随着Nevanlinna理论很好地应用到复域差分，微分的很多经典理论被推广到差分领域，这为复域差分和差分方程的研究提供了强有力的工具。复域差分有自己的数学物理背景。特征函数是亚纯函数最重要的特征。基于亚纯函数与其导数的特征函数之间的关系,一些学者建立了Nevanlinna理论的一些经典结果,如亚纯函数与其导数有相同的级与下级,导数的亏量和的估计等等。而关于函数与高阶差分的特征函数之间的关系的结果却很少。因此高阶差分的特征函数的研究有着非常重要的意义和应用前景。然后我们研究高阶差分的Nevanlinna例外值，补充并完善值分布理论。差分Painlevé 方程是当前的热门研究课题，具有几何物理背景。本项目研究高阶差分的值分布，包括其特征函数和Nevanlinna 例外值,及差分 Painlevé IV 方程的亚纯解的性质，包括解的亏量，解的差分和均差分的值分布。

As Nevanlinna theory well applying in complex difference, many classical results in complex differential theory are extended in the field of difference, which provided powerful tools in the study of complex difference and difference equations. Complex difference has its own mathematical and physical background.The characteristic function is the most important character of meromorphic function. Based on the relationships between the characteristic functions of meromorphic function and its derivative, some scholars establish some classical results in Nevanlinna theory, such as a meromorphic function and its derivative have the same order and lower order, the estimations of the sum of Nevanlinna exceptional values of derivative of meromorphic function, etc. But it has few results on the relationships between characteristic functions of meromorphic function and its forward differences. So, the study of the characteristic functions of the forward differences has very important ignificance and application prospect. Further more, we study the Nevanlinna exceptional value of forward difference, which will make up and improve the value distribution of Nevanlinna theory. The topic on difference Painlevé equations is a hot one in the current. They have geometric and physical ackgrounds. In this project, we study value distribution of forward difference,contain the characteristic functions of forward differences and Nevanlinna exceptional value, and properties of meromorphic solutions of difference PainlevéIV equations, including Nevanlinna exceptional value of meromorphic solutions, value distribution of their difference and divided difference.