A study on entire functions sharing one function with their difference operators and its application
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913151481544704 |
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| author | Xiang, Xuxu Long, Jianren |
| author_facet | Xiang, Xuxu Long, Jianren |
| contents | Let $f$ be a transcendental entire function with hyper-order strictly less than 1 and having a Borel exceptional small function. If $f$ and $Δ^n f$, or $f'$ and $f(z+1)$, share a function CM, then the exact form of $f$ is determined, which improves the previous results given by Lü et al. [Results Math. 74, article number 30 (2019)] and Liu et al. [Bull. Korean Math. Soc. 51, 1453-1467 (2014)]. As an application, the relationship between $f$ and $Δ^n f$ is established under the condition that they share a finite set, which partially resolves Liu's question raised in [J. Math. Anal. Appl. 359, 384-393 (2009)]. Furthermore, several examples are presented to demonstrate these results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_10056 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A study on entire functions sharing one function with their difference operators and its application Xiang, Xuxu Long, Jianren Complex Variables 30D35, 30D20 Let $f$ be a transcendental entire function with hyper-order strictly less than 1 and having a Borel exceptional small function. If $f$ and $Δ^n f$, or $f'$ and $f(z+1)$, share a function CM, then the exact form of $f$ is determined, which improves the previous results given by Lü et al. [Results Math. 74, article number 30 (2019)] and Liu et al. [Bull. Korean Math. Soc. 51, 1453-1467 (2014)]. As an application, the relationship between $f$ and $Δ^n f$ is established under the condition that they share a finite set, which partially resolves Liu's question raised in [J. Math. Anal. Appl. 359, 384-393 (2009)]. Furthermore, several examples are presented to demonstrate these results. |
| title | A study on entire functions sharing one function with their difference operators and its application |
| topic | Complex Variables 30D35, 30D20 |
| url | https://arxiv.org/abs/2509.10056 |