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
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.
[1] | T. M. Apostol, Introduction to analytic number theory, 1st Eds., New York: Springer, 1976. https://doi.org/10.1007/978-1-4757-5579-4 |
[2] | I. V. Blagouchine, E. Moreau, On a finite sum of cosecants appearing in various problems, J. Math. Anal. Appl., 539 (2024), 128515. https://doi.org/10.1016/j.jmaa.2024.128515 doi: 10.1016/j.jmaa.2024.128515 |
[3] | N. D. Bruijn, Asymptotic methods in analysis, 1958. |
[4] | E. T. Copson, Asymptotic expansions, Cambridge University Press, 2004. |
[5] | S. Elaydi, An introduction to difference equations, 3rd Eds., New York: Springer, 2005. https://doi.org/10.1007/0-387-27602-5 |
[6] | V. Lampret, Simple derivation of the Euler–Boole type summation formula and examples of its use, Mediterr. J. Math. 19 (2022), 77. https://doi.org/10.1007/s00009-022-02000-x doi: 10.1007/s00009-022-02000-x |
[7] | D. B. Grünberg, On asymptoticsm, Stirling numbers, gamma function and polylogs, Result. Math., 49 (2006), 89–125. https://doi.org/10.1007/s00025-006-0211-7 doi: 10.1007/s00025-006-0211-7 |
[8] | C. Heuberger, D. Krenn, G. F. Lipnik, Asymptotic analysis of $q$-recursive sequences, Algorithmica, 84 (2022), 2480–2532. http://doi.org/10.1007/s00453-022-00950-y doi: 10.1007/s00453-022-00950-y |
[9] | H. K. Hwang, S. Janson, T. H. Tsai, Exact and asymptotic solutions of a divide-and-conquer recurrence dividing at half: Theory and applications, ACM Trans. Algorithms, 13 (2017), 1–43. https://dl.acm.org/doi/10.1145/3127585 doi: 10.1145/3127585 |
[10] | C. Mortici, A. Vernescu, Some new facts in discrete asymptotic analysis, Math. Balkanica (N.S.), 21 (2007), 301–308. |
[11] | J. D. Murray, Asymptotic analysis, New York: Springer, 1984. https://doi.org/10.1007/978-1-4612-1122-8 |
[12] | F. Olver, Asymptotics and special functions, 1st Eds., New York: A K Peters/CRC Press, 1997. https://doi.org/10.1201/9781439864548 |
[13] | D. Popa, Asymptotic expansions for the recurrence $x_{n+1} = \frac{1}{n}\sum_{k = 1}^{n}f(\frac{x_{k}}{k})$, Math. Methods Appl. Sci., 46 (2023), 2165–2173. https://doi.org/10.1002/mma.8634 doi: 10.1002/mma.8634 |
[14] | M. Z. Spivey, The Euler-Maclaurin formula and sums of powers, Math. Mag., 79 (2006), 61–65. http://doi.org/10.1080/0025570X.2006.11953378 doi: 10.1080/0025570X.2006.11953378 |
[15] | A. Vernescu, C. Mortici, New results in discrete asymptotic analysis, Gen. Math., 16 (2008), 179–188. https://eudml.org/doc/117876 |
[16] | X. S. Wang, R. Wong, Discrete analogues of Laplace's approximation, Asymptot. Anal., 54 (2007), 165–180. |
[17] | R. Wong, Y. Q. Zhao, Recent advances in asymptotic analysis, Anal. Appl., 20 (2022), 1103–1146. https://doi.org/10.1142/S0219530522400012 doi: 10.1142/S0219530522400012 |
[18] | A. Xu, Approximations of the generalized-Euler-constant function and the generalized Somos' quadratic recurrence constant, J. Inequal. Appl., 2019 (2019), 198. https://doi.org/10.1186/s13660-019-2153-0 doi: 10.1186/s13660-019-2153-0 |
[19] | A. Xu, Asymptotic expansion related to the generalized Somos recurrence constant, Int. J. Number Theory, 15 (2019), 2043–2055. https://doi.org/10.1142/S1793042119501112 doi: 10.1142/S1793042119501112 |
[20] | L. Zhu, Asymptotic expansion of a finite sum involving harmonic numbers, AIMS Mathematics, 6 (2021), 2756–2763. https://doi.org/10.3934/math.2021168 doi: 10.3934/math.2021168 |