Asymptotics on a heriditary recursion

Yong-Guo Shi; Xiaoyu Luo; Zhi-jie Jiang; Yong-Guo Shi; Xiaoyu Luo; Zhi-jie Jiang

doi:10.3934/math.20241469

AIMS Mathematics

2024, Volume 9, Issue 11: 30443-30453. doi: 10.3934/math.20241469

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Research article

Asymptotics on a heriditary recursion

1.
Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China
2.
College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 64300, China

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.
- heriditary recursion,
- asymptotic expansion,
- Euler–Maclaurin formula
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

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Abstract

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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