Some Researches On Unicity Of Complex Differential Difference And Solutions Of Complex Differential Difference Equations

In 1925,Finnish mathematician R.Nevanlinna introduced characteristic functions and established two basic theorems of Nevanlinna,which marked the beginning of modern meromorphic function theory.In this thesis,we use the value distribution theory of meromorphic functions as the main research tool.On the one hand,the uniqueness of meromorphic solutions of a class of nonlinear complex differential difference equations is studied.On the other hand,we research the unicity of meromorphic functions and some kind of difference polynomials.We divide this paper into four chapters.The four chapters are arranged as follows:In chapter 1,we briefly introduce the Nevanlinna value distribution theory,meromorphic function value distribution complex difference analogues theory,the unicity theory of meromorphic function and some related basic symbols.In chapter 2,we mainly study the uniqueness of transcendental meromorphic solutions for a class of complex linear differential-difference equations.Specially,suppose that f（z）is a finite order transcendental meromorphic solution of complex linear differential-difference equation:W₁（z）f’（z+1）+W₂（z）f（z）=W₃（z）,where W₁（z）,W₂（z）,W₃（z）are nonzero meromorphic functions,with their orders of growth being less than one such that W₁（z）+W₂（z）（?）0.If f（z）and a meromorphic function g（z）share 0,1,∞ CM,then either f（z）≡ g（z）or f（z）+g（z）≡ f（z）g（z）or f²（z）（g（z）-1）²+g²（z）（f（z）-1）²≡ f（z）g（z）（f（z）g（z）-1）or there exists a polynomial φ（z）=a_z+b₀ such that f（z）=(1-e^φ（z）)/e^φ（z）(e^a₀-b₀-1),g（z）=(1-e^φ（z）)/(1-e^b₀-a₀),where a（≠0）,a0,b0 are constants with ea0≠eb0.In chapter 3,we study the unicity of meromorphic function f（z）and its difference polynomial M（f）,where M（f）=γ₀f（z）+γ₁f（z+η）+…+γ_pf（z+pη）and γ₀+γ₁+…+γ_p=0,η,γ₀,γ₁,…,γ_p（≠0）are constants.We obtain that the uniqueness theorems of f（z）and M（f）CM sharing a and satisfing E（∞,M（f））（?） E(∞,f（z）,E（0,M（f））（?） E（0,f（z））or IM sharing a,b,where a,b are distinct finite complex constants.The results in this section generalize some theorems obtained by S.J.Chen,A.Z.Xu,X.G.Qi,L.Z.Yang and so on.In chapter 4,we summarize the main research content of this paper and raise some problems for further study.