arXiv 2009.080662020-09math.CV

整函数关于其微分-差分算子的唯一性

Unicity on entire function concerning its differential-difference operators

Xiao Huang

备注：: 15 pages, Comments Welcomed

摘要：

本文研究了$\mathbb{C}^{n}$上整函数微分差分多项式的唯一性。我们证明了以下结果：设$f(z)$是$\mathbb{C}^{n}$上超阶小于$1$和$g(z)=b_{-1}+\sum_{i=0}^{n}b_{i}f^{(k_{i})}(z+η_{i})$的超越整函数，其中$b_{-1}$和$b_{i} (i=0\ldots,n)$是$\mathbb{C}^{n}$上$f$的小亚纯函数，$k_{i}\geq0 (i=0\ldots,n)$是整数，$η_{i} (i=0\ldots,n)$是有限值.设$a_{1}(z)\not\equiv\infty, a_{2}(z)\not\equiv\infty$是$f(z)$在$\mathbb{C}^{n}$上的两个不同的小亚纯函数.如果$f(z)$和$g(z)$共享$a_{1}(z)$ CM和$a_{2}(z)$ IM。然后是$f(z)\equiv g(z)$或$a_{1}=2a_{2}=2$，$$f(z)\equiv e^{2p}-2e^{p}+2,$$和$$g(z)\equiv e^{p},$$，其中$p(z)$是$\mathbb{C}^{n}$上的非常数整函数。特别地，在$g(z)=(Δ_η^{n}f(z))^{k}$的情况下，我们得到了$f(z)\equiv (Δ_η^{n}f(z))^{k}$。

英文摘要：

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 $f(z)\equiv (Δ_η^{n}f(z))^{k}$.

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摘要：

英文摘要：

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