Solutions of certain Fermat-type partial differential-difference equations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908686402715648 |
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| author | Majumder, Sujoy Pramanik, Debabrata |
| author_facet | Majumder, Sujoy Pramanik, Debabrata |
| contents | The purpose of this paper is to investigate the non-constant entire as well as meromorphic solutions of the Fermat-type partial differential-difference equation: \[\left(\sum_{j=1}^m\frac{\partial f(z_1, z_2, \ldots, z_m)}{\partial z_j}\right)^{m_1} + f^{m_2}(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m ) = 1,\] where $m_1$ and $m_2$ are positive integers such that $m_1+m_2>2$ and $(c_1, c_2, \ldots, c_m)\in \mathbb{C}^m$. The results of our paper generalize the result of Xu and Wang \cite {XW1} from $\mathbb{C}^2$ to $\mathbb{C}^m$. Also in the paper we give positive answer of the open problem addressed by Xu and Wang \cite {XW1}. Moreover plenty of examples are provided to illustrate our findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02042 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solutions of certain Fermat-type partial differential-difference equations Majumder, Sujoy Pramanik, Debabrata Complex Variables 39A45, 32H30, 39A14 and 35A20 The purpose of this paper is to investigate the non-constant entire as well as meromorphic solutions of the Fermat-type partial differential-difference equation: \[\left(\sum_{j=1}^m\frac{\partial f(z_1, z_2, \ldots, z_m)}{\partial z_j}\right)^{m_1} + f^{m_2}(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m ) = 1,\] where $m_1$ and $m_2$ are positive integers such that $m_1+m_2>2$ and $(c_1, c_2, \ldots, c_m)\in \mathbb{C}^m$. The results of our paper generalize the result of Xu and Wang \cite {XW1} from $\mathbb{C}^2$ to $\mathbb{C}^m$. Also in the paper we give positive answer of the open problem addressed by Xu and Wang \cite {XW1}. Moreover plenty of examples are provided to illustrate our findings. |
| title | Solutions of certain Fermat-type partial differential-difference equations |
| topic | Complex Variables 39A45, 32H30, 39A14 and 35A20 |
| url | https://arxiv.org/abs/2512.02042 |