Properties of meromorphic solutions of Painlevé III difference equations
© Zhang and Yi; licensee Springer 2013
Received: 17 June 2013
Accepted: 6 August 2013
Published: 21 August 2013
In this paper, we investigate the properties of meromorphic solutions of Painlevé III difference equations. In particular, we study the existence of Borel exceptional value, the exponent of convergence of zeros, poles and fixed points of a transcendental meromorphic solution.
the field of small functions with respect to w. A meromorphic solution w of a difference equation is called admissible if all coefficients of the equation are in . For example, if a difference equation has only rational coefficients, then all non-rational meromorphic solutions are admissible; if an admissible solution is rational, then all the coefficients must be constants.
where R is rational in w and meromorphic in z with slow growth coefficients, and the z-dependence is supposed by writing and . They proved that if (1.1) has an admissible meromorphic solution of finite order, then either w satisfies a difference Riccati equation or equation (1.1) can be transformed to eight simple difference equations. These simple difference equations include Painlevé I, II difference equations and linear difference equations.
where () are constants with .
Theorem A ()
w has at most one non-zero finite Borel exceptional value;
Furthermore, assume that a rational function is a solution of (1.2), where and are relatively prime polynomials with degrees p and q respectively, then , while (1.3) has no rational solution.
As for the family including Painlevé III difference equations, we recall the following theorem.
Theorem B ()
and either or ;
, , ;
In (1.5d), and , .
The first author and Yang  studied the difference Painlevé III equations (1.5b)-(1.5d) with the constant coefficients. In particular, they improved (i) of Theorem A: w does not have Borel exceptional value, and got the following two results.
where η (≠0), λ are constants, then and , where .
where λ (≠1) is a constant, then and , where .
It is natural to ask: What are the properties of a transcendental meromorphic solution w of equations (1.6) and (1.7)? Does w have Borel exceptional value? We will give the answers in Section 3. The remaining equation (1.5a) will be discussed in Section 4.
2 Some lemmas
Halburd and Korhonen  and Chiang and Feng  investigated the value distribution theory of difference expressions. A key result, which is a difference analogue of the logarithmic derivative lemma, reads as follows.
With the help of Lemma 2.1, the difference analogues of the Clunie and Mohon’ko lemmas are obtained.
Lemma 2.2 ()
possibly outside of an exceptional set of finite logarithmic measure.
We conclude this section by the following lemma.
Lemma 2.4 (See, e.g., [, pp.79-80])
is not a constant for ;
for and ,
3 Equations (1.6) and (1.7)
If , then w has at most one non-zero Borel exceptional value for .
which means , and thus .
where and .
which yield . Thus by (3.8), a contradiction. □
If , then ;
w has at most one non-zero Borel exceptional value.
Combining (3.11) and the last equation gives a contradiction, and (iii) follows. □
4 Equation (1.5a)
where λ and μ are constants.
, , where ;
, and is a constant.
Let . We discuss the following three cases.
which is a contradiction as r tends to infinite.
Noting that as r tends to infinite, the above equation also leads to a contradiction by the same reasoning as in Case 1.
It is easy to find that and have the same zeros, which means that must be a constant. We get (ii).
Case 3. . We suppose that as r tends to infinite. The conclusion (i) follows from (4.2). □
Example 4.2 The rational function is a solution of the difference equation . This shows that the conclusion (ii) of Theorem 4.1 may occur.
By the same reasoning as in Section 3, we obtain the following result.
If , then .
Example 4.4 The function is a solution of the difference equation . 0 is a Picard exceptional value of w, which shows that is necessary in (ii) of Theorem 4.3.
The authors would like to thank the referee for valuable suggestions to the present paper. This research was supported by the NNSF of China no. 11201014, 11171013, 11126036 and the YWF-ZY-302854 of Beihang University. This research was also supported by the youth talent program of Beijing.
- Hayman WK: Meromorphic Functions. Clarendon, Oxford; 1964.Google Scholar
- Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.View ArticleGoogle Scholar
- Yang CC, Yi HX: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.View ArticleGoogle Scholar
- Chiang YM, Feng SJ:On the Nevanlinna characteristic of and difference equations in the complex plane. Ramanujan J. 2008, 16: 105–129. 10.1007/s11139-007-9101-1MathSciNetView ArticleGoogle Scholar
- Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 2006, 314: 477–487. 10.1016/j.jmaa.2005.04.010MathSciNetView ArticleGoogle Scholar
- Laine I, Yang CC: Clunie theorems for difference and q -difference polynomials. J. Lond. Math. Soc. 2007, 76: 556–566. 10.1112/jlms/jdm073MathSciNetView ArticleGoogle Scholar
- Halburd RG, Korhonen RJ: Finite order solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. 2007, 94: 443–474.MathSciNetView ArticleGoogle Scholar
- Chen ZX, Shon KH: Value distribution of meromorphic solutions of certain difference Painlevé equations. J. Math. Anal. Appl. 2010, 364: 556–566. 10.1016/j.jmaa.2009.10.021MathSciNetView ArticleGoogle Scholar
- Ronkainen, O: Meromorphic solutions of difference Painlevé equations. Ann. Acad. Sci. Fenn. Diss. 155 (2010), 59ppGoogle Scholar
- Zhang, JL, Yang, LZ: Meromorphic solutions of Painlevé III difference equations. Acta Math. Sin. (to appear)Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.