Representation of finite order solutions to linear differential equations with exponential sum coefficients
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912163072835584 |
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| author | Li, Xing-Yu Wang, Jun Wen, Zhi-Tao |
| author_facet | Li, Xing-Yu Wang, Jun Wen, Zhi-Tao |
| contents | We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations
$$
f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+)
$$ where $a_i$ are exponential sums for $i=0,\ldots,n-1$ with all positive (or all negative) rational frequencies and constant coefficients. Moreover, under the condition that there exists a finite order solution of (+) with exponential sum coefficients having rational frequencies and constant coefficients, we give the precise form of all finite order solutions, which are exponential sums. It is a partial answer to Gol'dberg-Ostrovskiǐ Problem and Problem 5 in \cite{HITW2022} since exponential sums are of completely regular growth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15613 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representation of finite order solutions to linear differential equations with exponential sum coefficients Li, Xing-Yu Wang, Jun Wen, Zhi-Tao Complex Variables Classical Analysis and ODEs Primary 34M05, Secondary 30D35 We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations $$ f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+) $$ where $a_i$ are exponential sums for $i=0,\ldots,n-1$ with all positive (or all negative) rational frequencies and constant coefficients. Moreover, under the condition that there exists a finite order solution of (+) with exponential sum coefficients having rational frequencies and constant coefficients, we give the precise form of all finite order solutions, which are exponential sums. It is a partial answer to Gol'dberg-Ostrovskiǐ Problem and Problem 5 in \cite{HITW2022} since exponential sums are of completely regular growth. |
| title | Representation of finite order solutions to linear differential equations with exponential sum coefficients |
| topic | Complex Variables Classical Analysis and ODEs Primary 34M05, Secondary 30D35 |
| url | https://arxiv.org/abs/2412.15613 |