Research article

Asymptotics on a heriditary recursion

  • Received: 21 July 2024 Revised: 10 October 2024 Accepted: 16 October 2024 Published: 25 October 2024
  • MSC : 03D99, 11B37, 41A60, 65B15

  • 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\geq1 \end{equation*} $

    is studied, where $ f $ is decreasing, continuous on $ (a, \infty) $ ($ 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.

    Citation: Yong-Guo Shi, Xiaoyu Luo, Zhi-jie Jiang. Asymptotics on a heriditary recursion[J]. AIMS Mathematics, 2024, 9(11): 30443-30453. doi: 10.3934/math.20241469

    Related Papers:

  • 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\geq1 \end{equation*} $

    is studied, where $ f $ is decreasing, continuous on $ (a, \infty) $ ($ 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.



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