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# Meromorphic Solutions of Some Complex Difference Equations

*Advances in Difference Equations*
**volume 2009**, Article number: 982681 (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 .

## 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.

If a complex difference equation

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].

In this paper, we mention the above details, used in [4, 10, 15], with 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 .

## 2. Main Results

In [10], Heittokangas et al. considered the complex difference equations of the form

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.

Suppose that are distinct, nonzero complex numbers and that is a transcendental meromorphic solution of

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.

Suppose that are distinct, nonzero complex numbers and that is a transcendental meromorphic solution of

where the coefficients 's are nonvanishing small functions relative to and where and are relatively prime polynomials in over the field of small functions relative to . Moreover, one assumes that ,

and that, without restricting generality, is a monic polynomial. If there exists such that for all sufficiently large,

where then either the order , or

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.

Suppose that are distinct, nonzero complex numbers and that is a transcendent meromorphic solution of

where the coefficients 's are nonvanishing small functions relative to and where are relatively prime polynomials in over the field of small functions relative to . Moreover, one assumes that ,

and that, without restricting generality, is a monic polynomial. If there exists such that for all sufficiently large,

where . Then either the order or

where is a small meromorphic function relative to .

## 3. The Proofs of Theorems

Let be a meromorphic function. Then for all irreducible rational functions in ,

with meromorphic coefficients and , the characteristic function of satisfies

where and

In the particular case when

we have

Lemma 3.2.

Given distinct complex numbers a meromorphic function and meromorphic functions 's, one has

where . In the particular case when

one has

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]).

Let be a nonconstant meromorphic function of finite order, and . Then

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

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

Let be a nondecreasing continuous function, and let be the set of all such that

If the logarithmic measure of is infinite, that is, then

Proof of Theorem 2.1.

Since the coefficients 's, and in (2.2) are small functions relative to , that is,

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

Let be a finite order meromorphic solution of (2.2). According to Lemma 3.5, we have, for any ,

where the exceptional set associated to is of finite logarithmic measure

It follows from Lemma 3.6 that

for any .

Now, equating the Nevanlinna characteristic function on both sides of (2.2), and applying Lemmas 3.1 and 3.2, we have

where .

Therefore, by (3.13) and (3.14), it follows that

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.

Let be a constant such that where , and let . We see that solves

This shows that the equality is arrived in Theorem 2.1 if

Example 3.8.

Let . We see that solves

This shows that the case may occur in Theorem 2.1 if

Lemma 3.9 (see [17]).

Let be a meromorphic function and let be given by

Then either

or

Lemma 3.10 (see [15]).

Let be a nonconstant meromorphic function and let , be two polynomials in with meromorphic coefficients small functions relative to . If and have no common factors of positive degree in over the field of small functions relative to , then

Proof of Theorem 2.3.

Suppose that the second alternative of the conclusion is not correct. Then we have, by using Lemmas 3.9, 3.10, 3.2, (2.7), and (2.9),

where

Thus, we have

Now assuming that , we have and for all

It follows from Lemmas 3.1, 3.2, (3.23), and (2.9) we have

From this, we have

Together with (3.25)–(3.27), we can use method of induction and obtain, for ,

Moreover, we immediately obtain from (3.28) that

and for sufficiently large , we have

It also follows from Lemma 3.6 that

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.

## References

- 1.
Gao S-A, Chen Z-X, Chen T-W:

*Oscillation Theory of Linear Differential Equation*. University of Science and Technology Press, Huazhong, China; 1998. - 2.
Hayman WK:

*Meromorphic Functions, Oxford Mathematical Monographs*. Clarendon Press, Oxford, UK; 1964:xiv+191. - 3.
Laine I:

*Nevanlinna Theory and Complex Differential Equations, Studies in Mathematics*.*Volume 15*. de Gruyter, Berlin, Germany; 1993:viii+341. - 4.
Ablowitz MJ, Halburd R, Herbst B:

**On the extension of the Painlevé property to difference equations.***Nonlinearity*2000,**13**(3):889-905. 10.1088/0951-7715/13/3/321 - 5.
Bergweiler W, Langley JK:

**Zeros of differences of meromorphic functions.***Mathematical Proceedings of the Cambridge Philosophical Society*2007,**142**(1):133-147. 10.1017/S0305004106009777 - 6.
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.2008.02.048 - 7.
Chen Z-X, Huang Z-B, Zheng X-M:

**A note "Value distribution of difference polynomials".**to appear in*Acta Mathematica Sinica* - 8.
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/s11139-007-9101-1 - 9.
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-z - 10.
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. - 11.
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.2005.04.010 - 12.
Halburd RG, Korhonen RJ:

**Nevanlinna theory for the difference operator.***Annales Academiæ Scientiarium Fennicæ. Mathematica*2006,**31**(2):463-478. - 13.
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.2006.05.005 - 14.
Ishizaki K, Yanagihara N:

**Wiman-Valiron method for difference equations.***Nagoya Mathematical Journal*2004,**175:**75-102. - 15.
Laine I, Rieppo J, Silvennoinen H:

**Remarks on complex difference equations.***Computational Methods and Function Theory*2005,**5**(1):77-88. - 16.
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. - 17.
Weissenborn G:

**On the theorem of Tumura and Clunie.***The Bulletin of the London Mathematical Society*1986,**18**(4):371-373. 10.1112/blms/18.4.371

## 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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Huang, ZB., Chen, ZX. Meromorphic Solutions of Some Complex Difference Equations.
*Adv Differ Equ* **2009, **982681 (2009). https://doi.org/10.1155/2009/982681

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### Keywords

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