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

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 ${\left(\sum _{k=n}^{\infty }\frac{1}{W_{mk+l}^d}\right)}^{-1}$ and ${\left(\sum _{k=n}^{\infty }\frac{{\left(-1\right)}^k}{W_{mk+l}^d}\right)}^{-1}$, which generalize the asymptotic estimation results of Yuan et al. \cite{A14} in 2025.