The Uniqueness Of Several Meromorphic Functions And Their Difference Sharing Values

Complex differential difference equations have a mathematical field with many branches.In these branches,Local theory is also the most studied content.Through the in-depth study of complex differential equations based on Nevanlinna value distribution theory,a large number of conclusions about local existence,uniqueness theory and the basic structure of solutions of linear differential equations are obtained.Some mathematicians are more interested in its research in the broad sense.They consider such problems from many angles.Among them,Matsuda considers the research of complex differential equations from the algebraic angle,Jurkat consider the research of differential equations from the angle of differential equations or also from the angle of function theory.In the early 1920 s,R.Nevanlinna first introduction the characteristic function of meromorphic functions,and created two basic theorems,which pioneered the theory of distribution of meromorphic function values and later generations called Nevanlinna's theory of distribution of meromorphic function values.The famous Picard and Borel theorems would be a simple deduction of it.Through the unremitting efforts of the scholars,the theory has been continuously improved and developed rapidly,and has been widely used in the field of compound analysis.Among them,many far-reaching consequences have been obtained in the fields of uniqueness theory,complex difference equation and complex differential equation.This paper mainly uses Nevanlinna value distribution theory and its main conclusions,we study the properties of solutions of certain types of non-linear differential and difference equations and generalize some important results.Firstly,Nevanlinna value distribution theory,basic results and common symbols are summarized,as well as some basic concepts and related knowledge about the properties of functions.Secondly,we study the uniqueness of meromorphic functions and their difference sharing values,and improve the theorem condition of Lü [3],extending the condition of sharing three constants to sharing three polynomials,and proving the theorem.Once again,consider adding conditional meromorphic functions and its n-order difference to share three polynomials,and obtain a new theorem.