Some results on a certain type of difference equation originated from difference Painlevé I equation
- Qian Li^{1} and
- Zhi-Bo Huang^{2}Email author
https://doi.org/10.1186/s13662-015-0618-0
© Li and Huang 2015
Received: 19 December 2014
Accepted: 25 August 2015
Published: 4 September 2015
Abstract
In this paper, by using Nevanlinna value distribution theory, we consider a certain type of difference equation, which originates with the difference Painlevé I equation, \(f(z+1)+f(z-1)=\frac{A(z)}{f(z)}+C(z)\), where \(A(z)\), \(C(z)\) are small meromorphic functions relative to \(f(z)\), and we obtain the existence and the forms of rational solutions. We also discuss the properties of the Borel exceptional value, zeros, poles, and fixed points of finite order transcendental meromorphic solutions.
Keywords
MSC
1 Introduction
Theorem 1.A
(Theorem 1.1 of [3])
Recently, due to the difference analog of the lemma on the logarithmic derivative given by Halburd and Korhonen in [5], and Chiang and Feng in [6] independently, many authors focused their interest on the complex difference analogs of Nevanlinna theory and complex difference equations (see [5–13]). But most of them mainly dealt with the growth of order of meromorphic solutions of difference equations (see e.g. [6, 9, 13]).
Though there are few papers on the existence of finite order meromorphic solution of difference equations (see [3, 9, 11, 12]), there is only one paper concerning with the existence of rational solution of difference Painlevé I equation (see [9]). In this paper, we will discuss the existence and forms of rational solutions, and investigate the properties on finite order transcendental meromorphic solutions of a certain type of difference equation originating with the difference Painlevé I equation.
2 The existence and forms of rational solutions
Theorem 2.A
(Theorem 4 of [9])
- (i)
if \(a\neq0\), then (2.1) has no rational solution;
- (ii)if \(a=0\), and \(b\neq0\), then (2.1) has a nonzero constant solution \(f(z)=A\), where A satisfies$$ 2A^{2} -cA-b=0. $$
What will happen if we consider a certain type of difference equation originating with the difference Painlevé I equation (2.1)? Here, we obtain the following result.
Theorem 2.1
- (i)Suppose that \(m\geq n\) and \(m-n\) is an even number or zero. If the difference equationhas an irreducible rational solution \(f(z)=\frac{P(z)}{Q(z)}\), where \(P(z)\) and \(Q(z)\) are polynomials with \(\deg P(z)=p\) and \(\deg Q(z)=q\), then$$ f(z+1)+f(z-1)=\frac{A(z)}{f(z)}+C $$(2.2)$$ p-q=\frac{m-n}{2}. $$
- (ii)Suppose that \(m< n\). If the difference equation (2.2) has an irreducible rational solution \(f(z)=\frac{P(z)}{Q(z)}\), then$$ q-p=n-m\geq1 \quad \textit{or} \quad q-p=0. $$
- (iii)
Suppose that \(m>n\) and \(m-n\) is an odd number.
Remark 2.1
We know that (1.2), (1.3), and (1.4) are difference Painlevé I equations. Why do we only consider the existence and forms of rational solutions of (1.3)? We cannot know the limits of the type \(0\cdot\infty\) when we investigate (1.2) and (1.4) by using the same method below.
Example 2.1, Example 2.2, and Example 2.3 show that the difference equations have rational solutions satisfying Theorem 2.1(i), and Example 2.4 and Example 2.5 show that the difference equations have rational solutions satisfying Theorem 2.1(ii).
Example 2.1
Example 2.2
Example 2.3
Example 2.4
Example 2.5
Proof of Theorem 2.1
If \(\deg P(z)=p< q=\deg Q(z)\), then \(\frac{P(z)}{Q(z)}\rightarrow0\), \(\frac{P(z+1)}{Q(z+1)}\rightarrow0\), and \(\frac{P(z-1)}{Q(z-1)}\rightarrow0\) as \(z\rightarrow\infty\), while \(\frac{m(z)}{n(z)}\rightarrow\infty\) as \(z\rightarrow\infty\). Thus, (2.4) is a contradiction.
If \(\deg P(z)=p=q=\deg Q(z)\), then \(\frac{P(z)}{Q(z)}\rightarrow a\), \(\frac{P(z+1)}{Q(z+1)}\rightarrow a\), and \(\frac{P(z-1)}{Q(z-1)}\rightarrow a\) as \(z\rightarrow\infty\), where a is a nonzero constant, while \(\frac{m(z)}{n(z)}\rightarrow \infty\) as \(z\rightarrow\infty\). Thus, (2.4) is also a contradiction.
If \(\deg P(z)=p< q=\deg Q(z)\), then using the same method as above, we get a contradiction.
(iii) Suppose that \(n>m\) and \(n-m\) is an odd number, and that (2.2) has a rational solution \(f(z)=\frac{P(z)}{Q(z)}\). By the proof in (i), we also get \(p-q=\frac{m-n}{2}\). This is a contradiction. Thus, (2.2) has no rational solution. The proof of Theorem 2.1 is completed. □
3 Value distribution of finite order meromorphic solutions
The above facts imply that it is possible that (3.1) has finite order transcendental meromorphic solutions. Thus, we consider (3.1) and obtain the following.
Theorem 3.1
- (i)
\(\lambda(f)=\lambda (\frac{1}{f} )=\sigma(f)\).
- (ii)
\(f(z)\) has no Borel exceptional value.
- (iii)
If \(A(z)\not\equiv2z^{2} -z C(z)\), then the exponent of convergence of fixed points of \(f(z)\) satisfies \(\tau(f)=\sigma(f)\).
We need some lemmas to prove Theorem 3.1.
Lemma 3.1
(Theorem 3.2 of [5])
Lemma 3.2
(Theorem 2.3 of [13])
Lemma 3.3
(Theorem 2.2 of [6])
Lemma 3.4
(Theorem 1.51 of [14])
- (1)
\(\sum_{j=1}^{n} f_{j}(z) e^{g_{j}(z)}=0\).
- (2)
\(g_{j}(z)-g_{k}(z)\) are not constants for \(1\leq j< k \leq n\).
- (3)For \(1\leq j \leq n\), \(1\leq h < k\leq n\),where \(E\subset(1, +\infty)\) is of finite linear measure or finite logarithmic measure.$$ T(r, f_{j})=o \bigl\{ T\bigl(r, e^{g_{h}-g_{k}}\bigr) \bigr\} \quad (r \rightarrow +\infty, r\notin E), $$
Proof of Theorem 3.1
Hence, together with the result of (i), \(f(z)\) has no Borel exceptional value.
Declarations
Acknowledgements
Our research is supported by the Guangdong National Natural Science Foundation (No. 2014A030313422) and also partly supported by the National Natural Science Foundation of China (No. 11171119).
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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