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Properties of meromorphic solutions of Painlevé III difference equations
Advances in Difference Equations volume 2013, Article number: 256 (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.
We assume that the reader is familiar with the standard notations and results of Nevanlinna value distribution theory (see, e.g., [1–3]). Let w be a meromorphic function in the complex plane. , and denote the order, the exponents of convergence of zeros and poles of w, respectively. The exponent of convergence of fixed points of w is defined by
Furthermore, we denote by any quantity satisfying for all r outside of a set with finite logarithmic measure and by
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.
In 2010, Chen and Shon started the topic of researching the properties of finite-order meromorphic solutions of difference Painlevé I and II equations. In fact, they studied the equations
where () are constants with .
Theorem A ()
If w is a transcendental finite-order meromorphic solution of (1.2) or (1.3), then
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 ()
Assume that the equation
has an admissible meromorphic solution w of hyper-order less than one, where is rational and irreducible in w and meromorphic in z, then either w satisfies the difference Riccati equation
where are algebroid functions, or equation (1.4) can be transformed to one of the following equations:
In (1.5a), the coefficients satisfy , , , and one of the following:
In (1.5b), and . In (1.5c), the coefficients satisfy one of the following:
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.
Theorem C Let be given as in Theorem A. If is a solution of
where η (≠0), λ are constants, then and , where .
Theorem D Let be given as in Theorem A. If is a solution of
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.
Lemma 2.1 Let f be a meromorphic function of finite order, and let c be a non-zero complex constant. Then
With the help of Lemma 2.1, the difference analogues of the Clunie and Mohon’ko lemmas are obtained.
Lemma 2.2 ()
Let f be a transcendental meromorphic solution of finite order ρ of a difference equation of the form
where , and are difference polynomials such that the total degree in and its shifts, and . If contains just one term of maximal total degree in and its shifts, then, for each ,
possibly outside of an exceptional set of finite logarithmic measure.
Let w be a transcendental meromorphic solution of finite order of the difference equation
where is a difference polynomial in . If for a meromorphic function , then
We conclude this section by the following lemma.
Lemma 2.4 (See, e.g., [, pp.79-80])
Let () () be meromorphic functions, () be entire functions. If
is not a constant for ;
for and ,
3 Equations (1.6) and (1.7)
Theorem 3.1 If w is a transcendental finite-order meromorphic solution of (1.6), then
If , then w has at most one non-zero Borel exceptional value for .
Proof Denote . So, is a transcendental meromorphic function and . Substituting in (1.6), we obtain
We get , Lemma 2.3 gives
which means , and thus .
If , Lemma 2.3 tells us . If , we rewrite (1.6) as
Noting that , by applying Lemma 2.2 to (1.6), we deduce from Lemma 2.1 and the above equation that
Next, we prove the conclusion (ii). Some ideas here come from . Assume to the contrary that w has two non-zero Borel exceptional values a and b (≠a). Set
Then , and . Since f is of finite order, we suppose that
where d (≠0) is a constant, n (≥1) is an integer, is meromorphic and satisfies
where and .
We get from (3.1) and (3.2) that . Noting that , by (1.6) and (3.4), we have
From (3.3), we apply Lemma 2.4 on (3.5), resulting in vanishing of all the coefficients. Since a and b are non-zero constants, we deduce from and that
which means that a and b are distinct zeros of the equation
Denote , and . From , and (3.6), we have
It follows from (3.7) that
Since the last two equations are both homogeneous, there exist two non-zero constants α and β such that and . Then
On the other hand, combining (3.2) with (3.4) yields
which yield . Thus by (3.8), a contradiction. □
Theorem 3.2 If w is a transcendental meromorphic solution of (1.7) with finite order , then
If , then ;
w has at most one non-zero Borel exceptional value.
Proof The conclusions (i) and (ii) can be discussed by the same reasoning as in the proof of Theorem 3.1, we only prove the conclusion (iii) here. Assume to the contrary that w has two non-zero Borel exceptional values a and b (≠a). Let f be given by (3.1). Then we still have (3.2)-(3.4). Substituting in (1.7), we obtain
From (3.3), we apply Lemma 2.4 to (3.9), which results in vanishing of all the coefficients. By a similar way to that above, we deduce from and that a and b are distinct zeros of the equation
Noting this, from , we have or . If , then (3.10) gives , a contradiction. Therefore, , i.e.,
We get from and that
If , we have from (3.10) that , which is a contradiction. Then , (3.11) yields
Combining (3.11) and the last equation gives a contradiction, and (iii) follows. □
4 Equation (1.5a)
Assume that the coefficients are constants in (1.5a). In this section, we consider the equation
where λ and μ are constants.
Theorem 4.1 Let be given as in Theorem A. If is a solution of (4.1), then one of the following holds:
, , where ;
, and is a constant.
Proof Substituting w by in (4.1), we get
Let . We discuss the following three cases.
Case 1. . Then as r tends to infinite, and (4.2) gives
which is a contradiction as r tends to infinite.
Case 2. . Now, and . By (4.2), we obtain . If , equation (4.2) turns into
Noting that as r tends to infinite, the above equation also leads to a contradiction by the same reasoning as in Case 1.
Then . We have
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.
Theorem 4.3 If w is a transcendental meromorphic solution of (4.1) with finite order , then
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.
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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.
The authors declare that they have no competing interests.
All authors drafted the manuscript, read and approved the final manuscript.