陶磊,龙见仁.某类$q$差分方程亚纯解的性质[J].数学研究及应用,2023,43(1):83~90
某类$q$差分方程亚纯解的性质
On Properties of Meromorphic Solutions for Certain $q$-Difference Equation
投稿时间:2021-11-23  修订日期:2022-05-07
DOI:10.3770/j.issn:2095-2651.2023.01.009
中文关键词:  复域差分方程  超越亚纯函数  增长级  存在性
英文关键词:complex $q$-difference equation  transcendental meromorphic function  order of growth  existence
基金项目:国家自然科学基金(Grant Nos.12261023; 11861023; 贵州省科学技术基金(Grant No.[2018]5769-05).
作者单位
陶磊 贵州师范大学数学科学学院, 贵州 贵阳 550025
贵州师范学院数学与大数据学院, 贵州 贵阳 550018 
龙见仁 贵州师范学院数学与大数据学院, 贵州 贵阳 550018 
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中文摘要:
      对于一个有穷非零复数$q$, 若下列$q$差分方程存在一个非常数亚纯解$f$, $$f(qz)f(\frac{z}{q})=R(z,f(z))=\frac{P(z,f(z))}{Q(z,f(z))}=\frac{\sum_{j=0}^{\tilde{p}}a_j(z)f^{j}(z)}{\sum_{k=0}^{\tilde{q}}b_k(z)f^{k}(z)},\eqno(\dag)$$ 其中 $\tilde{p}$和$\tilde{q}$是非负整数, $a_j$ ($0\leq j\leq \tilde{p}$)和$b_k$ ($0\leq k\leq \tilde{q}$)是关于$z$的多项式满足$a_{\tilde{p}}\not\equiv 0$和$b_{\tilde{q}}\not\equiv 0$使得$P(z,f(z))$和$Q(z,f(z))$是关于$f(z)$互素的多项式, 且$m=\tilde{p}-\tilde{q}\geq 3$. 则在$|q|=1$时得到方程$(\dag)$不存在亚纯解, 在$m\geq 3$和$|q|\neq 1$时得到方程$(\dag)$解$f$的下级的下界估计.
英文摘要:
      Let $q$ be a finite nonzero complex number, let the $q$-difference equation $$f(qz)f(\frac{z}{q})=R(z,f(z))=\frac{P(z,f(z))}{Q(z,f(z))}=\frac{\sum_{j=0}^{\tilde{p}}a_j(z)f^{j}(z)}{\sum_{k=0}^{\tilde{q}}b_k(z)f^{k}(z)}\eqno(\dag)$$ admit a nonconstant meromorphic solution $f,$ where $\tilde{p}$ and $\tilde{q}$ are nonnegative integers, $a_j$ with $0\leq j\leq \tilde{p}$ and $b_k$ with $0\leq k\leq \tilde{q}$ are polynomials in $z$ with $a_{\tilde{p}}\not\equiv 0$ and $b_{\tilde{q}}\not\equiv 0$ such that $P(z, f(z))$ and $Q(z, f(z))$ are relatively prime polynomials in $f(z)$ and let $m=\tilde{p}-\tilde{q}\geq 3$. Then, $(\dag)$ has no transcendental meromorphic solution when $|q|=1$, and the lower bound of the lower order of $f$ is obtained when $m \geq 3$ and $|q|\neq 1$.
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