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Existence and properties of meromorphic solutions of some q-difference equations
- Na Xu^{1}Email author and
- Chun-Ping Zhong^{1}
https://doi.org/10.1186/s13662-014-0346-x
© Xu and Zhong; licensee Springer 2015
Received: 14 October 2014
Accepted: 25 December 2014
Published: 30 January 2015
Abstract
In this paper, we investigate the existence and growth of solutions of the q-difference equation \(\prod_{i=1}^{n}f(q_{i}z)=R(z,f(z))\), where \(R(z,f(z))\) is an irreducible rational function in \(f(z)\). We also give an estimation of the growth of transcendental meromorphic solutions of the equation \(\prod_{i=1}^{n}f(q_{i}z)=f(z)^{m}\).
Keywords
- q-difference equation
- meromorphic solution
- growth
MSC
- 30D35
- 39A05
1 Introduction and main results
In the last two decades, the existence and growth of meromorphic solutions of difference equations have been investigated in many papers [3–9]. Recently, with the development of the q-difference analog of Nevanlinna theory, there has been a renewed interest in studying meromorphic solutions of q-difference equations. For instance, Zheng and Chen [8] considered the growth problem of transcendental meromorphic solutions of some q-difference equations.
Theorem A
[8, Theorem 2]
In [10], Heittokangas et al. first considered meromorphic solutions with Borel exceptional zeros and poles of some type of difference equations and obtained the result as follows.
Theorem B
[10, Theorem 13]
Recently, Zheng and Chen [9] considered a q-difference equation under a condition similar to Theorem B on meromorphic solution, and they obtained the following result.
Theorem C
[9, Theorem 1]
For the q-difference equation (1.1), Theorem C only considered the case when solutions have Borel exceptional zeros and poles. But how about the existence and growth of meromorphic solutions of (1.1)? Theorem 1.1 considers under what conditions (1.1) will not have solutions with zero order.
Theorem 1.1
As many papers (see [8, 11]) obtained the lower bound of the order of solutions of difference equations. The natural question arises of the upper bound of the order of the solutions of (1.1). The following theorem answers this question partly.
Theorem 1.2
Corollary 1.3
The following theorem gives an estimation of the growth of the meromorphic solutions of (1.2), where \(q_{i}\in\mathbb{C}\), \(|q_{i}|>1 \) (\(i=1,\ldots,n\)).
Theorem 1.4
Remark
In the following, we consider transcendental entire solutions with \(\lambda(f)<\sigma(f)\) of (1.2).
Theorem 1.5
The following example shows that the case of Theorem 1.5 can occur.
Example 1
2 Lemmas
To prove our results, we need some lemmas.
Lemma 2.1
[12] (Valiron-Mohon‘ko)
Lemma 2.2 gives us the relationship of the characteristic function between \(f(z)\) and \(f(qz)\), provided that \(f(z)\) is a non-constant meromorphic function of zero order.
Lemma 2.2
[7]
Lemma 2.3
[3]
The next lemma is the relationship between \(T(r,f(qz))\) and \(T(|q|r,f(z))\).
Lemma 2.4
[4]
Lemma 2.5
[13]
The following are the well-known Weierstrass factorization theorem and Hadamard factorization theorem.
Lemma 2.6
[14]
3 The proofs
3.1 Proof of Theorem 1.1
Thus if \(d>n\), (1.1) has no transcendental meromorphic solution of zero order.
Using the same method of the proof of (1), we can get if \(d>n\), (1.1) has no transcendental entire solution of zero order.
Thus if \(d\neq n\), (1.1) has no transcendental entire solution of zero order.
3.2 Proof of Theorem 1.2
3.3 Proof of Theorem 1.4
We will divide the argument into two cases.
Case 1. If \(n\geq m\), it is easily to see \(\sigma(f)\geq \mu(f)\geq\frac{\log m-\log n}{\log|q|}\) is obviously true.
3.4 Proof of Theorem 1.5
Declarations
Acknowledgements
This research was partly supported by National Natural Science Foundation of China (11271304); the Natural Science Foundation of Fujian Province of China for Distinguished Young Scholars (2013J06001); and the Program for New Century Excellent Talents in University (NCET-13-0510).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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