Uniqueness of Meromorphic Solutions for a Class of Complex Linear Differential-Difference Equations
Received:April 02, 2021  Revised:January 11, 2022
Key Words: meromorphic solution   complex differential-difference equation   shared value   uniqueness   finite order
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12001211) and the Natural Science Foundation of Fujian Province, China (Grant No.2021J01651).
 Author Name Affiliation Hongjin LIN School of Mathematics and Statistics, Fujian Normal University, Fujian 350117, P. R. China Junfan CHEN School of Mathematics and Statistics, Fujian Normal University, Fujian 350117, P. R. China Shuqing LIN School of Mathematics and Statistics, Fujian Normal University, Fujian 350117, P. R. China
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In this paper, we mainly study the uniqueness of transcendental meromorphic solutions for a class of complex linear differential-difference equations. Specially, suppose that $f(z)$ is a finite order transcendental meromorphic solution of complex linear differential-difference equation: $W_{1}(z)f'(z+1)+W_{2}(z)f(z)=W_{3}(z)$, where $W_{1}(z)$, $W_{2}(z)$, $W_{3}(z)$ are nonzero meromorphic functions, with their orders of growth being less than one, such that $W_{1}(z)+W_{2}(z)\not\equiv0$. If $f(z)$ and a meromorphic function $g(z)$ share 0, 1, $\infty$ CM, then either $f(z)\equiv g(z)$ or $f(z)+g(z)\equiv f(z)g(z)$ or $f^{2}(z)(g(z)-1)^{2}+g^{2}(z)(f(z)-1)^{2}\equiv f(z)g(z)(f(z)g(z)-1)$ or there exists a polynomial $\varphi(z)=az+b_{0}$ such that $f(z)=\frac{1-e^{\varphi(z)}}{e^{\varphi(z)}(e^{a_{0}-b_{0}}-1)}$, $g(z)=\frac{1-e^{\varphi(z)}}{1-e^{b_{0}-a_{0}}}$, where $a\,(\neq 0)$, $a_{0}$, $b_{0}$ are constants with $e^{a_{0}}\neq e^{b_{0}}$.