Studies On Complex Differential-Difference Polynomials And Analytical Properties Of Equations

Applying Nevanlinna value distribution theory for meromorphic functions and its latest achievements in this thesis,we make exploration on growth and value distribution of linear shifts polynomials and differentialdifference polynomials of meromorphic functions,establish corresponding uniqueness theorems based on divergent types of sharing conditions,as well as study the uniqueness of meromorphic solutions to partial differential-difference equations.The structure of this thesis is arranged as follows.In chapter 1,the academic history of Nevanlinna value distribution theory is retrospected first;then development contexts of various research directions are sorted out,including theories of deficiency,value distribution of derivatives,unicity of meromorphic functions and differential-difference equations;finally fundamental concepts,standard notations and important theorems are introduced which will be used later.In chapter 2,value distribution properties of a class of linear shift polynomials with constant coefficients are studied,such as their characteristic functions,exceptional values,deficiency relations and the convergent exponent of poles of divided difference.Some existing results of higher order difference operators are partially generalized.In chapter 3,first order or higher order difference operators are replaced by linear shift polynomials with small function coefficients,both the characteristic functions of such polynomials and the convergent exponent of zeros of the difference between such polynomials and small functions are studied under certain deficiency conditions.Higher derivative operators are replaced by linear differential polynomials with small function coefficients and the value distribution of such polynomials is analyzed.In chapter 4,the value distribution of a class of q-shifts differentialdifference polynomials is studied,the uniqueness of meromorphic functions that generate such polynomials is discussed based on several types of sharing assumptions including weighted sharing,weakly weighted sharing,relaxed weighted sharing and partially sharing.In chapter 5,a class of nonhomogeneous meromorphic differentialdifference polynomials with small function coefficients is being concerned about,and the uniqueness of such polynomials when sharing small functions is studied under different types of sharing conditions.In chapter 6,a class of homogeneous linear difference equations with entire coefficients that are of finite order is being concerned about.For two classes of linear shift polynomials of meromorphic solutions to such equations,their value distribution properties are studied,such as exceptional values,distribution of zeros,the convergent exponent of zeros of the difference between them and small functions.In chapter 7,a class of nonlinear partial differential-difference equations with small function coefficients is studied.The conclusion indicates that meromorphic solutions of finite order to such equations could be uniquely determined by their poles and the zeros of the difference between them and two distinct small functions.In chapter 8,the differences between this study and previous studies are concluded in terms of several perspectives including the subjects,the methods and the conclusions.The inadequacies of this study are pointed out,and potential directions for further studies are analyzed.