Meromorphic Solutions of Some Complex Difference Equations
© Z.-B. Huang and Z.-X. Chen. 2009
Received: 27 January 2009
Accepted: 28 May 2009
Published: 29 June 2009
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 .
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 , 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 , Heittokangas et al. extended and improved the above result to higher-order difference equations of more general type. However, by inspecting the proofs in , 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.
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 , 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.
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 , 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.
where is a small meromorphic function relative to .
3. The Proofs of Theorems
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].
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 .
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.
This shows that the equality is arrived in Theorem 2.1 if
This shows that the case may occur in Theorem 2.1 if
Lemma 3.9 (see ).
Lemma 3.10 (see ).
Proof of Theorem 2.3.
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.
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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