线性与非线性复微-差分方程和复微-差分多项式的若干性质

Complex differential-difference equations and complex differential-difference polynomials becomes to be a new research hotspot in the field of complex analysis. This research is the extension and development of the ones of the case of complex differentiates and the case of complex differences, and has wide space for further research. In this research, by combining Nevanlinna theory’s classic applications in the field of complex differentiates and Nevanlinna theory’s new applications in the field of complex differences, we will generalize Nevanlinna theory into the field of complex differential-differences. Hence, this research has important practical meaning and pure mathematical meaning. By the tool of Nevanlinna theory and its (differential-)difference analogues, we will carry out our research work in this project on the following aspects. We will investigate the properties of meromorphic solutions of the homogeneous and non-homogeneous complex linear differential, difference, differential-difference equations and more general complex linear equations on composite functions; we will investigate the properties of meromorphic solutions of (the systems of) non-linear complex (differential-)difference equations of Malmquist type, Fermat type and other special types; we will investigate complex (differential-)difference analogues of the Hayman’s classic results on the Picard exceptional values of the meromorphic function and its derivatives; we will investigate the properties of complex (differential-)difference polynomials of general or special forms; we will construct complex (differential-)difference analogues of some classic results of Nevanlinna theory and apply these results to investigate the properties of meromorphic solutions of complx (differential-)difference equations and complex (differential-)difference polynomials.

复微-差分方程和复微-差分多项式是复分析领域的新研究热点，是复微分情形和复差分情形的必然延伸与发展，具有广泛的研究空间。本项目研究将结合Nevanlinna理论在复微分领域的经典应用和在复差分领域的新应用，并进一步应用至复微-差分领域，具有重要的实际意义和纯数学理论意义。在本项目中，我们将运用Nevanlinna理论及其复(微-)差分模拟结果在以下方面进行研究：研究齐次与非齐次复线性微分、差分、微-差分方程和更一般的复线性方程亚纯解的性质；研究Malmquist型、Fermat型和其他特殊类型的非线性复(微-)差分方程(组)亚纯解的性质；研究Hayman关于亚纯函数及其导数的Picard例外值的经典结果的复(微-)差分模拟；研究复(微-)差分多项式的性质；建立某些Nevanlinna理论经典结果的复(微-)差分模拟，并应用于研究复(微-)差分方程亚纯解和复(微-)差分多项式的性质。

我们对本项目的研究基本按预期研究计划进行，主要在复线性微分、差分和微-差分方程亚纯解的解析性质、几类特殊的非线性复差分方程(组)亚纯解(组)的解析性质、复(微-)差分多项式的解析性质、多复变量Fermat型偏(微-)差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：研究了复线性微分、差分、微-差分方程和关于复合函数的复线性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(组)亚纯解(组)的存在性、具体形式、增长性和值分布；研究了一般形式和某些特殊形式的复(微-)(q-)(平移)差分多项式的值分布；研究了多复变量Fermat型偏(微-)差分方程亚纯解的存在性和具体形式。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，得到了一系列的新研究成果，并在SCI源刊及国内外核心刊物上发表了22篇学术论文，超过了预定目标，且培养了4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，复微-差分方程与复微-差分多项式理论依然是复分析方向及数理方程方向的学者研究的热点之一。我们紧跟本领域国际研究发展趋势，重点研究复微-差分情形的相关内容，顺利完成了预计的研究目标，为后续研究打下了良好的基础。同时，我们还将本项目研究从单复变量情形拓宽至多复变量情形，取得了一些进展，为后续相关研究开了一个好头。