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Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences

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  • Renato Fiorenza

    (Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università di Napoli Federico II, Via Cintia, I-80126 Napoli, Italy)

Abstract

Let ( F n ) n = 1 ∞ be the classical Fibonacci sequence. It is well known that the lim F n + 1 / F n exists and equals the Golden Mean. If, more generally, ( F n ) n = 1 ∞ is an order- k linear recurrence with real constant coefficients, i.e., F n = ∑ j = 1 k λ k + 1 − j F n − j with n > k , λ j ∈ R , j = 1 , … , k , then the existence of the limit of ratios of consecutive terms may fail. In this paper, we show that the limit exists if the first k elements F 1 , F 2 , … , F k of ( F n ) n = 1 ∞ are positive, λ 1 , … , λ k − 1 are all nonnegative, at least one being positive, and max ( λ 1 , … , λ k ) = λ k ≥ k . The limit is characterized as fixed point, bounded below by λ k and bounded above by λ 1 + λ 2 + ⋯ + λ k .

Suggested Citation

  • 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.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2065-:d:839207
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    References listed on IDEAS

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    1. Tanackov, Ilija & Kovačević, Ilija & Tepić, Jovan, 2015. "Formula for Fibonacci sequence with arbitrary initial numbers," Chaos, Solitons & Fractals, Elsevier, vol. 73(C), pages 115-119.
    2. 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.
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