- Research
- Open Access
Existence of zero-order meromorphic solutions of certain q-difference equations
- Yunfei Du^{1},
- Zongsheng Gao^{1},
- Jilong Zhang^{1} and
- Ming Zhao^{2}Email author
https://doi.org/10.1186/s13660-018-1790-z
© The Author(s) 2018
- Received: 24 March 2018
- Accepted: 21 July 2018
- Published: 20 August 2018
Abstract
Keywords
- Painlevé equations
- q-Difference
- Meromorphic solution
MSC
- 39B32
- 34M05
- 30D35
1 Introduction
In this paper, we use the basic notions of Nevanlinna theory [1–4] such as the characteristic function \(T(r,f)\), counting function \(N(r,f)\), and proximity function \(m(r,f)\). We also use \(S(r,f)\) to denote any quantity satisfying \(S(r,f)=o(T(r,f))\) as \(r\rightarrow \infty \), possibly outside a set with finite logarithmic measure and \(S(f)\) to denote the field of small functions with respect to f, which is defined as \(S(f)=\{\alpha \text{ meromorphic} : T(r, \alpha )=S(r,f)\}\).
In what follows, we use the short notation \(\bar{f}\equiv f(z+1)\) and \(\underline{f}\equiv f(z-1)\). A meromorphic solution f of a difference equation is called admissible if all the coefficients of the equation are in \(S(f)\). In particular, if the coefficients are rational, then an admissible solution must be transcendental, and if an admissible solution is rational, then the coefficients must be constants.
An ordinary differential equation is said to possess the Painlevé property if all of its solutions are single-valued about all movable singularities, see [5]. In 1895–1910, Painlevé [6, 7], Fuchs [8], and Gambier [9] completed substantial classification work, which comprised sieving through a large class of second-order differential equations by making use of a criterion proposed by Picard [10], now known as the Painlevé property. Painlevé and his colleagues discovered six new equations, later named Painlevé equations, which were not solvable in terms of known functions. Actually, the Painlevé equations are six nonlinear ordinary differential equations denoted traditionally by \(P_{I}\), \(P _{II}\), … , \(P_{VI}\).
Painlevé equations is a fascinating subject in mathematics, they possess many special features [11]. One of them is that, given a solution of a Painlevé equation \((P_{II}, \ldots , P_{VI})\) with a choice of some parameter, a special method based on Bäcklund transformations can be used for deriving a new solution with a different value of the parameter, either for the same Painlevé equation or for another. Symmetry is a word used frequently to refer to a mechanism of constructing new solutions by transformation. Specially, Painlevé equation appeared in many applications and fields such as hydrodynamics, plasma physics, nonlinear optics, solid state physics, etc.
2 Some lemmas
We introduce some lemmas for the proofs of our theorems in this section. The logarithmic derivative lemma [2] plays an important role in difference equations. As for q-difference equation, Barnett et al. [19] gave the analogue of the logarithmic derivative lemma, we recall it as follows.
Lemma 2.1
Lemma 2.2
([19])
Lemma 2.3
([14])
The next lemma is an essential result about the Nevanlinna characteristic and plays an important role in the difference equations.
Lemma 2.4
([27])
The following lemma, i.e., the Valiron–Mohon’ko identity [29, 30], is a useful tool in the theory of complex difference equations.
Lemma 2.5
3 Main result
Combining the above discussion, it may happen \(p\leq 4\) and \(q\leq 3\). If \(q=3\), the degree of \(P(z,f)-f^{2}(z)Q(z,f)\) will be 5, which contradicts \(\deg_{f} (K)\leq 3\). Hence, it must have \(p\leq 4\) and \(q\leq 2\).
If \(q=2\), it follows that the degree of \(P(z,f)-f^{2}(z)Q(z,f)\) cannot be more than 3, that \(q=4\), and the coefficients of the highest degree of \(P(z,f)\) and \(Q(z,f)\) are identical.
Therefore, we conclude the following result.
Theorem 3.1
- (i)
If \(q=0\), then \(p\leq 2\);
- (ii)
If \(q=1\), then \(p\leq 3\);
- (iii)
If \(q=2\), then \(p=4\) and the coefficients of the highest degree of \(P(z,f)\) and \(Q(z,f)\) are identical.
Proof
In the following, we discuss the cases \(p=4\) and \(q=2\). Actually, we obtain the result as follows.
Theorem 3.2
- (a)
\(b_{1}(q^{2}z)=b_{1}(z)\) and \(b_{2}(q^{2}z)=b_{2}(z)\);
- (b)
\(b_{1}(q^{2}z)=b_{2}(z)\) and \(b_{2}(q^{2}z)=b_{1}(z)\).
Proof
- (I)
\(f(z_{0})=a_{i}(z_{0})\) (\(i=1,2,3,4\));
- (II)
\(f(qz_{0})=\infty\) or \(f(z_{0}/q)=\infty \).
In what follows, we adopt the notation \(a_{\star }\) and \(b_{\ast }\) to represent \(a_{i}\) (\(i=1,\ldots,4\)) and \(b_{j}\) (\(j=1,2\)), respectively.
Let \(n_{A}(r,\frac{1}{f-b_{j}})\) (\(j=1,2\)) be the counting function of the multi-set \(A \cap \{z \in C: |z|\leq r\}\) and \(N_{A}(r,\frac{1}{f-b _{j}})\) (\(j=1,2\)) represents the integrated counting function. Similarly, we use \(N_{I}\) to represent the corresponding integrated counting function which satisfies condition (I).
- (a)
\(f(z_{0})=b_{1}(z_{0})\) and \(f(q^{2}z_{0})=b_{2}(q^{2}z _{0})\);
- (b)
\(f(z_{0})=b_{2}(z_{0})\) and \(f(q^{2}z_{0})=b_{1}(q^{2}z _{0})\);
- (c)
\(f(z_{0})=b_{2}(z_{0})\) and \(f(q^{2}z_{0})=b_{2}(q^{2}z _{0})\);
- (d)
\(f(z_{0})=b_{1}(z_{0})\) and \(f(q^{2}z_{0})=b_{1}(q^{2}z _{0})\).
- (i)
\(N_{B_{a}} ( r, \frac{1}{f-b_{\ast }} ) > S(r,f)\) and \(N_{B_{b}} ( r, \frac{1}{f-b_{\ast }} ) > S(r,f)\);
- (ii)
\(N_{B_{c}} ( r, \frac{1}{f-b_{\ast }} ) >S(r,f)\) and \(N_{B_{d}} ( r, \frac{1}{f-b_{\ast }} ) > S(r,f)\);
- (iii)
\(N_{B_{a}} ( r, \frac{1}{f-b_{\ast }} ) >S(r,f)\) and \(N_{B\setminus B _{a}} ( r, \frac{1}{f-b_{\ast }} ) = S(r,f)\);
- (iv)
\(N_{B_{b}} ( r, \frac{1}{f-b_{\ast }} ) > S(r,f)\) and \(N_{B\setminus B_{b}} ( r, \frac{1}{f-b_{ \ast }} ) = S(r,f)\).
Obviously, \(\alpha_{i} (i=1,\ldots,4), \beta_{j}(j=1,2)\in S(f)\).
Combining (37)–(40) yields \(b_{1}(q^{2}z)+b_{1}(z)=b _{2}(q^{2}z)+b_{2}(z)\) and \(b_{1}(q^{2}z)b_{1}(z)= [4]b_{2}(q^{2}z)b_{2}(z)\). From \(b_{1}(z)\) and \(b_{2}(z)\) are distinct, it follows that \(b_{1}(q^{2}z)=b_{2}(z)\) and \(b_{2}(q^{2}z)=b_{1}(z)\)
Declarations
Funding
This research was supported by the National Natural Science Foundation of China (No: 11171013, 11371225) and it was also supported by the Fundamental Research Funds for the Central Universities.
Authors’ contributions
The main idea of this paper was proposed by YD, ZG, JZ, and MZ. YD, ZG, JZ, and MZ prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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