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*} $</p><p>is studied, w...
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AIMS Press
2024-10-01
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| author | Yong-Guo Shi Xiaoyu Luo Zhi-jie Jiang |
| author_facet | Yong-Guo Shi Xiaoyu Luo Zhi-jie Jiang |
| author_sort | Yong-Guo Shi |
| collection | DOAJ |
| description | The asymptotic behavior for a heriditary recursion</p><p class="disp_formula">$ \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 $. Some examples and applications are provided. |
| format | Article |
| id | doaj-art-bc3e000c31e64ab2a1c12042da6de6be |
| institution | DOAJ |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-bc3e000c31e64ab2a1c12042da6de6be2025-08-20T02:49:49ZengAIMS PressAIMS Mathematics2473-69882024-10-01911304433045310.3934/math.20241469Asymptotics on a heriditary recursionYong-Guo Shi0Xiaoyu Luo1Zhi-jie Jiang21. Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China2. College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 64300, China2. College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 64300, ChinaThe asymptotic behavior for a heriditary recursion</p><p class="disp_formula">$ \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. https://www.aimspress.com/article/doi/10.3934/math.20241469?viewType=HTMLheriditary recursionasymptotic expansioneuler–maclaurin formula |
| spellingShingle | Yong-Guo Shi Xiaoyu Luo Zhi-jie Jiang Asymptotics on a heriditary recursion AIMS Mathematics heriditary recursion asymptotic expansion euler–maclaurin formula |
| title | Asymptotics on a heriditary recursion |
| title_full | Asymptotics on a heriditary recursion |
| title_fullStr | Asymptotics on a heriditary recursion |
| title_full_unstemmed | Asymptotics on a heriditary recursion |
| title_short | Asymptotics on a heriditary recursion |
| title_sort | asymptotics on a heriditary recursion |
| topic | heriditary recursion asymptotic expansion euler–maclaurin formula |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241469?viewType=HTML |
| work_keys_str_mv | AT yongguoshi asymptoticsonaheriditaryrecursion AT xiaoyuluo asymptoticsonaheriditaryrecursion AT zhijiejiang asymptoticsonaheriditaryrecursion |