On properties of meromorphic solutions for difference Painlevé equations
- Chang-Wen Peng^{1} and
- Zong-Xuan Chen^{2}Email author
https://doi.org/10.1186/s13662-015-0463-1
© Peng and Chen; licensee Springer. 2015
Received: 8 October 2014
Accepted: 8 April 2015
Published: 22 April 2015
Abstract
In this paper, we mainly investigate properties of finite order transcendental meromorphic solutions of difference Painlevé equations. If f is a finite order transcendental meromorphic solution of difference Painlevé equations, then we get some estimates of the order and the exponent of convergence of poles of \(\Delta f(z)\), where \(\Delta f(z)=f(z+1)-f(z)\).
Keywords
difference Painlevé equation transcendental meromorphic solution fixed pointsMSC
30D35 39B121 Introduction and main results
Let f be a function transcendental and meromorphic in the plane. The forward difference is defined in the standard way by \(\Delta f(z)=f(z+1)-f(z)\). In what follows, we assume that the reader is familiar with the basic notions of Nevanlinna value distribution theory (see [1–3]). In addition, we use the notations \(\sigma(f)\) to denote the order of growth of the meromorphic function \(f(z)\), and \(\lambda(f)\) and \(\lambda(\frac{1}{f})\) to denote, respectively, the exponents of convergence of zeros and poles of \(f(z)\).
Some results on the existence of meromorphic solutions for certain difference equations were obtained by Shimomura [6] and Yanagihara [7] 30 years ago.
Recently, a number of papers (see [8–20]) focused on complex difference equations and difference analogues of Nevanlinna theory. As the difference analogues of Nevanlinna theory are being investigated, many results on the complex difference equations are rapidly obtained.
Theorem A
(see [8])
From above, we see that difference Painlevé I and II equations are the development of the differential and discrete Painlevé I and II equations. So they are an important class of difference equations.
Chen and Chen [10] investigated some properties of meromorphic solutions of difference Painlevé I equation and proved the following Theorem B.
Theorem B
(see [10])
- (i)
\(\lambda(\frac{1}{f})=\lambda(f)=\sigma(f)\);
- (ii)
if \(p(z)\) is a non-constant polynomial, then \(f(z)-p(z)\) has infinitely many zeros and \(\lambda(f-p)=\sigma(f)\);
- (iii)
if \(a\neq0\), then \(f(z)\) has no Borel exceptional value;
if \(a=0\), then Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 3z^{2}-cz-b=0\}\).
The main aims of this paper are to consider the properties of finite order transcendental meromorphic solutions of difference Painlevé I and II equations (2)-(5), and we obtain the following results.
Theorem 1.1
- (i)
if \(a\neq0\) and \(p(z)\) is a polynomial, then \(f(z)-p(z)\) has infinitely many zeros and \(\lambda(f-p)=\sigma(f)\);
if \(a=0\), then Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 2z^{2}-cz-b=0\}\);
- (ii)
\(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} ) =\sigma(\Delta f)=\sigma(f)\).
Theorem 1.2
- (i)
if \(a\neq0\) and \(p(z)\) is a nonzero polynomial, then \(f(z)-p(z)\) has infinitely many zeros and \(\lambda(f-p)=\sigma(f)\);
if \(a=0\), then Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 2z^{3}+(b-2)z+c=0\}\);
if \(c\neq0\), then \(\lambda(f)=\sigma(f)\);
- (ii)
\(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} )=\sigma(\Delta f)=\sigma(f)\).
Remark 1.1
By Theorem 1.2, we conclude that if \(ac\neq0\) and \(f(z)\) is a finite order transcendental meromorphic solution of the difference Painlevé II equation (5), then \(f(z)\) has no Borel exceptional value.
Theorem 1.3
- (i)
\(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} )=\sigma(\Delta f)=\sigma(f)\);
- (ii)
if \(a=0\), then Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 2z^{3}-bz-c=0\}\).
Theorem 1.4
Let a, b, c be constants with \(|a|+|b|\neq0\). Suppose that \(f(z)\) is a finite order transcendental meromorphic solution of the difference Painlevé I equation (4). Then \(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} )=\sigma(\Delta f)=\sigma(f)\).
Remark 1.2
The following examples show that \(\lambda (\Delta f )=\sigma(f)\) may not hold in above several theorems.
Example 1.1
Example 1.2
Example 1.3
2 The proof of Theorem 1.1
We need the following lemmas to prove Theorem 1.1.
Lemma 2.1
Lemma 2.2
Remark 2.1
Lemma 2.3
(see [18])
Lemma 2.4
(Valiron-Mohon’ko, see [2])
Lemma 2.5
(see [12])
Lemma 2.6
(see [21])
Let \(g:(0,+\infty)\rightarrow R\), \(h:(0,+\infty)\rightarrow R\) be non-decreasing functions. If (i) \(g(r)\leq h(r)\) outside of an exceptional set of finite linear measure, or (ii) \(g(r)\leq h(r)\), \(r \notin{H\cup(0,1]}\), where \(H\subset(1,\infty)\) is a set of finite logarithmic measure, then for any \(\alpha>1\), there exists \(r_{0}>0\) such that \(g(r)\leq h(\alpha r)\) for all \(r>r_{0}\).
Proof of Theorem 1.1
Suppose that \(f(z)\) is a transcendental meromorphic solution of finite order \(\sigma(f)\) of equation (2).
3 The proof of Theorem 1.2
Suppose that \(f(z)\) is a transcendental meromorphic solution of finite order \(\sigma(f)\) of equation (5).
4 Proofs of Theorems 1.3 and 1.4
Proof of Theorem 1.3
Suppose that \(f(z)\) is a transcendental meromorphic solution of finite order \(\sigma(f)\) of equation (3).
Proof of Theorem 1.4
Declarations
Acknowledgements
The authors thank the referee for his/her valuable suggestions. This work is supported by the State Natural Science Foundation of China (No. 61462016), the Science and Technology Foundation of Guizhou Province (Nos. [2014]2125; [2014]2142) and the Doctoral Foundation of Guizhou Normal University (No. 11904-0514021).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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