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

Fiorenza, Renato

doi:10.3390/math10122065

Open AccessArticle

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

by

Renato Fiorenza

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

Mathematics 2022, 10(12), 2065; https://doi.org/10.3390/math10122065

Submission received: 18 May 2022 / Revised: 9 June 2022 / Accepted: 13 June 2022 / Published: 15 June 2022

(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)

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Abstract

Let

{(F_{n})}_{n = 1}^{\infty}

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}^{\infty}

is an order-k linear recurrence with real constant coefficients, i.e.,

F_{n} = \sum_{j = 1}^{k} λ_{k + 1 - j} F_{n - j}

with

n > k

,

λ_{j} \in R

,

j = 1, \dots, 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}, \dots, F_{k}

of

{(F_{n})}_{n = 1}^{\infty}

are positive,

λ_{1}, \dots, λ_{k - 1}

are all nonnegative, at least one being positive, and

max (λ_{1}, \dots, λ_{k}) = λ_{k} \geq k

. The limit is characterized as fixed point, bounded below by

λ_{k}

and bounded above by

λ_{1} + λ_{2} + \dots + λ_{k}

.

Keywords: Fibonacci sequence; asymptotic analysis; Kepler limit; consecutive terms; linear recurrence sequences; contraction mapping theorem; lower limit; upper limit

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MDPI and ACS Style

Fiorenza, R. Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences. Mathematics 2022, 10, 2065. https://doi.org/10.3390/math10122065

AMA Style

Fiorenza R. Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences. Mathematics. 2022; 10(12):2065. https://doi.org/10.3390/math10122065

Chicago/Turabian Style

Fiorenza, Renato. 2022. "Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences" Mathematics 10, no. 12: 2065. https://doi.org/10.3390/math10122065

APA Style

Fiorenza, R. (2022). Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences. Mathematics, 10(12), 2065. https://doi.org/10.3390/math10122065

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

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