Meromorphic Solutions of Some Complex Difference Equations

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doi:10.1155/2009/982681

Research Article

Meromorphic Solutions of Some Complex

Difference Equations

Zhi-Bo Huang and Zong-Xuan Chen

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

Correspondence should be addressed to Zong-Xuan Chen,[email protected]

Received 27 January 2009; Accepted 28 May 2009

Recommended by Binggen Zhang

The main purpose of this paper is to present the properties of the meromorphic solutions of complex diﬀerence equations of the form_{_J_}αJzj∈Jfzcj Rz, fz, where{J}is

a collection of all subsets of{1,2, . . . , n},cj j ∈Jare distinct, nonzero complex numbers,fzis

a transcendental meromorphic function,αJz’s are small functions relative tofz, andRz, fz

is a rational function infzwith coeﬃcients which are small functions relative tofz.

Copyrightq2009 Z.-B. Huang and Z.-X. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

We assume that the readers are familiar with the basic notations of Nevanlinna’s value distribution theory; see1–3 .

Recent interest in the problem of integrability of difference equations is a consequence of the enormous activity on Painlevé differential equations and their discrete counterparts during the last decades. Many people study this topic and obtain some results; see4–15. In

4 , Ablowitz et al. obtained a typical result as follows.

Theorem A. If a complex diﬀerence equation

fz1 fz−1 Rz, fz a0z a1zfz · · ·apzfz p

b0z b1zfz · · ·bqzfzq, 1.1

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In10 , Heittokangas et al. extended and improved the above result to higher-order diﬀerence equations of more general type. However, by inspecting the proofs in 4 , we can find a more general class of complex diﬀerence equations by making use of a similar technique; see10,15 .

In this paper, we mention the above details, used in4,10,15 , with equations of the form numbers,fzis a transcendental meromorphic function,αJz’s are small functions relative tofzandRz, fzis a rational function infzwith coeﬃcients which are small functions relative tofz.

2. Main Results

In10 , Heittokangas et al. considered the complex diﬀerence equations of the form

n

It is obvious that the left-hand side of2.1is just a product only. If we consider the left-hand side of2.1is a product sum, we also have the following theorem.

Theorem 2.1. Suppose thatc1, c2, . . . , cnare distinct, nonzero complex numbers and thatfzis a

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Corollary 2.2. Suppose thatc1, c2, . . . , cnare distinct, nonzero complex numbers and thatfzis a

transcendental meromorphic solution of 2.2with rational coeﬃcientsαJz’s,aiz i0,1, . . . , p

andbjz j0,1, . . . , q. Ifdmax{p, q}> n, then the orderρfis infinite.

In15 , when the left-hand side of2.1is just a sum, Laine et al. obtained the following theorem.

Theorem C. Suppose thatc1, c2, . . . , cn are distinct, nonzero complex numbers and thatfzis a

transcendental meromorphic solution of

n

j1

αjzfzcjRz, fz P

z, fz

Qz, fz, 2.3

where the coeﬃcientsαjz’s are nonvanishing small functions relative tofzand wherePz, fz

andQz, fzare relatively prime polynomials infzover the field of small functions relative to

fz. Moreover, one assumes thatqdeg_fQz, fz>0,

nmaxp, qmaxdeg_fPz, fz,deg_fQz, fz, 2.4

and that, without restricting generality,Qz, fzis a monic polynomial. If there existsα∈0, n

such that for allrsuﬃciently large,

N

⎛ ⎝_r,n

j1

αjzfzcj

⎞

⎠_≤_αN_r_{C, f}_z_S_{r, f}_, 2.5

whereCmax1≤j≤n{|cj|},then either the orderρf ∞, or

Qz, fz≡fz hzq, 2.6

wherehzis a small meromorphic function relatively tofz.

They obtained Theorem C and presented a problem that whether the result will be correct if we replace the left-hand side of2.3by a product sum as inTheorem 2.1. Here, under the new hypothesis, we consider the left-hand side of2.3is a product sum and obtain what follows.

Theorem 2.3. Suppose thatc1, c2, . . . , cnare distinct, nonzero complex numbers and thatfzis a

transcendent meromorphic solution of

{J} αJz

⎛ ⎝

j∈J

fzcj

⎞

⎠_R_{z, f}_z P

z, fz

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where the coeﬃcientsαJz’s are nonvanishing small functions relative tofzand wherePz, fz,

Qz, fzare relatively prime polynomials infzover the field of small functions relative tofz. Moreover, one assumes thatqdeg_fQz, fz>0,

nmaxp, qmaxdeg_fPz, fz,deg_fQz, fz, 2.8

and that, without restricting generality,Qz, fzis a monic polynomial. If there existsα∈0, n

such that for allrsuﬃciently large,

n

j1

Nr, fzcj≤αNrC, fzSr, f, 2.9

whereCmax{|c1|,|c2|, . . . ,|cn|}. Then either the orderρf ∞,or

Qz, fz≡fz hzq, 2.10

wherehzis a small meromorphic function relative tofz.

3. The Proofs of Theorems

Lemma 3.1 see 3, 9 . Let fz be a meromorphic function. Then for all irreducible rational functions infz,

Rz, fz a0z a1zfz · · ·apzfz p

b0z b1zfz · · ·bqzfzq, 3.1

with meromorphic coeﬃcientsaiz i 0,1, . . . , pand bjz j 0,1, . . . , q, the characteristic function ofRz, fzsatisfies

Tr, Rz, fzdTr, fOΨr, 3.2

wheredmax{p, q}and

Ψr max i,j

Tr, ai, Tr, bj. 3.3

In the particular case when

Tr, ai Sr, f, i0,1, . . . , p,

Tr, bjSr, f, j0,1, . . . , q,

3.4

we have

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Lemma 3.2. Given distinct complex numbers c1, c2, . . . , cn, a meromorphic function fz and

meromorphic functionsαJz’s, one has

T inspection of the proof of16, Proposition B.15, Theorem B.16 .

Remark 3.4. Note that the inequality 3.6 remains true, if we replace the characteristic functionT by the proximity functionmor by the counting functionN.

Lemma 3.5 see12, Theorem 2.1 . Let fz be a nonconstant meromorphic function of finite order,c∈C,and0< δ <1. Then

for allroutside of a possible exceptional setEwith finite logarithmic measure_Edr/r <∞.

Lemma 3.6see12, Lemma 2.2 . LetT : 0,∞ → 0,∞be a nondecreasing continuous function,s >0, 0< α <1,and letF⊂Rbe the set of allrsuch that

Tr≤αTrs. 3.10

If the logarithmic measure ofFis infinite, that is,_Fdr/r ∞,then

lim r→ ∞

logTr

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Proof ofTheorem 2.1. Since the coeﬃcients αJz’s, aiz i 0,1, . . . , p and bjz j

hold for all r outside of a possible exceptional set E1 with finite logarithmic measure

where the exceptional setE2 associated toSr, fis of finite logarithmic measure

E2dr/r < ∞.

It follows fromLemma 3.6that

Nrs, fNr, fSr, f, 3.14

for anys >0.

Now, equating the Nevanlinna characteristic function on both sides of 2.2, and applying Lemmas3.1and3.2, we have

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Therefore, by3.13and3.14, it follows that

dTr, f≤nNr, fnmr, fSr, fSr, f

nTr, fSr, fSr, f,

3.16

for allroutside of a possible exceptional setE1∪E2with finite logarithmic measure. Dividing

this byTr, fand lettingr → ∞outside of the exceptional setE1 andE2 ofSr, fand

Sr, f, respectively, we haved≤n.The proof ofTheorem 2.1is completed.

Example 3.7. Letc∈Cbe a constant such that_{c /} π/2m,wherem∈Z, and letAtanc, B tanc/2. We see thatfz tanzsolves

fzc

2

fzc fz−c

2

fz−c

2ABfz

4₂₁ _A_B2_A2_B2_f_z2₂_AB

A2_B2_f_z4₋_A2_B2_f_z2_AB .

3.17

This shows that the equalitydn4 is arrived inTheorem 2.1ifρf 1<∞.

Example 3.8. Letμe−1/e, νe1/e. We see thatfz zez_solves

fz−1fz2−fz1fz−2

μfz2μν−3z−ν22ν2fz−μν−2zν2−2ν. 3.18

This shows that the cased2< n4 may occur inTheorem 2.1ifρf 1<∞.

Lemma 3.9see17 . Letfzbe a meromorphic function and letφbe given by

φfn_an

−1fn−1· · ·a0,

Tr, ajSr, f, j0,1, . . . , n−1. 3.19

Then either

φ≡

fan−1

n

, 3.20

or

Tr, f≤N

r, 1 φ

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Lemma 3.10 see 15 . Let fz be a nonconstant meromorphic function and let Pz, fz,

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It follows from Lemmas3.1,3.2,3.23, and2.9we have

Moreover, we immediately obtain from3.28that

Nr2mC, f≥ nm

It also follows fromLemma 3.6that

Nrs, fNr, fSr, f, 3.31

for anys >0, assuming thatfzis of finite order.

Now 3.31 combined with 3.29 and 3.30 yields an immediate contradiction if ρf < ∞. Therefore the only possibility is that fz is of infinite order. The proof of

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Acknowledgments

The authors are very grateful to the referee for his her many valuable comments and suggestions which greatly improved the presentation of this paper. The project was supposed by the National Natural Science Foundation of China no. 10871076, and also partly supposed by the School of Mathematical Sciences Foundation of SCNU, China.

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