Zongxuan CHEN

School of Mathematical Sciences,South China Normal University,Guangzhou 510631,China

E-mail:chzx@vip.sina.com

Kwang Ho SHON

Department of Mathematics,College of Natural Sciences,Pusan National University,Busan,609-735,Korea

E-mail:khshon@pusan.ac.kr

Abstract Consider the di ff erence Riccati equation f(z+1)=where A,B,C,D are meromorphic functions,we give its solution family with one-parameter

Key words di ff erence Riccati equation;solution family;order of growth

1 Introduction and Results

In this article,we assume that the reader is familiar with basic notions of Nevanlinna’s value distribution theory(see[9,12]).In addition,we use the notation σ(f)to denote the order of growth of a meromorphic function f.

30 years ago,Yanagihara studied meromorphic solutions of nonlinear di ff erence equations,and obtained the following di ff erence analogue of Malmquist’s theorem.

Theorem A(see[15]) If the fi rst order di ff erence equation

where R(z,w)is rational in both arguments,admits a nonrational meromorphic solution of fi nite order,then degw(R)=1.

Recently,a number of articles(including[1,3–8,10,11,13,17,18])focus on complex di ff erence equations and di ff erences analogues of Nevanlinna’s theory.

Equation(1.1)with degw(R)=1 is called the di ff erence Riccati equation

Halburd and Korhonen[6]showed that(1.2)possesses a continuum limit to the di ff erential Riccati equation.If γ (z)6≡ 0,then equation(1.2)is linearized by the transformation

where u satis fi es the second order linear di ff erence equation

Halburd and Korhonen[8]used value distribution theory to single out the di ff erence Painlev′e II equation from a large class of di ff erence equations of the form

where A is a polynomial,δ=±1(see[8,p.197]).

From this,we see that the di ff erence Riccati equation is an important class of di ff erence equations,it will play an important role for research of di ff erence Painlev′e equations.

The main goals of this article are to consider existence of meromorphic solutions of di ff erence Riccati equations.Bank,Gundersen and Laine[2]considered existence of meromorphic solutions of Riccati di ff erential equation

where A is a nonconstant meromorphic function,and obtained the following results.

Theorem B(see[2]) If the Riccati di ff erential equation(1.4)with A(z)meromorphic possesses at least three distinct meromorphic solutions u1(z),u2(z),u3(z),then equation(1.4)possesses a one-parameter family{uc:c∈C}of distinct meromorphic solutions with the property that any meromorphic solution u(z)6≡u(píng)1(z)of(1.4)satis fi es u(z)=uc(z)for some c∈C.

Theorem C(see[2]) If A(z)is a transcendental entire function of fi nite order,then(1.4)admits at most two distinct meromorphic solutions of fi nite order.

Theorem D(see[11]) Suppose that A(z)is a meromorphic function,that di ff erence equation

possesses three distinct meromorphic solutions f1(z),f2(z),f3(z).Then any meromorphic solution of(1.5)can be represented by

where Q(z)is a periodic function of period 1.Conversely,if for any periodic function Q(z)of period 1,we de fi ne a function f(z)by(1.6),then f(z)is a meromorphic solution of(1.5).

From Theorem D,it is natural to ask

(1)Does a solution family exist for a more general di ff erence Riccati equation than(1.5)?

(2)In the solution family(1.6),the parameter function Q(z)appears four times,then,can the number of times be reduced?

In this aeticle,we answer these questions and use the totally distinct method applied in the proof of Theorem D,to obtain the following results.

Theorem 1.1Let A,B,C,D be meromorphic functions,AC 6≡0 and AD?BC 6≡0.If a di ff erence Riccati equation

possesses at least three distinct meromorphic solutions f0(z),f1(z),f2(z),then all solutions of(1.7)constitute a one-parameter family

where Q(z)is any constant in C or any periodic meromorphic function with period 1,as Q(z)≡0,f(z)=f1(z);as Q(z)≡ ?1,f(z)=f2(z).

Remark 1.1(1) Comparing Theorems 1.1 and D,Theorem 1.1 suits the most general di ff erence Riccati equation.But Theorem D only suits the special di ff erence Riccati equation.

(2)The parameter function Q(z)only appears one time in(1.8),thus,we can obtain more information on solutions of(1.7)from(1.8).For example,if fj(j=0,1,2)are rational solutions,then σ(f)= σ(Q)by(1.8).But we do not obtain the corresponding result by Theorem D and(1.6).

(3)From Theorem 1.1,we can prove the following Corollaries 1.1 and 1.2.But from Theorem D,we can not obtain the corresponding results of Corollaries 1.1 and 1.2 since the parameter function Q(z)appears four times in(1.6).

Corollary 1.1Under conditions of Theorem 1.1,if fj(j=0,1,2)are rational solutions,then

(i)equation(1.7)has in fi nitely many rational solutions;

(ii)for any σ ≥ 1,(1.7)has a meromorphic solution f(z)satisfying σ(f)= σ.

Corollary 1.2Under conditions of Theorem 1.1,for any σ (>max{σ(fj):j=0,1,2}),equation(1.7)has a meromorphic solution f(z)satisfying σ(f)= σ.

Theorem C and Corollaries 1.1 and 1.2 show that the growth of meromorphic solutions of di ff erence Riccati equation(1.7)is quite di ff erent from that of Riccati di ff erential equation(1.4).

Theorem 1.2Let a(z),b(z),c(z),d(z)be polynomials,ac 6≡ 0,ad?bc 6≡ 0 and degc>max{dega,degd}.If the di ff erence Riccati equation

has a rational solution f0(z)satisfying f0(z)6→ 0(z → ∞),then(1.9)has at most two distinct rational solutions.

Corollary 1.3Let a(z),b(z),c(z),d(z)be polynomials,ac 6≡ 0 and ad ? bc 6≡ 0.Then the di ff erence Riccati equation(1.9)either has at most two distinct rational solutions,or has in fi nitely many rational solutions.

Example 1.1The di ff erence Riccati equation

possesses three distinct meromorphic solutions,and all solutions of(1.10)constitute a one-parameter family

where Q(z)is any constant in C or any periodic meromorphic function with period 1,as Q(z)≡0,f(z)=f1(z);as Q(z)≡?1,f(z)=f2(z).Clearly,(1.10)has in fi nitely many rational solutions.

Example 1.2The di ff erence Riccati equation

2 Proofs of Theorem 1.1,Corollaries 1.1 and 1.2

We need the following lemmas for proof of Theorem 1.1.

Lemma 2.1Suppose that A,B,C,D are meromorphic functions,AC 6≡0 and AD?BC 6≡0.If f(z)is a meromorphic solution of di ff erence Riccati equation(1.7),then

ProofIf C(z)f(z+1)?A(z)≡0,then f(z+1)=.Substituting it into(1.7),we obtain

Then C(z+1)≡0,which contradicts our condition. ?

Lemma 2.2Suppose that A0and A1are nonzero meromorphic functions.If the equation

has a nonzero meromorphic solution y0(z),then all solutions of(2.1)constitute a one-parameter family

where Q(z)is any constant in C or any periodic meromorphic function with period 1.

ProofSuppose that y0(z)is a nonzero meromorphic solution of(2.1).Clearly,y(z)=Q(z)y0(z)is a meromorphic solution of(2.1)for any constant Q(z)in C or any periodic meromorphic function Q(z)with period 1.

If y1(z)is a meromorphic solution of(2.1),then by(2.1),we obtain

Hence,any meromorphic solution f(z)(6≡f0(z))of(1.7)is of form(2.5).

In what follows,we prove that for any constant in C or any periodic meromorphic function with period 1,Q(z),the function f(z)denoted by(2.5)must be solution of(1.7).To this aim,we only need to prove that for every solution of(2.2)

3 Proofs of Theorem 1.2 and Corollary 1.3

We also have that

and c(z)h(z+1)q(z+1)H(z)Q(z)and c(z)h(z)q(z)H(z+1)Q(z+1)have the same leading coefficient.Thus,the left side of(3.3)is a polynomial with degree=deg(c(z)h(z+1)q(z+1)H(z)Q(z))>0.It is a contradiction.

Hence,(1.9)has at most two rational solutions.

Thus,Theorem 1.2 is proved.?

Proof of Corollary 1.3Combining Theorem 1.2 with Corollary 1.1(i),we can obtain Corollary 1.3. ?