Some Researches On Meromorphic Solutions Of Painlevé Ⅲ Difference Equations

In the 1980 s,Bank S.B.,Kaufman R.P.,Shimomura S.,Yanagihara N.and others proved the existence of meromorphic solutions for some complex difference equations.In recent years,with the rise of the complex domain difference analogues theory of value distribution,it has promoted the research process of complex domain difference and difference equations,which has received widespread attention and has become a new hot topic.Many beautiful results have been achieved in the study of the Panlevé difference equation,and many scholars are still exploring it.In this paper,we mainly study the meromorphic solutions of Painlevé III difference equation.Firstly,we discuss the uniqueness of the meromorphic solutions of Painlevé III difference equation sharing values(sets)with another meromorphic function.Secondly,we study the value distribution properties of the meromorphic solutions of the Painlevé III difference equation.The paper structure is divided into the following four chapters:In chapter 1: preliminary knowledge.we briefly describe the value distribution theory and common symbols of meromorphic functions,the main results of the uniqueness theory of meromorphic functions,and some results of the theory of complex domain difference analogues of value distribution.In chapter 2: we first study the uniqueness of the finite order transcendental meromorphic solution of Painlevé difference equation sharing two values and one set with the general meromorphic function,and then we generalize the results of Chen Baoqin and Li Sheng about the finite order transcendental meromorphic solution of the Painlevé III difference equation sharing three values with the general meromorphic function.In chapter 3: In this chapter,we mainly study the relationship between the zero,pole,fixed point convergence index and mean deviation,difference,etc.and the order of the meromorphic solutions of the Painlevé III difference equation under the condition of equation coefficient is restricted.In addition,we also study the existence of Borel exceptional values and Nevanlinna exceptional values of meromorphic solutions.In chapter 4: The summary and prospect.we analyze and summarize the research contents and results of this paper,and put forward the questions and ideas for the next step.