Asymptotics on a heriditary recursion
The asymptotic behavior for a heriditary recursion</p><p class="disp_formula">$ \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*} $</p><p>is studied, w...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-10-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241469?viewType=HTML |
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| Summary: | The asymptotic behavior for a heriditary recursion</p><p class="disp_formula">$ \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*} $</p><p>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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| ISSN: | 2473-6988 |