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Some properties of algebraic difference equations of first order
- Yong Liu^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1395-8
© The Author(s) 2017
Received: 11 June 2017
Accepted: 10 October 2017
Published: 17 October 2017
Abstract
Keywords
- meromorphic functions
- difference equations
- value distribution
- finite order
MSC
- 30D35
- 39B12
1 Introduction
We assume that the reader is familiar with the basic notions of Nevanlinna theory (see, e.g., [4, 5]). Of late, several scholars [3, 6–14] studied the properties of finite-order meromorphic solutions of algebraic difference equations and obtained many interesting results.
In 2014, Liu [12] considered the Nevanlinna growth of an equation related to (1.2). It is interesting to consider some properties of (1.2), and our results will be stated in Section 2.
2 Main results
Theorem 2.1
Remark
It is a curious problem to construct a transcendental meromorphic solution of (2.1) for the case \(\deg A>0\).
Theorem 2.2
- (i)If \(d\geq f\) and \(d-f\) is zero or an even number, then$$h-k=\frac{d-f}{2}. $$
- (ii)
If \(d< f\), then \(h-k=\frac{d-f}{2}\).
Further, Example 2.3 shows that there exist rational solutions satisfying Theorem 2.2(i), and Example 2.4 shows that there exist rational solutions satisfying Theorem 2.2(ii).
Example 2.3
Example 2.4
3 Proof of Theorem 2.1
Lemma 3.1
([11])
The following result obtained by Chiang and Feng [16] and Halburd and Korhonen [9, 17] independently. We state here the form stated in [16, Theorem 8.2(b)].
Lemma 3.2
([16])
Firstly, we prove that \(\rho(g)=\rho\geq1\). We consider the following two cases separately.
4 Proof of Theorem 2.2
Declarations
Acknowledgements
The author would like to thank the referee for his/her helpful suggestions and comments. The work was supported by the NNSF of China (No. 10771121, 11401387), the NSF of Zhejiang Province, China (No. LQ 14A010007), the NSFC Tianyuan Mathematics Youth Fund (No. 11226094), the NSF of Shandong Province, China (No. ZR2012AQ020 and No. ZR2010AM030), and the Fund of Doctoral Program Research of Shaoxing College of Art and Science (20135018).
Author’s contributions
Author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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Authors’ Affiliations
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