Uniqueness theorems of meromorphic functions with their differential-difference operators in several complex variables

Abstract

An example in the article shows that the first derivative of $f(z)=\frac{2}{1-e^{-2z}}$ sharing $0$ CM and $1,\infty$ IM with its shift $\pi i$ cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its $k-th$ derivatives. We improves the author's result \cite{h} from entire function to meromorphic function, the first derivative to its differential-difference polynomial, and also finite values to small functions. As for $k=0$, we obtain: Let $f(z)$ be a transcendental meromorphic function of $\rho_{2}(f)<1$, let $c$ be a nonzero finite value, and let $a_{1}(z)\not\equiv\infty, a_{2}(z)\not\equiv\infty\in \hat{S}(f)$ be two distinct small functions of $f(z)$ such that $a(z)$ is a periodic function with period $c$ and $b(z)$ is any small function of $f(z)$. If $f(z)$ and $f(z+c)$ share $a_{1}(z),\infty$ CM, and share $a_{2}(z)$ IM, then either $f(z)\equiv f(z+c)$ or $$e^{p(z)}\equiv \frac{f(z+c)-a_{1}(z+c)}{f(z)-a_{1}(z)}\equiv \frac{a_{2}(z+c)-a_{1}(z+c)}{a_{2}(z)-a_{1}(z)},$$ where $p(z)$ is a non-constant entire function of $\rho(p)<1$ such that $e^{p(z+c)}\equiv e^{p(z)}$.