亚纯函数与其差分的唯一性
Unicity of Meromorphic Functions and Their Difference Operators

研究了亚纯函数与其差分算子分担多项式的唯一性问题, 证明了: 设$f$是一个有穷级非常数亚纯函数, $p(z)(\not\equiv 0)$ 是一个多项式. 如果$f, \Delta_cf$ 与$\Delta_{c}^{2}f$ CM 分担$\infty$, $ p(z)$, 则$f\equiv\Delta_cf$ 或 $f(z)=\rme^{Az+B}+b$, 其中$p(z)\equiv b\neq 0$, $A\neq 0$ 满足$\rme^{Ac}=1$. 本文结果是对Chang, Fang(Chang J M, Fang M L. Uniqueness of entire functions and fixed points [J]. {\it Kodai Math J}, 2002, 25(1): 309--320.)结果的差分模拟, 并且完整回答了Chen, Chen(Chen B Q, Chen Z X, Li S. Uniqueness theorems on entire functions and their difference operators or shifts [J]. {\it Abstr Appl Anal}, 2012, Art. ID 906893, 8 pp.)的问题.
This paper deals with the unicity of meromorphic functions and their difference operators and proves: Let $f$ be a nonconstant meromorphic function of finite order, and let $p(z)(\not\equiv 0)$ be a polynomial. If $f, \Delta_cf$ and $\Delta_{c}^{2}f$ share $\infty$ and $ p(z)$ CM, then either $f\equiv\Delta_cf$ or $f(z)=\rme^{Az+B}+b$, where $p(z)\equiv b\neq 0$, $A\neq 0$ satisfying $\rme^{Ac}=1$. Our result provides a difference analogue of a result of Chang and Fang (Chang J M, Fang M L. Uniqueness of entire functions and fixed points [J]. {\it Kodai Math J}, 2002, 25(1): 309--320.), and answers the question of Chen and Chen (Chen B Q, Chen Z X, Li S. Uniqueness theorems on entire functions and their difference operators or shifts [J]. {\it Abstr Appl Anal}, 2012, Art ID 906893, 8 pp.).