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Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients

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  • Fiorenza, Alberto
  • Vincenzi, Giovanni

Abstract

For complex linear homogeneous recursive sequences with constant coefficients we find a necessary and sufficient condition for the existence of the limit of the ratio of consecutive terms. The result can be applied even if the characteristic polynomial has not necessarily roots with modulus pairwise distinct, as in the celebrated Poincaré’s theorem. In case of existence, we characterize the limit as a particular root of the characteristic polynomial, which depends on the initial conditions and that is not necessarily the unique root with maximum modulus and multiplicity. The result extends to a quite general context the way used to find the Golden mean as limit of ratio of consecutive terms of the classical Fibonacci sequence.

Suggested Citation

  • Fiorenza, Alberto & Vincenzi, Giovanni, 2011. "Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 145-152.
    • Handle: RePEc:eee:chsofr:v:44:y:2011:i:1:p:145-152
    DOI: 10.1016/j.chaos.2011.01.003
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    References listed on IDEAS

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    Cited by:

    1. Renato Fiorenza, 2022. "Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences," Mathematics, MDPI, vol. 10(12), pages 1-8, June.
    2. Ivana Matoušová & Pavel Trojovský, 2020. "On Coding by (2, q )-Distance Fibonacci Numbers," Mathematics, MDPI, vol. 8(11), pages 1-24, November.
    3. Alberto Fiorenza & Giovanni Vincenzi, 2013. "From Fibonacci Sequence to the Golden Ratio," Journal of Mathematics, Hindawi, vol. 2013, pages 1-3, March.
    4. Florek, Wojciech, 2018. "A class of generalized Tribonacci sequences applied to counting problems," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 809-821.
    5. Anatriello, Giuseppina & Németh, László & Vincenzi, Giovanni, 2022. "Generalized Pascal’s triangles and associated k-Padovan-like sequences," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 278-290.

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