Asymptotics on a heriditary recursion - Yong-Guo Shi; Xiaoyu Luo; Zhi-jie Jiang (AIMS Mathematics)
The asymptotic behavior for a heriditary recursion $ \begin{equation*} x_1>a \, \, \text{and} \, \, x_{n+1} = \frac{1}{n^s}\sum\limits_{k = 1}^nf\left(\frac{x_k}k\right)\text{ for every }n≥1 \end{equation*} $ is studied, where $ f $ is decreasing, continuous on $ (a, ∞) $ ($ a < 0 $), and twice differentiable at $ 0 $. The result has been known for the case $ s = 1 $. This paper analyzes the case $ s > 1 $. We obtain an asymptotic sequence that is quite different from the case $ s = 1 $. Some examples and applications are provided.