C上微-差分多项式和微-差分方程亚纯解的解析性质

Recently, the theory of complex differences and difference equations develops rapidly and becomes a new research hotspot. The theory has wide applications in Mathematics and Physics. By the tools of Nevanlinna theory as well as its complex difference analogies and the theory of differential field, we will carry out our research work in this project on the following aspects. We will investigate analytic properties, such as the existence, the growth, the value distribution and so on, of meromorphic solutions of some difference equations, which include homogeneous and non-homogeneous linear difference equations, difference Painlevé equations, non-linear q-difference equations, non-linear differential-difference equations; and we investigate the growth of meromorphic solutions of some systems of difference equations, such as systems of difference Painlevé equations and systems of difference equations of Malmquist type. We will also explore analytic properties of complex differences and divided differences of meromorphic functions, as well as the value distribution and uniqueness of general difference polynomials and differential- difference polynomials. In addition, we will probe into difference analogies of some results of Nevanlinna theory and their applications. Furthermore, we will seek the essential relations between complex differential equations and complex difference equations, consequently find similar results and common research methods. In conclusion, the research work in this project will not only enrich the research on the theory of complex differences and difference equations, but also push forward the research and applications of Nevanlinna theory and the theory of differential field in the difference case.

复差分与差分方程理论近年来发展迅速，是新的研究热点之一。该理论在数学与物理等领域应用广泛。在本项目中，我们将应用Nevanlinna理论及其复差分模拟结果和微分域理论等工具在以下方面进行研究。研究某些复差分方程亚纯解的存在性、增长性和值点分布等解析性质，方程类型包括齐次和非齐次线性差分方程、差分Painlevé方程、非线性q-差分方程和非线性微-差分方程等；研究差分Painlevé方程组和Malmquist型差分方程组等方程组的亚纯解的增长性。研究亚纯函数差分和均差分的解析性质；研究一般形式的差分多项式和微-差分多项式的值分布和唯一性。研究Nevanlinna理论中某些结果的差分模拟并加以应用。探索复微分方程和复差分方程之间的本质联系，寻求相似的研究结果和共同的研究方法。本项目研究将丰富复差分与差分方程理论的研究，同时也将推动Nevanlinna理论和微分域理论在差分领域中的研究与应用。

我们对本项目的研究基本按预期研究计划进行，主要在复(微-)差分方程(组)亚纯解的解析性质、复(微-)差分多项式的解析性质、Nevanlinna理论的复(微-)差分模拟和复线性微分方程亚纯解的解析性质等四个方面开展了研究，具体情况如下：研究了齐次与非齐次复线性(微-)(q-)差分方程亚纯解的增长性和值分布；研究了Malmquist型、Painlevé型和Riccati型等非线性复(微-)(q-)差分方程亚纯解的存在性、增长性和值分布；研究了Malmquist型等复(微-)q-差分方程组的退化定理和亚纯解的增长性；研究了一般形式和某些特殊形式的复(微-)差分多项式的值分布；研究了Valiron-Mohon’ko定理的复(微-)差分模拟，并应用于研究复微-差分多项式的值分布；研究了复线性微分方程亚纯解的增长性和值分布。我们在上述四个方面对已有相关结果进行了推广、补充和改进，得到了一系列重要的新研究成果，并在SCI源刊物及国内外核心刊物上发表或录用了28篇学术论文，大大超过了预定目标(即发表学术论文10篇左右)，且培养了6名硕士研究生，顺利地完成了本项目的研究。. 在本项目研究中，我们紧跟本领域国际发展趋势，改变了独立研究复微分情形和复差分情形的研究模式，将复微分和复差分结合起来研究，顺利过渡到了复微-差分领域，为进一步深入研究打下了良好的基础。