Unicity on entire function concerning its differential-difference operators

Abstract

In this paper, we study the uniqueness of the differential-difference polynomials of entire functions on $\mathbb{C}^{n}$. We prove the following result: Let $f(z)$ be a transcendental entire function on $\mathbb{C}^{n}$ of hyper-order less than $1$ and $g(z)=b_{-1}+\sum_{i=0}^{n}b_{i}f^{(k_{i})}(z+\eta_{i})$, where $b_{-1}$ and $b_{i} (i=0\ldots,n)$ are small meromorphic functions of $f$ on $\mathbb{C}^{n}$, $k_{i}\geq0 (i=0\ldots,n)$ are integers, and $\eta_{i} (i=0\ldots,n)$ are finite values. Let $a_{1}(z)\not\equiv\infty, a_{2}(z)\not\equiv\infty$ be two distinct small meromorphic functions of $f(z)$ on $\mathbb{C}^{n}$. If $f(z)$ and $g(z)$ share $a_{1}(z)$ CM, and $a_{2}(z)$ IM. Then either $f(z)\equiv g(z)$ or $a_{1}=2a_{2}=2$, $$f(z)\equiv e^{2p}-2e^{p}+2,$$ and $$g(z)\equiv e^{p},$$ where $p(z)$ is a non-constant entire function on $\mathbb{C}^{n}$. Especially, in the case of $g(z)=(\Delta_{\eta}^{n}f(z))^{k}$, we obtain $f(z)\equiv (\Delta_{\eta}^{n}f(z))^{k}$.