Research On Uniqueness Of Meromorphic Solutions Of Malmquist Type Difference Equations And Zeros Of One Class Of Differential-difference Polynomials

In this note,we study a uniqueness theorem meromorphic solutions of a class of difference equation of Malmquist type and the zeros of complex differential-difference polynomial on transcendental meromorphic functions.Firstly,by using the Nevanlinna theory,we prove that uniqueness theorem meromorphic solutions of a more general class of difference equation of Malmquist type.Secondly,analysing the zeros and poles,we show that when n takes a certain value,the complex difference-differential polynomials take infinitely many zeros,which can be regarded as the differential-difference analogues of Hayman conjecture.The main results are as follows:Theorem1.2.1 Let f be a finite-order transcendental meromorphic solution of here,a_??z?????0?,bu?z?,dl?z?are small functions respect to f?z?,c?j are pairwise distinct constants,???Define ???,can be rewritten as H?z,f?=0?1.3?Let e₁,,e₂ be two distinct finite numbers such that H?z,e₁?,H?z,e₂?? 0.If f and another meromorphic function g share the values e₁,e₂ and ? CM,then f?g.Theorem3,2.1 Let f?z?be a transcendental meromorphic function of hyper-order P2?f?<1.If n?k+6,then f n?z?f?k??z?+f?z+c?-a?z?has infinitely many zeros,where a?z?is a non-zero small function with respect of f?z?.Theorem3.2.1 Let f?z?be a transcendental meromorphic function of hyper-order P2?f?<1-If n?2k+8,then fn?z?f?k??z+c?+f?z?-a?z?has infinitely many zeros,where a?z?is a non-zero small function with respect of f?z?.The structure of this paper is as follows:In chapter 1,we introduce the background,some concepts and main theorems.In chapter 2,we prove a uniqueness theorem meromorphic solutions of a more general class of difference equation of Malmquist type.In chapter 3,we prove the zeros of complex differential-difference polynomial on transcendental meromorphic functions.