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_λ（z）（（?）0）,bu（z）,dl（z）are small functions respect to f（z）,cλj 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.