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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.
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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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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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DOI: https://doi.org/10.1155/2009/982681
Keywords
- Meromorphic Function
- Finite Order
- Monic Polynomial
- Meromorphic Solution
- Small Function