The asymptotic estimation for two classes of generalized Fibonacci sub-sequences

TL;DR

The paper addresses the problem of accurately estimating the inverse tail sums of reciprocal powers of generalized Fibonacci subsequences $W_n(a,b,p,q)$. By leveraging the Binet-type representation $W_n=c_1\alpha^n-c_2\beta^n$ and careful asymptotic expansions, it derives explicit forms for $\left(\sum_{k=n}^{\infty}\frac{1}{W_{mk+l}^d}\right)^{-1}$ and, for the first time in this context, $\left(\sum_{k=n}^{\infty}\frac{(-1)^k}{W_{mk+l}^d}\right)^{-1}$, including detailed corollaries for $d=1,2,3,4$. The results extend Yuan et al.’s 2025 A14 work to a broad generalized Fibonacci class and provide high-precision asymptotic expansions with quantified error terms, under standard positivity and dominance assumptions ensuring $|\beta|<1$. These findings offer precise asymptotic tools for analyses involving Fibonacci-like sequences and their reciprocal-tail sums.

Abstract

Since the $\mathrm{Fibonacci}$ sequence has good properties, it's important in theory and applications, such as in combinatorics, cryptography, and so on. In this paper, for the generalized Fibonacci sequence $\left\{W_n\left(a,b,p,q\right)\right\}$, by using elementary methods and techniques, we respectively give the asymptotic estimation values of $\left(\sum\limits_{k=n}^{\infty}\frac{1}{W_{mk+l}^d}\right)^{-1}$ and $\left(\sum\limits_{k=n}^{\infty}\frac{\left(-1\right)^k}{W_{mk+l}^d}\right)^{-1}$, which generalize the asymptotic estimation results of Yuan et al. \cite{A14} in 2025.

Theorems & Definitions (19)

• Definition 2.1: A10
• Definition 2.2: A15
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 3.1
• Corollary 3.1
• Remark 3.1
• Corollary 3.2
• Remark 3.2