A Result On Transcendental Entire Solutions To Certain Nonlinear Difference-differential Equations

In the 1920s,Rolf Nevanlinna expanded upon the early research of Picard,Borel,and other scholars in the field of entire functions,and established the value distribution theory of meromorphic functions.Nevanlinna’s value distribution theory has had a profound and far-reaching impact.In honor of Nevanlinna’s outstanding contributions,the value distribution theory is also known as Nevanlinna Theory,with its core being the First and Second Fundamental Theorems of Nevanlinna.Later,Ahlfors provided a geometric interpretation for the theory,further solidifying its theoretical foundation.Nevanlinna Theory has been widely applied in various complex analysis areas such as the uniqueness of meromorphic functions,normal families,complex differential equations,and complex dynamical systems.One significant application is using Nevanlinna Theory to study the existence and other analytic properties of solutions to differential equations in the complex domain.In recent years,some researchers have applied the theory to the study of difference equations in the complex domain.Among them,Halburd and Korhonen established the difference form of Nevanlinna’s fundamental theorem.The difference form of the logarithmic derivative lemma plays a crucial role in studying the properties of solutions to difference-differential(or difference)equations in the complex domain.This paper mainly studies the properties of transcendental entire function solutions for a class of differential-difference equations,and the obtained results are an extension of existing conclusions.The structure of the paper is arranged as follows.In Chapter 1,the basic knowledge of Nevanlinna theory is mainly presented,including some fundamental definitions,theorems and symbols that will be used in the paper.In Chapter 2,we mainly study the properties of transcendental entire solutions of differential-difference equations in the following form:fn(z)+ωfn-1(z)f’(z)+q(z)eQ(z)f(z+c)=p1eα1(z)+p2eα2(z),n≥ 3,where ω is a constant,c,λ1,λ2,p1,p2 are non-zero constants,q,Q,α1,α2 is polynomials.