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Results on meromorphic solutions of linear difference equations
Advances in Difference Equations volume 2012, Article number: 203 (2012)
In this paper, we investigate meromorphic solutions of linear difference equations and prove a number of results. We give estimates for the growth of meromorphic solutions under some special cases and provide some examples to show that the answer to a question of Laine and Yang is not always positive. The zeros, poles and fixed points of finite order solutions are also studied.
MSC:39A13, 39A22, 30D35.
1 Introduction and results
In this paper, a meromorphic function means meromorphic in the complex plane, and we assume the reader is familiar with the basic notions of Nevanlinna theory (see, e.g., [1–3]). We use the notations , , to denote the order of growth of f, the exponent of convergence of the poles of f and the exponent of convergence of the zeros of f, respectively, and we define them as follows:
Recently, numbers of papers (see, e.g., [6–16]) are devoted to considering the complex difference equations and difference analogues of Nevanlinna theory. For the growth of meromorphic solutions of difference equations, Chiang and Feng [8, 9] considered the polynomial coefficients case and got
Let be polynomials such that there exists an integer l, , such that
Suppose that is a meromorphic solution to
then we have .
The following result shows that the polynomial coefficients in Theorem A can be extended to rational functions.
Theorem 1.1 Let be rational functions having no common zeros or poles. For , set , where , are irreducible polynomials. If there exists an integer l, , such that
then, for any meromorphic solution to (1.1), we have .
Remark Set , then we see that are all polynomials and
And hence Theorem 1.1 follows from Theorem A. We omit the details of its proof.
Theorem B 
Let be entire functions such that there exists an integer l, , such that
If is a meromorphic solution to (1.1), then .
Theorem C 
Let be entire functions of finite order such that among those coefficients having the maximal order , exactly one has its type strictly greater than the others. Then, for any meromorphic solution to (1.1), we have .
In Theorems B and C, there is always some dominating coefficient such that . A natural question is what happens if the dominating coefficient is of order zero? Another question raised by Laine and Yang in  is whether all meromorphic solutions of (1.1) satisfy , even if there is no dominating coefficient. For the first question, we get the following result.
Theorem 1.2 Let be meromorphic functions such that there exists an integer l, , such that is a transcendental entire function, while , , are all rational functions. If is a meromorphic solution to (1.1), then .
Considering Laine and Yang’s question, we get the following example which indicates that the answer to their question is not always positive.
Example For a given positive integer k, is an entire solution of the equation
In this example, the relationship between and exactly depends on k.
However, the answer may be positive in some special case. In fact, we prove the following results, in which there is still some coefficient dominating in some angle.
Theorem 1.3 Let k be a positive integer, p be a nonzero real number and be a nonconstant meromorphic solution of the difference equation
where are all entire functions such that and . Then we have .
Theorem 1.4 Under exactly one of assumptions for the coefficients of (1.1) in Theorems A-C and Theorems 1.1 and 1.2, if is a finite order meromorphic solution to (1.1), then . What is more, either or , where .
Theorem 1.5 Under the assumption for the coefficients of (1.2) in Theorem 1.3, if is a finite order meromorphic solution to (1.2), then . What is more, either or .
The following examples show the sharpness of the estimates in Theorems 1.4 and 1.5.
Examples (1) The gamma function is a meromorphic solution to the equation
which satisfies the assumptions in Theorem A and Theorem 1.1. We see that and .
(2) and are entire solutions to the equation
which satisfies the assumptions in Theorems B and C and Theorem 1.2. We have , and , .
(3) and are entire solutions of the equation
which satisfies the assumptions in Theorem 1.3. We have , and and .
2 Proof of Theorem 1.2
We first recall a key lemma used to prove Theorems A and B and the pointwise estimates for difference quotient which are counterparts to Gundersen’s logarithmic derivative estimates  (see , Corollary 2.6, Theorem 8.3).
Lemma 2.1 
Let be a meromorphic function of finite order ρ, ε be a positive constant, and be two distinct nonzero complex constants. Then
and there exists a subset of finite logarithmic measure such that, for all z satisfying , and as sufficiently large,
Proof of Theorem 1.2 If , the assertion follows from Theorem B. We next consider the case that .
Assume that , then by Lemma 2.1 (or , Lemma 3.3), it is easy to deduce that for , there exists a subset of finite logarithmic measure such that, for all , , , and as ,
For , set , where , are irreducible polynomials. Denote . Since is a transcendental entire function, for sufficiently large r, we have
Now we choose a sequence , , such that , as . Combining (1.1), (2.1) and (2.2), we get
a contradiction. Our proof is thus finished. □
3 Proof of Theorem 1.3
Let be a meromorphic function with finite order ρ. Then, for any given , there exists a set of finite linear measure such that, for all z satisfying and r sufficiently large,
Proof of Theorem 1.3 Without loss of generality, we assume that . Suppose that (1.2) admits a nontrivial entire solution such that . Then by Lemma 2.1, for any given ε such that , we have
for all r outside of a possible exceptional set with finite logarithmic measure.
Applying Lemma 3.1, we have
for all r outside of a possible exceptional set with finite linear measure.
Choose an infinite positive real sequence such that as , and we get from (1.2), (3.1)-(3.3) that
a contradiction. And hence we have . □
4 Proofs of Theorems 1.4 and 1.5
Lemma 4.1 
Let (, ) be meromorphic functions and (, ) be entire functions such that
are not constant functions for .
(, ), where E is an exceptional set of finite linear measure, and .
Proofs of Theorems 1.4 and 1.5 In fact, we only give the proof of Theorem 1.5 since the proof of Theorem 1.4 is similar.
Suppose that is a nonconstant meromorphic solution of (1.2) such that . We firstly prove that . Submitting into (1.2), we get
Obviously, we have .
Now, for any given , applying Lemma 2.1, we can deduce that
This implies that
Next, we assert that either or . If the assertion does not hold, we have .
Assume that is a zero (or a pole) of of order m. Applying the Hadamard factorization of a meromorphic function, we write as follows:
where , are entire functions such that , and is a polynomial such that .
Now, we obtain from (1.2) that
for . Notice that for and . Thus, Lemma 4.1 is valid for (4.1) and hence we get that for , a contradiction to our assumption. This completes our proof. □
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The authors would like to thank the editor and the referees for their constructive comments to improve the readability of our paper. This work is supported by the National Natural Science Foundation of China (No. 11226091) and the Natural Science Research Projects of GDOU (No. 1212331).
The authors declare that they have no competing interests.
All authors drafted the manuscript, read and approved the final manuscript.