Meromorphic Solutions of a Type of Complex Differential-Difference Equations

REN Guozhen(任国珍),GAO Lingyun(高凌云)

(Department of Mathematics,Jinan University,Guangzhou 510632,China)

Abstract: Using Nevanlinna theory of the value distribution of meromorphic functions,we investigate the properties and expression of meromorphic solutions of a type of complex differential-difference equation,and we obtain some results,which are the extensions of complex difference equations to differential-difference equations.Example shows that our results are meaningful.

Key words: Complex differential-difference equation;Meromorphic solution;Value distribution

1.Introduction

Letf(z) be a meromorphic function in the whole complex plan C.We assume that the reader is familiar with the standard notation and results of the Nevanlinna theory of meromorphic functions[1−2],such as the characteristic functionT(r,f),proximity functionm(r,f),counting functionN(r,f).

The notationS(r,f)denotes any quantity that satisfies the condition:S(r,f)=o(T(r,f))asr →∞possibly outside an exceptional setEofrof finite linear measure∞.A meromorphic functiona(z) is called a small function off(z) if and only ifT(r,a(z))=S(r,f).Moreover,we use the notation degf Pfor the degree ofP(z,f) with respect tofand the order offis defined by

Many papers investigate solutions of some types of complex differential equations,and obtain some results[3−8].Recently,there has been renewed interests in meromorphic solutions of complex difference equations,in addition to the complex differential equation.Many authors have investigated complex difference equations,and obtained some results[9−15].

In 2005,Laine and Rieppo[14]considered a type of complex differential equations of the following form

wherePandQare relatively prime polynomials infover the field of rational functions,the coefficientsαJare rational functions andq=degf Q >0.They obtained the following results.

Theorem A[14]Assumef(z) is a transcendental meromorphic solution of the above difference equation,andf(z) has finitely many poles,it must be of the form

f(z)=r(z)eg(z)+s(z),

wherer(z) ands(z) are rational functions andg(z) is a transcendental entire function satisfying a difference equation of the form either

2.Some Lemmas

In order to prove our results,we need the following lemmas.

Lemma 2.1[14]Letfbe a non-constant meromorphic function and letP(z,f),Q(z,f)be two polynomials infwith meromorphic coefficients small relative tof.IfPandQhave no common factors of positive degree infover the field of small functions relative tof,then

Remark 2.1[14]Iff(z) is a transcendental meromorphic function andP(z,f),Q(z,f)have rational coefficients,thenP(z,f(z))andQ(z,f(z))has only finitely many common zeros.

Lemma 2.2[16]Letf(z) be a meromorphic function and letΦbe given by

3.Main Results and Proofs

In this paper,letting

P(z,f)=ap(z)fp+ap−1(z)fp−1+···+a0(z),

where