论文标题
有关其差分差异操作员的整个功能的独立性
Unicity on entire function concerning its differential-difference operators
论文作者
论文摘要
在本文中，我们研究了$ \ mathbb {c}^{n} $的整个功能的差分差异多项式的唯一性。我们证明了以下结果：让$ f（z）$是$ \ mathbb {c}^{n} $ hystrorder of throfl-ordore的$ hyprorder of hyper-order的$ bys hyper-order by Baty by $ 1 $和$ g（z）= b _ { - 1}+\ sum_ {i = 0}^{n} b_ {i} f^{（k_ {i}）}（z+η_{i}）$ $ \ mathbb {c}^{n} $，$ k_ {i} \ geq0（i = 0 \ ldots，n）$是整数，$η_{i}（i = 0 \ ldots，n）$是有限的值。令$ a_ {1}（z）\ not \ equiv \ infty，a_ {2}（z）\ not \ equiv \ equiv \ infty $为$ \ mathbb {c}^{n} $的$ f（z）$的两个独特的小毛素函数。如果$ f（z）$和$ g（z）$ share $ a_ {1}（z）$ cm和$ a_ {2}（z）$ im。那么要么$ f（z）\ equiv g（z）$或$ a_ {1} = 2a_ {2} = 2 $，$$ f（z）\ equiv e^{2p} -2p} -2e^{p} +2，$$和$ g（$ g（z）和$ g（Z） $ \ mathbb {c}^{n} $。特别是在$ g（z）=（δ_η^{n} f（z））^{k} $的情况下，我们获得$ f（z）\ equiv（δ_η^{n} f（z）f（z））^{k} $。
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 $g(z)=b_{-1}+\sum_{i=0}^{n}b_{i}f^{(k_{i})}(z+η_{i})$, 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 $η_{i} (i=0\ldots,n)$ 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 $g(z)=(Δ_η^{n}f(z))^{k}$, we obtain $f(z)\equiv (Δ_η^{n}f(z))^{k}$.
