Studies On The Value Distribution Theory Of Difference Operators And Complex Equations

In the 1920s,the Finnish mathematician R.Nevanlinna established the value distribution theory of meromorphic functions,which is also called Nevanlinna theory of meromorphic functions.After a long periods’researching and exploring,scholars established the value distribution theory of meromorphic functions,which not only been extended to holomorphic curves,but also promoted the development of difference operators.On the basis of the value distribution theory of meromorphic functions,the difference operators and holomorphic curve,this dissertation studies the value distribution of holomorphic curves for Jackson difference operators and Askey-Wilson difference operators respectively.In addition,we study some properties of Jackson difference operators of entire solutions to complex differential-difference equations.The structure of this dissertation is as follows:In Chapter 1,the background,research status and main results of our work are introduced.Chapter 2 is the preliminary knowledge.Definitions and theorems that will be used later are introduced,including the value distribution theory of meromorphic function,the difference operators and holomorphic curve.Chapter 3 is devote to investigating the value distribution theory of holomorphic curves for Jackson difference operators.For a holomorphic curve with zero order,the Jackson difference analogue of Cartan’s second theorem and its truncated form are obtained by using the redefined Jackson q-Casorati determinant.Additionally,we introduce the concept of the Jackson difference radical and prove the Mason theorem.Furthermore,the Mason theorem for m+1 polynomials is extended.In Chapter 4,the value distribution of holomorphic curves for Askey-Wilson difference operators is studied.For a holomorphic curve with finite logarithmic order,firstly,we establish the second main theorem of holomorphic curves for Askey-Wilson difference operators and its truncated form.Secondly,we obtain the Picard type theorem of holomorphic curves for Askey-Wilson difference operators with the concept of AW-invariant.Finally,we introduce the concept of Askey-Wilson difference polynomial and give the Tumura-Clunie theorem.In Chapter 5,we estimate the measure of the limiting direction of the entire function solutions to two kinds of complex differential-difference equations.Firstly,we consider the complex differential-difference equation where pλk(z,f)(λ∈N,k=0,1,...,n)are distinct differential-difference monomials,aλk(z)are entire functions of growth smaller than that of the transcendental entire h(z).We mainly estimate the measure of Julia limiting directions and transcendental directions of Jackson difference operators for non-trivial transcendental entire solutions to the above equation.In addition,for non-trivial entire solutions f to where Pλ(z,f)(λ=1,2)are differential-difference polynomials,when the Petrenko’s deviation of entire coefficient A0(z)satisfies certain conditions,the measure of common transcendental directions of Jackson difference operators and shift difference operators of f can be eatimated.In Chapter 6,the main work of our research is summarized and the future research directions are expected.