abd-Elhameed & N. A. Zeyada, 2022. "New formulas including convolution, connection and radicals formulas of k-Fibonacci and k-Lucas polynomials," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(4), pages 1006-1016, December.
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