Uniqueness of Meromorphic Solutions for a Class of Complex Linear Differential-Difference Equations

Abstract:

      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 $e^{a_{0}}\neq e^{b_{0}}$.