Unicity on entire function concerning its differential polynomials in Several complex variables

XiaoHuang Huang

Published 2021-03-03Version 1

In this paper, we study the uniqueness of the differential polynomials of entire functions. We prove the following result: Let $f(z)$ be a nonconstant entire function on $\mathbb{C}^{n}$ and $g(z)=b_{-1}+\sum_{i=0}^{n}b_{i}f^{(k_{i})}(z)$, where $b_{-1}$ and $b_{i} (i=0\ldots,n)$ are small meromorphic functions of $f$, $k_{i}\geq0 (i=0\ldots,n)$ are integers. Let $a_{1}(z)\not\equiv\infty, a_{2}(z)\not\equiv\infty$ be two distinct small meromorphic functions of $f(z)$. If $f(z)$ and $g(z)$ share $a_{1}(z)$ CM, and $a_{2}(z)$ IM. Then either $f(z)\equiv g(z)$ or $a_{1}=2a_{2}$, $$f(z)\equiv a_{2}(e^{2p(z)}-2e^{p(z)}+2),$$ and $$g(z)\equiv a_{2}e^{p(z)},$$ where $p(z)$ is a nonzero entire function satisfying $a_{2}(z)p'(z)=1$.