Meromorphic solutions of Fermat type differential and difference equations of certain types. (English) Zbl 1527.30025
Summary: In this paper, we mainly consider the Fermat type differential equation
\[
f(z)^n+f'(z)^n=\varphi (z),
\]
where \(\varphi (z)=e^{h(z)}\) or \(1-e^{2h(z)}\), and \(h(z)\) is any entire function, and the Fermat type difference equation
\[
f(z)^n+f(z+c)^m=e^{P(z)},
\]
where \(P(z)\) is any entire function and \(c\) is a non-zero constant. We also provide short proofs for some existence results without complicated computations using elliptic functions.
MSC:
| 30D30 | Meromorphic functions of one complex variable (general theory) |
| 34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |
| 39A45 | Difference equations in the complex domain |
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| [25] | Yinhao Guo, Kai Liu (corresponding author) Department of Mathematics Nanchang University |
| [26] | Nanchang, P.R. China E-mail: 312045660@qq.com liukai418@126.com liukai@ncu.edu.cn |
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