DETECTING THE SLOWLY GROWING SOLUTIONS OF SECOND ORDER LINEAR DIFFERENCE EQUATIONS

doi:10.1007/s10473-025-0306-4

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (3): 837-854.doi: 10.1007/s10473-025-0306-4

DETECTING THE SLOWLY GROWING SOLUTIONS OF SECOND ORDER LINEAR DIFFERENCE EQUATIONS

Zongxuan CHEN¹, Zhibo HUANG^1,†, Jun WANG², Xiumin ZHENG³

1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
3. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

收稿日期:2023-12-14 修回日期:2024-02-29 出版日期:2025-05-25 发布日期:2025-09-30

DETECTING THE SLOWLY GROWING SOLUTIONS OF SECOND ORDER LINEAR DIFFERENCE EQUATIONS

Zongxuan CHEN¹, Zhibo HUANG^1,†, Jun WANG², Xiumin ZHENG³

1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
3. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

Received:2023-12-14 Revised:2024-02-29 Online:2025-05-25 Published:2025-09-30
Contact: ^†Zhibo HUANG, E-mail: huangzhibo@scnu.edu.cn
About author:Zongxuan CHEN, E-mail: chzx@vip.sina.com; Jun WANG, E-mail: majwang@fudan.edu.cn; Xiumin ZHENG, E-mail: zhengxiumin2008@sina.com
Supported by:
NSFC (12471072, 12171050).

摘要/Abstract

摘要： By using asymptotic method, we verify the existence on the slowly growing solutions to second order difference equations discussed by Ishizaki-Yanagihara's Wiman-Valiron method and Ishizaki-Wen's binomial series method. The classical problem on finding conditions on the polynomial coefficients $P_{j}(z) ~(j=0,1,2)$ and $F(z)$ to guarantee that all nontrivial solutions of complex second order difference equation $ P_2(z)f(z+2)+P_1(z)f(z+1)+P_0(z)f(z)=F(z)$ has slowly growing solutions with order $1/2$ is detected.

关键词: complex difference equation, slowly growing solution, asymptotic method, Wiman-Valiron method, binomial Series method

Abstract: By using asymptotic method, we verify the existence on the slowly growing solutions to second order difference equations discussed by Ishizaki-Yanagihara's Wiman-Valiron method and Ishizaki-Wen's binomial series method. The classical problem on finding conditions on the polynomial coefficients $P_{j}(z) ~(j=0,1,2)$ and $F(z)$ to guarantee that all nontrivial solutions of complex second order difference equation $ P_2(z)f(z+2)+P_1(z)f(z+1)+P_0(z)f(z)=F(z)$ has slowly growing solutions with order $1/2$ is detected.

Key words: complex difference equation, slowly growing solution, asymptotic method, Wiman-Valiron method, binomial Series method

中图分类号:

30D35

Zongxuan CHEN, Zhibo HUANG, Jun WANG, Xiumin ZHENG. DETECTING THE SLOWLY GROWING SOLUTIONS OF SECOND ORDER LINEAR DIFFERENCE EQUATIONS[J]. 数学物理学报(英文版), 2025, 45(3): 837-854.

Zongxuan CHEN, Zhibo HUANG, Jun WANG, Xiumin ZHENG. DETECTING THE SLOWLY GROWING SOLUTIONS OF SECOND ORDER LINEAR DIFFERENCE EQUATIONS[J]. Acta mathematica scientia,Series B, 2025, 45(3): 837-854.

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DETECTING THE SLOWLY GROWING SOLUTIONS OF SECOND ORDER LINEAR DIFFERENCE EQUATIONS

DETECTING THE SLOWLY GROWING SOLUTIONS OF SECOND ORDER LINEAR DIFFERENCE EQUATIONS

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