Meromorphic Solutions of Some Complex Difference Equations

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

(1.1)

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

(1.2)

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

(2.1)

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

(2.2)

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

(2.3)

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 ,

(2.4)

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

(2.5)

where then either the order , or

(2.6)

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

(2.7)

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 ,

(2.8)

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

(2.9)

where . Then either the order or

(2.10)

where is a small meromorphic function relative to .

3. The Proofs of Theorems

Lemma 3.1 (see [3, 9]).

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

(3.1)

with meromorphic coefficients and , the characteristic function of satisfies

(3.2)

where and

(3.3)

In the particular case when

(3.4)

we have

(3.5)

Lemma 3.2.

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

(3.6)

where . In the particular case when

(3.7)

one has

(3.8)

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

(3.9)

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

(3.10)

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

(3.11)

Proof of Theorem 2.1.

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

(3.12)

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 ,

(3.13)

where the exceptional set associated to is of finite logarithmic measure

It follows from Lemma 3.6 that

(3.14)

for any .

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

(3.15)

where .

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

(3.16)

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

(3.17)

This shows that the equality is arrived in Theorem 2.1 if

Example 3.8.

Let . We see that solves

(3.18)

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

(3.19)

Then either

(3.20)

or

(3.21)

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

(3.22)

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),

(3.23)

where

Thus, we have

(3.24)

Now assuming that , we have and for all

(3.25)

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

(3.26)

From this, we have

(3.27)

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

(3.28)

Moreover, we immediately obtain from (3.28) that

(3.29)

and for sufficiently large , we have

(3.30)

It also follows from Lemma 3.6 that

(3.31)

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.