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

Mathematics

Renato Fiorenza¹ 

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

关键词: Fibonacci sequence;    asymptotic analysis;    Kepler limit;    consecutive terms;    linear recurrence sequences;    contraction mapping theorem;   

DOI  :  10.3390/math10122065

来源: DOAJ

【 摘 要 】

Let ${\left(F_n\right)}_{n=1}^{\infty }$ 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, ${\left(F_n\right)}_{n=1}^{\infty }$ is an order-k linear recurrence with real constant coefficients, i.e., $F_n={\sum }_{j=1}^k{\lambda }_{k+1-j}{F}_{n-j}$ with $n>k$, ${\lambda }_j\in \mathbb{R}$, $j=1,\dots ,k$, 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 ${\left(F_n\right)}_{n=1}^{\infty }$ are positive, ${\lambda }_1,\dots ,{\lambda }_{k-1}$ are all nonnegative, at least one being positive, and $max({\lambda }_1,\dots ,{\lambda }_k)={\lambda }_k\ge k$. The limit is characterized as fixed point, bounded below by ${\lambda }_k$ and bounded above by ${\lambda }_1+{\lambda }_2+\cdots +{\lambda }_k$.

【 授权许可】

Unknown