Meromorphic solutions of Painlevé III difference equations with Borel exceptional values
© Zhang; licensee Springer. 2014
Received: 15 April 2014
Accepted: 8 August 2014
Published: 2 September 2014
In this paper, we investigate the properties of meromorphic solutions of Painlevé III difference equations. In particular, the difference equation with μ being a non-zero constant is studied. We show that the rational solutions of the equation assume only one form and the transcendental solutions have at most one Borel exceptional value. We also show that the difference equation does not have nonconstant rational solution, where λ () is a constant.
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.
Recently, with the development of Nevanlinna value distribution theory on difference expressions [4–6], Halburd and Korhonen  gave the full classification of the family including Painlevé I and II difference equations. As for the family including Painlevé III difference equations, we recall the following.
Theorem A ()
and either or ;
, , ;
where and . In (1.2d), and , .
where () are constants with , then w has at most one non-zero finite Borel exceptional value.
The present author and Yang  improved the above result and verified that w does not have any Borel exceptional value. And they also considered the difference Painlevé III equations (1.2d) with the constant coefficients. The difference equations (1.2b) and (1.2c) are studied by the present author in the following.
Theorem B ()
w has at most one non-zero Borel exceptional value for .
The purpose of this paper is to study the remaining difference equation (1.2a) with constant coefficients. As it is complicated to discuss meromorphic solutions of (1.2a) when , we will consider the following special cases.
with any constant b.
Remark Equation (1.3) is a special case of (1.2a) in the option (2) as . From the proof of Theorem 1.1, we shall see that the degrees of numerator and denominator of must be 2. Here, the coefficient is determined. We think that it is because there is a −1 in (1.3).
w has at most one Borel exceptional value for .
Example 1.3 The function is a solution of the difference equation . It is easy to see that . Since , then is a Picard exceptional value of . This shows that the conclusions of Theorem 1.2 may occur.
Remark In in Example 1.3, the function can be replaced by any finite-order function with period 1. For example, , and so on.
where λ () is a constant. Then w must be transcendental.
w satisfies the difference Riccati equation .
Example 1.6 The transcendental function is a solution of both the difference equation and the Riccati equation . Noting that has two Picard exceptional values 3 and −1, we see that the conclusions in Theorem 1.5 may occur.
2 Some lemmas
Halburd and Korhonen  and Chiang and Feng  investigated the value distribution theory of difference expressions, a key result of which is a difference analogue of the logarithmic derivative lemma. With the help of the lemma, the difference analogues of the Clunie and Mohon’ko lemmas are obtained.
Lemma 2.1 ()
possibly outside of an exceptional set of finite logarithmic measure.
We conclude this section by the following lemma.
Lemma 2.3 (see, e.g., [, pp.79-80])
is not a constant for ;
for and ,
3 Proofs of theorems
Without loss of generality, we assume that the coefficients of the highest degree terms of and are a and 1 respectively. Let .
which is a contradiction since the left-hand side of the above equation goes to infinity as r tends to infinity, while the right-hand side is a constant.
which contradicts .
caused by . It means that and are both zeros of with the order at least k.
Without loss of generality, assume that . Obviously, and . Then is a zero of with the order , which is a contradiction by (3.4). Therefore, all the zeros of have even orders.
and . □
Suppose that is a zero of with the order k, and k is an odd integer. Then is a zero of with the order k. However, since all the zeros of have even orders by (3.8), must be a zero of with the odd order l. Thus, is a zero of with the odd order l. Therefore, are all zeros of by induction, which is impossible. Then all the zeros of have even orders. Similarly, all the zeros of have even orders too.
Combining the above equation with (3.8), we get that or , a contradiction. □
we have from Lemma 2.1 that , then and ∞ is not the Borel exceptional value of w. Thus, a and b are finite complex numbers.
and then .
where and .
Define , and . We discuss the following three cases.
which yield , a contradiction since .
which is . Thus . Substituting a by in the first equation of (3.20), we get that or , both are impossible. □
Proof of Theorem 1.2 Let . Then . We deduce from Lemma 2.2 that , and then . Thus .
which means , a contradiction. □
The author would like to thank the referee for his or her 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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