| 崔宁,陈宗煊.一类线性差分方程的亚纯解与一个亚纯函数分担3个值的唯一性[J].数学年刊A辑,2017,38(1):013~22 |
| 一类线性差分方程的亚纯解与一个亚纯函数分担3个值的唯一性 |
| Uniqueness for Meromorphic Solutions Sharing Three Values with a Meromorphic Function to Some Linear Difference Equations |
| Received:November 06, 2015 Revised:May 10, 2016 |
| DOI：10.16205/j.cnki.cama.2017.0002 |
| 中文关键词: Meromorphic function, Difference equation, Shared values, Uniqueness |
| 英文关键词:Meromorphic function, Difference equation, Shared values, Uniqueness |
| 基金项目:本文受到广东省自然科学基金(No.2014A030313422,No.2016A030310106,No.2016A030313745)的资助. |
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| 中文摘要: |
| 主要研究差分方程~$a_{1}(z)f(z+1)+a_{0}(z)f(z)=F(z)$~的一个有穷级超越亚纯解
$f(z)$~与亚纯函数~$g(z)$~分担~$0, 1, \infty$~CM~时的唯一性问题~(其中~$a_{1}(z)$,
$a_{0}(z), F(z)$~为非零多项式, 且满足~$a_{1}(z)+a_{0}(z)\not\equiv0$),
得到~$f(z)\equiv g(z)$, 或$f(z)+g(z)\equiv f(z)g(z)$, 或存在一个多项式
$\beta(z)=az+b_{0}$~和一个常数~$a_{0}$~满足~$\rme^{a_{0}}\neq \rme^{b_{0}}$,~使得
$f(z)=\frac{1-\rme^{\beta(z)}}{\rme^{\beta(z)}(\rme^{a_{0}-b_{0}}-1)}$~与
$g(z)=\frac{1-\rme^{\beta(z)}}{1-\rme^{b_{0}-a_{0}}}$, 其中~$a(\neq0), b_{0}$~为常数. |
| 英文摘要: |
| This paper deals with the uniqueness of a finite-order
meromorphic solution $f(z)$ of some linear difference equation
$a_{1}(z)f(z+1)+a_{0}(z)f(z)=F(z)$ sharing $0, 1, \infty$ CM with
meromorphic function $g(z)$ (where $a_{1}(z)$, $a_{0}(z)$ and $ F(z)$ are
nonzero polynomials satisfying $a_{1}(z)+a_{0}(z)\not\equiv0$), and
obtain either $f(z)\equiv g(z)$ or $f(z)+g(z)\equiv f(z)g(z)$ or
there exists a polynomial $\beta(z)=az+b_{0}$ and a constant $a_{0}$
satisfying $\rme^{a_{0}}\neq \rme^{b_{0}}$, such that
$f(z)=\frac{1-\rme^{\beta(z)}}{\rme^{\beta(z)}(\rme^{a_{0}-b_{0}}-1)}$
and $g(z)=\frac{1-\rme^{\beta(z)}}{1-\rme^{b_{0}-a_{0}}}$, where
$a(\neq0), b_{0}$ are constants. |
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