论文标题
杂形函数的独特性相对于它们有关衍生物的变化
Uniqueness of Meromorphic Functions With Respect To Their Shifts Concerning Derivatives
论文作者
论文摘要
文章中的一个示例表明，$ f（z）= \ frac {2} {1-e^{ - 2z}} $共享$ 0 $ cm和$ 1，\ infty $ im at shift $πi$无法获得的第一个衍生物。在本文中，我们研究了共同函数共享小功能的独特性，以及其有关其$ k-th $衍生产品的转变。我们使用与Qi和Yang \ cite {Qy}的不同方法来改善Meromorormormormormormorphic函数的整个函数，这是$ K-TH $衍生词的第一个衍生物，也是对小函数的有限值。至于$ k = 0 $，我们获得：让$ f（z）$是$ρ_{2}（f）（f）<1 $的先验性异常功能$ f（z）$使得$ a（z）$是一个定期功能，带有$ c $，$ b（z）$是$ f（z）$的任何小函数。如果$ f（z）$和$ f（z+c）$ share $ a（z），\ infty $ cm，以及共享$ b（z）$ im，则$ f（z）\ equiv f（z+c）$或$$ e^{p（z）} \ equiv \ equiv \ equiv \ equiv \ equ \ frac {f（z+c）-a（z+c）-a（z+c）} - \ frac {b（z+c）-a（z+c）} {b（z）-a（z）}，$$，其中$ p（z）$是$ρ（p）<1 $的非恒定整体函数，以便$ e^{p（z+c）} \ equiv e equiv e e^p（z）$。
An example in the article shows that the first derivative of $f(z)=\frac{2}{1-e^{-2z}}$ sharing $0$ CM and $1,\infty$ IM with its shift $πi$ cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its $k-th$ derivatives. We use a different method from Qi and Yang \cite {qy} to improves entire function to meromorphic function, the first derivative to the $k-th$ derivatives, and also finite values to small functions. As for $k=0$, we obtain: Let $f(z)$ be a transcendental meromorphic function of $ρ_{2}(f)<1$, let $c$ be a nonzero finite value, and let $a(z)\not\equiv\infty, b(z)\not\equiv\infty\in \hat{S}(f)$ be two distinct small functions of $f(z)$ such that $a(z)$ is a periodic function with period $c$ and $b(z)$ is any small function of $f(z)$. If $f(z)$ and $f(z+c)$ share $a(z),\infty$ CM, and share $b(z)$ IM, then either $f(z)\equiv f(z+c)$ or $$e^{p(z)}\equiv \frac{f(z+c)-a(z+c)}{f(z)-a(z)}\equiv \frac{b(z+c)-a(z+c)}{b(z)-a(z)},$$ where $p(z)$ is a non-constant entire function of $ρ(p)<1$ such that $e^{p(z+c)}\equiv e^{p(z)}$.
