Borel exceptional values of meromorphic solutions of Painlevé III difference equations
© Zhang and Yi; licensee Springer. 2014
Received: 19 March 2014
Accepted: 29 April 2014
Published: 13 May 2014
In this paper, we investigate Borel exceptional values of meromorphic solutions of Painlevé III difference equations. In particular, let w be a transcendental meromorphic solution of with finite order, where η (≠0), λ () are constants. If a, b are two Borel exceptional values of w, then and .
Let w be a meromorphic function in the complex plane. The z-dependence is supposed by writing and . We assume the reader is familiar with the standard notation and results of Nevanlinna value distribution theory (see, e.g., [1–3]). , , and denote the order, the exponents of convergence of zeros and poles of w, respectively. We also denote by any quantity satisfying for all r outside of a set with finite logarithmic measure.
where R is rational in w and meromorphic in z with slow growth coefficients. They proved that if (1.1) has an admissible meromorphic solution of finite order, then either w satisfies a difference Riccati equation or (1.1) can be transformed into eight simple difference equations. These simple difference equations include Painlevé I, II difference equations and linear difference equations. Some of these, restricting the coefficients to be constants, are studied by Chen and Shon , leading to the following.
w has at most one non-zero finite Borel exceptional value for ;
In 2010, Ronkainen  gave the full classification of the family including Painlevé III difference equations in his dissertation. He showed that if the equation has an admissible meromorphic solution w of hyper-order less than one, then either w satisfies a difference Riccati equation or the equation can be transformed to four simple Painlevé III difference equations.
if , then w has at most one non-zero Borel exceptional value for .
where , then .
0 is a Picard exceptional value of w, this shows that is necessary in Theorem C.
The purpose of this paper is to study the Borel exceptional values of meromorphic solutions of difference equations (1.2) and (1.3). In fact, one may ask: what happens if we remove the restriction in the second conclusion in Theorem B? The two solutions and of difference equation (1.3) given by Example 1.1 both have a Picard exceptional value (also a Borel exceptional value) 0. It is natural to ask can the solutions of (1.3) have two Borel exceptional values? Corresponding to these questions, we obtain the following results as the complement of researching of Painlevé III difference equations.
w satisfies the difference Riccati equation .
Example 1.4 The transcendental function is a solution of both difference equation (1.3) and the Riccati equation . Noting that is a period function and has two Picard exceptional values 0 and 2, we see that case (i) in Theorem 1.3 may occur.
2 Some lemmas
Halburd-Korhonen  and Chiang-Feng  investigated the value distribution theory of difference expressions. A key result, which is a difference analog of the logarithmic derivative lemma, reads as follows.
With the help of Lemma 2.1, the difference analogs 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 Proofs of theorems
Proof of Theorem 1.2 Rewriting (1.2) as , we get from Lemma 2.2 that and then . Therefore, a and b are not infinity.
which means and thus .
where and .
Noting that and , we obtain . This is the conclusion (ii).
Combining this with (3.7), we get , then (i) follows.
which yields the conclusion (iii) by . □
which means that , and thus . Therefore, a and b are not infinity.
The authors would like to thank the referee for valuable suggestions to the present paper. This research was supported by the NNSF of China Nos. 11201014, 11171013, 11126036 and the YWF-14-SXXY-008, YWF-ZY-302854 of Beihang University. This research was also supported by the youth talent program of Beijing No. 29201443.
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