Analytic Properties Of Meromorphic Solutions Of Complex Difference And Difference Equations

As early as 30 years ago,several initial results on the existence of meromorphic solutions of some complex difference equations have been obtained.Later on,the development was relatively slow due to the lack of useful research tools.Around 2000,Ablowitz,Halburd,Korhonen,Chiang Yikman,Feng Shaoji and et al made significant breakthroughs by using the value distribution theory of meromorphic function to investigate the properties of solutions of complex difference equations,and many researchers are devoted to studying in this field.In this thesis,we mainly use the value distribution theory to investigate the properties of difference and difference equations.The thesis includes the following five parts.In the first part,we introduce the research status and significance of complex differential and difference equations,meanwhile the basic definitions,theorems and symbols of theory of value distribution will be given in this part.In the second part,starting from the Hayman's conjecture,we mainly discuss the relation between small functions and a class of difference polynomials.In the third part,we study the uniqueness of meromorphic difference operators sharing rational functions and the zero distribution of difference operators with respect to rational functions.In the fourth part,by using the theory of uniqueness of meromorphic functions,we invstigate the uniqueness of meromorphic solutions of differential-difference equations.In the fifth part,we are devoted to studying the growth of meromorphic solutions of higher order linear difference equations without domaining coefficients,where the coefficients are meromorphic functions.With some additional conditions on coefficients,we obtain the precise estimates of growth of meromorphic solutions of such equation.