- Research Article
- Open Access

# Meromorphic Solutions of Some Complex Difference Equations

- Zhi-Bo Huang
^{1}and - Zong-Xuan Chen
^{1}Email author

**2009**:982681

https://doi.org/10.1155/2009/982681

© Z.-B. Huang and Z.-X. Chen. 2009

**Received:**27 January 2009**Accepted:**28 May 2009**Published:**29 June 2009

## Abstract

The main purpose of this paper is to present the properties of the meromorphic solutions of complex difference equations of the form , where is a collection of all subsets of , are distinct, nonzero complex numbers, is a transcendental meromorphic function, 's are small functions relative to , and is a rational function in with coefficients which are small functions relative to .

## Keywords

- Meromorphic Function
- Finite Order
- Monic Polynomial
- Meromorphic Solution
- Small Function

## 1. Introduction

We assume that the readers are familiar with the basic notations of Nevanlinna's value distribution theory; see [1–3].

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; see [4–15]. In [4], Ablowitz et al. obtained a typical result as follows.

Theorem 1 A.

with rational coefficients and admits a transcendental meromorphic solution of finite order, then

In [10], 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; see [10, 15].

where is a collection of all subsets of , are distinct, nonzero complex numbers, is a transcendental meromorphic function, 's are small functions relative to and is a rational function in with coefficients which are small functions relative to .

## 2. Main Results

with rational coefficients and . They obtained the following theorem.

Theorem 2 B.

Let . If the difference equation (2.1) with rational coefficients and admits a transcendental meromorphic solution of finite order , then , where .

It is obvious that the left-hand side of (2.1) is just a product only. If we consider the left-hand side of (2.1) is a product sum, we also have the following theorem.

Theorem 2.1.

with coefficients 's, and are small functions relative to . If the order is finite, then , where .

It seems that the equivalent proposition is a known fact. In [15], Laine et al. obtain the similar result to the following Corollary 2.2. Here, for the convenience for the readers, we list it, that is, we have the following corollary.

Corollary 2.2.

Suppose that are distinct, nonzero complex numbers and that is a transcendental meromorphic solution of (2.2) with rational coefficients 's, and . If , then the order is infinite.

In [15], when the left-hand side of (2.1) is just a sum, Laine et al. obtained the following theorem.

Theorem 2 C.

where is a small meromorphic function relatively to .

They obtained Theorem C and presented a problem that whether the result will be correct if we replace the left-hand side of (2.3) by a product sum as in Theorem 2.1. Here, under the new hypothesis, we consider the left-hand side of (2.3) is a product sum and obtain what follows.

Theorem 2.3.

where is a small meromorphic function relative to .

## 3. The Proofs of Theorems

Lemma 3.2.

Remark 3.3.

Observe that the term does not appear in (3.6). This follows by a careful inspection of the proof of [16, Proposition B.15, Theorem B.16].

Remark 3.4.

Note that the inequality (3.6) remains true, if we replace the characteristic function by the proximity function (or by the counting function ).

Lemma 3.5 (see [12, Theorem 2.1]).

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

Lemma 3.6 (see [12, Lemma 2.2]).

Proof of Theorem 2.1.

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

where the exceptional set associated to is of finite logarithmic measure

for any .

where .

for all outside of a possible exceptional set with finite logarithmic measure. Dividing this by and letting outside of the exceptional set and of and , respectively, we have The proof of Theorem 2.1 is completed.

Example 3.7.

This shows that the equality is arrived in Theorem 2.1 if

Example 3.8.

This shows that the case may occur in Theorem 2.1 if

Lemma 3.9 (see [17]).

Lemma 3.10 (see [15]).

Proof of Theorem 2.3.

where

for any , assuming that is of finite order.

Now (3.31) combined with (3.29) and (3.30) yields an immediate contradiction if . Therefore the only possibility is that is of infinite order. The proof of Theorem 2.3 is completed.

## Declarations

### 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.

## Authors’ Affiliations

## References

- Gao S-A, Chen Z-X, Chen T-W:
*Oscillation Theory of Linear Differential Equation*. University of Science and Technology Press, Huazhong, China; 1998.Google Scholar - Hayman WK:
*Meromorphic Functions, Oxford Mathematical Monographs*. Clarendon Press, Oxford, UK; 1964:xiv+191.Google Scholar - Laine I:
*Nevanlinna Theory and Complex Differential Equations, Studies in Mathematics*.*Volume 15*. de Gruyter, Berlin, Germany; 1993:viii+341.View ArticleGoogle Scholar - Ablowitz MJ, Halburd R, Herbst B:
**On the extension of the Painlevé property to difference equations.***Nonlinearity*2000,**13**(3):889-905. 10.1088/0951-7715/13/3/321MathSciNetView ArticleMATHGoogle Scholar - Bergweiler W, Langley JK:
**Zeros of differences of meromorphic functions.***Mathematical Proceedings of the Cambridge Philosophical Society*2007,**142**(1):133-147. 10.1017/S0305004106009777MathSciNetView ArticleMATHGoogle Scholar - Chen Z-X, Shon KH:
**On zeros and fixed points of differences of meromorphic functions.***Journal of Mathematical Analysis and Applications*2008,**344**(1):373-383. 10.1016/j.jmaa.2008.02.048MathSciNetView ArticleMATHGoogle Scholar - Chen Z-X, Huang Z-B, Zheng X-M: A note "Value distribution of difference polynomials". to appear in Acta Mathematica SinicaGoogle Scholar
- Chiang Y-M, Feng S-J:
**On the Nevanlinna characteristic of**and difference equations in the complex plane.*Ramanujan Journal*2008,**16**(1):105-129. 10.1007/s11139-007-9101-1MathSciNetView ArticleMATHGoogle Scholar - Gundersen GG, Heittokangas J, Laine I, Rieppo J, Yang D:
**Meromorphic solutions of generalized Schröder.***Aequationes Mathematicae*2002,**63**(1-2):110-135. 10.1007/s00010-002-8010-zMathSciNetView ArticleMATHGoogle Scholar - Heittokangas J, Korhonen R, Laine I, Rieppo J, Tohge K:
**Complex difference equations of Malmquist type.***Computational Methods and Function Theory*2001,**1**(1):27-39.MathSciNetView ArticleMATHGoogle Scholar - Halburd RG, Korhonen RJ:
**Difference analogue of the lemma on the logarithmic derivative with applications to difference equations.***Journal of Mathematical Analysis and Applications*2006,**314**(2):477-487. 10.1016/j.jmaa.2005.04.010MathSciNetView ArticleMATHGoogle Scholar - Halburd RG, Korhonen RJ:
**Nevanlinna theory for the difference operator.***Annales Academiæ Scientiarium Fennicæ. Mathematica*2006,**31**(2):463-478.MathSciNetMATHGoogle Scholar - Halburd RG, Korhonen RJ:
**Existence of finite-order meromorphic solutions as a detector of integrability in difference equations.***Physica D*2006,**218**(2):191-203. 10.1016/j.physd.2006.05.005MathSciNetView ArticleMATHGoogle Scholar - Ishizaki K, Yanagihara N:
**Wiman-Valiron method for difference equations.***Nagoya Mathematical Journal*2004,**175:**75-102.MathSciNetMATHGoogle Scholar - Laine I, Rieppo J, Silvennoinen H:
**Remarks on complex difference equations.***Computational Methods and Function Theory*2005,**5**(1):77-88.MathSciNetView ArticleMATHGoogle Scholar - Gromak V, Laine I, Shimomura S:
*Painlevé Differential Equations in the Complex Plane, Studies in Mathematics*.*Volume 28*. de Gruyter, New York, NY, USA; 2002.View ArticleGoogle Scholar - Weissenborn G:
**On the theorem of Tumura and Clunie.***The Bulletin of the London Mathematical Society*1986,**18**(4):371-373. 10.1112/blms/18.4.371MathSciNetView ArticleMATHGoogle Scholar

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