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On generalized Fibonacci and Lucas polynomials

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  • Nalli, Ayse
  • Haukkanen, Pentti

Abstract

Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Qh(x) that generalizes the Q-matrix 1110 whose powers generate the Fibonacci numbers.

Suggested Citation

  • Nalli, Ayse & Haukkanen, Pentti, 2009. "On generalized Fibonacci and Lucas polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3179-3186.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:5:p:3179-3186
    DOI: 10.1016/j.chaos.2009.04.048
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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.
    3. Falcón, Sergio & Plaza, Ángel, 2007. "On the Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1615-1624.
    4. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    5. Falcón, Sergio & Plaza, Ángel, 2007. "The k-Fibonacci sequence and the Pascal 2-triangle," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 38-49.
    6. Stakhov, A. & Rozin, B., 2006. "The “golden” algebraic equations," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1415-1421.
    7. 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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    Cited by:

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    3. Wang, Weiping & Wang, Hui, 2017. "Generalized Humbert polynomials via generalized Fibonacci polynomials," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 204-216.
    4. Ye, Xiaoli & Zhang, Zhizheng, 2017. "A common generalization of convolved generalized Fibonacci and Lucas polynomials and its applications," Applied Mathematics and Computation, Elsevier, vol. 306(C), pages 31-37.

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