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Borel exceptional values of meromorphic solutions of Painlevé III difference equations
Advances in Difference Equations volume 2014, Article number: 144 (2014)
Abstract
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 $\overline{w}\underline{w}(w1)=\eta (w\lambda )$ with finite order, where η (≠0), λ ($\ne 0,1$) are constants. If a, b are two Borel exceptional values of w, then $a+b=1+\eta $ and $ab=\lambda ={\eta}^{2}$.
MSC:30D35, 39A10.
1 Introduction
Let w be a meromorphic function in the complex plane. The zdependence is supposed by writing $\overline{w}\equiv w(z+1)$ and $\underline{w}\equiv w(z1)$. We assume the reader is familiar with the standard notation and results of Nevanlinna value distribution theory (see, e.g., [1–3]). $\rho (w)$, $\lambda (w)$, and $\lambda (1/w)$ denote the order, the exponents of convergence of zeros and poles of w, respectively. We also denote by $S(r,w)$ any quantity satisfying $S(r,w)=o(T(r,w))$ for all r outside of a set with finite logarithmic measure.
Meromorphic solutions of complex difference equations have become a subject of great interest recently [4–9], due to applications of value distribution theory to difference expressions [10–12]. In particular, Halburd and Korhonen [13] studied the difference equation
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 [4], leading to the following.
Theorem A If w is a transcendental finiteorder meromorphic solution of
then

(i)
w has at most one nonzero finite Borel exceptional value for $\rho (w)>0$;

(ii)
$\lambda (1/w)=\lambda (w)=\rho (w)$.
In 2010, Ronkainen [14] gave the full classification of the family including Painlevé III difference equations in his dissertation. He showed that if the equation $\overline{w}\underline{w}=R(z,w)$ has an admissible meromorphic solution w of hyperorder less than one, then either w satisfies a difference Riccati equation or the equation can be transformed to four simple Painlevé III difference equations.
Most recently, the first author and Yi [15] investigated the properties of meromorphic solutions of Painlevé III difference equations given by [14]. Especially, we recall the following.
Theorem B If w is a transcendental finiteorder meromorphic solution of
where η (≠0), $\lambda \ne 1$ are constants, then

(i)
$\lambda (w)=\rho (w)$;

(ii)
if $\lambda =0$, then w has at most one nonzero Borel exceptional value for $\rho (w)>0$.
Theorem C If w is a transcendental finiteorder meromorphic solution of
where $\lambda \mu \ne 0$, then $\lambda (w)=\rho (w)$.
Example 1.1 The rational function ${w}_{1}(z)=\frac{1}{{(z+1)}^{2}}$ and the transcendental function ${w}_{2}(z)={sec}^{2}\frac{\pi z}{2}$ are solutions of difference equation
0 is a Picard exceptional value of w, this shows that $\lambda \mu \ne 0$ 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 $\lambda =0$ in the second conclusion in Theorem B? The two solutions ${w}_{1}(z)$ and ${w}_{2}(z)$ 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.
Theorem 1.2 Let w be a transcendental meromorphic solution of (1.2) with finite order. If a, b are two Borel exceptional values of w and $\lambda \ne 0$, then

(i)
$a+b=1+\eta $;

(ii)
$ab=\lambda $;

(iii)
$\lambda ={\eta}^{2}$.
Theorem 1.3 Let w be a transcendental meromorphic solution of (1.3) with finite order. If a, b are two Borel exceptional values of w, then

(i)
$a=0$, $b=2$;

(ii)
$\overline{w}=\underline{w}$;

(iii)
w satisfies the difference Riccati equation $\overline{w}=\frac{w}{w1}$.
Example 1.4 The transcendental function ${w}_{3}(z)=\frac{2{e}^{i\pi z}}{{e}^{i\pi z}1}$ is a solution of both difference equation (1.3) and the Riccati equation $\overline{w}=\frac{w}{w1}$. Noting that ${w}_{3}(z)$ 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
HalburdKorhonen [11] and ChiangFeng [10] investigated the value distribution theory of difference expressions. A key result, which is a difference analog of the logarithmic derivative lemma, reads as follows.
Lemma 2.1 Let f be a transcendental meromorphic function of finite order and c be a nonzero complex constant. Then
With the help of Lemma 2.1, the difference analogs of the Clunie and Mohon’ko lemmas are obtained.
Lemma 2.2 ([12])
Let f be a transcendental meromorphic solution of finite order ρ of a difference equation of the form
where $U(z,f)$, $P(z,f)$, and $Q(z,f)$ are difference polynomials such that the total degree ${deg}_{f}U(z,f)=n$ in $f(z)$ and its shifts, and ${deg}_{f}Q(z,f)\le n$. If $U(z,f)$ contains just one term of maximal total degree in $f(z)$ and its shifts, then, for each $\epsilon >0$,
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 $P(z,w)$ is a difference polynomial in $w(z)$. If $P(z,a)\not\equiv 0$ for a meromorphic function a satisfying $T(r,a)=S(r,w)$, then
We conclude this section by the following lemma.
Lemma 2.4 (See, e.g., [[3], pp.7980])
Let ${f}_{j}$ ($j=1,\dots ,n$) ($n\ge 2$) be meromorphic functions, ${g}_{j}$ ($j=1,\dots ,n$) be entire functions. If

(i)
${\sum}_{j=1}^{n}{f}_{j}(z){e}^{{g}_{j}(z)}\equiv 0$;

(ii)
${g}_{h}(z){g}_{k}(z)$ is not a constant for $1\le h<k\le n$;

(iii)
$T(r,{f}_{j})=S(r,{e}^{{g}_{h}(z){g}_{k}(z)})$ for $1\le j\le n$ and $1\le h<k\le n$,
then ${f}_{j}(z)\equiv 0$ ($j=1,\dots ,n$).
3 Proofs of theorems
Proof of Theorem 1.2 Rewriting (1.2) as $\overline{w}\underline{w}w=\overline{w}\underline{w}+\eta (w\lambda )$, we get from Lemma 2.2 that $m(r,w)=S(r,w)$ and then $N(r,w)=T(r,w)+S(r,w)$. Therefore, a and b are not infinity.
Let $P(z,w)=\overline{w}\underline{w}(w1)\eta (w\lambda )$. We get $P(z,0)=\eta \lambda \ne 0$, Lemma 2.3 gives
which means $N(r,1/w)=T(r,w)+S(r,w)$ and thus $ab\ne 0$.
Now we find that a and b are two nonzero finite Borel exceptional values of w. Set
Then $\rho (f)=\rho (w)$, $\lambda (f)=\lambda (wa)<\rho (f)$, and $\lambda (1/f)=\lambda (wb)<\rho (f)$. Since f is of finite order, we suppose that
where d (≠0) is a constant, n (≥1) is an integer, $g(z)$ is meromorphic and satisfies
Then
where ${g}_{1}(z)={e}^{nd{z}^{n1}+\cdots +d}$ and ${g}_{2}(z)={e}^{nd{z}^{n1}+\cdots +{(1)}^{n}d}$.
We get from (3.1) and (3.2) that $w=\frac{bfa}{f1}$. By (1.2) and (3.4), we have
where
From (3.3), we apply Lemma 2.4 to (3.5), and as a result all the coefficients vanish. Since a and b are nonzero constants, we deduce from $A(z)=0$ and $D=0$ that
Then
Denote $G=g$, ${G}_{1}=\overline{g}{g}_{1}$, and ${G}_{2}=\underline{g}{g}_{2}$. From $B(z)=0$, $C(z)=0$ and (3.6), we have
Since the last two equations are both homogeneous, there exist two nonzero constants α and β such that ${G}_{1}=\alpha G$ and ${G}_{2}=\beta G$. Then
On the other hand, combining (3.2) with (3.4), we get
which yield $\alpha \beta =1$. It follows from (3.8) and (3.9) that
Combining (3.6) and (3.10) gives
Noting that $\eta \ne 0$ and $a\ne b$, we obtain $ab=\lambda $. This is the conclusion (ii).
Rewrite (3.10) as
Combining this with (3.7), we get $a+b=1+\eta $, then (i) follows.
On the other hand, we get from (3.11) ${a}^{2}+ab+{b}^{2}=2(a+b)1$, i.e.,
which yields the conclusion (iii) by $a+b=1+\eta $. □
Proof of Theorem 1.2 Denote
Then u is a transcendental function of finite order and $T(r,u)=T(r,w)+O(1)$. Substituting $w=1/u$ in (1.3), we obtain
Both sides of (3.13) are divided by ${u}^{2}$, giving
It follows from Lemma 2.1 that
which means that $m(r,w)=m(r,1/u)=S(r,w)$, and thus $N(r,w)=T(r,w)+S(r,w)$. Therefore, a and b are not infinity.
Let $P(z,w)=\overline{w}\underline{w}{(w1)}^{2}{w}^{2}$. Since a and b are Borel exceptional values of w, Lemma 2.3 tells us that
Solving the last two equations, it follows that $a=0$ and $b=2$. Set
Then $u=1/w=(1f)/2$. Substituting this in (3.13), it yields
Since now 0 and 2 are Borel exceptional values of w, we still have (3.2)(3.4). By (3.4) and the above equation, we have
where ${G}_{1}$, ${G}_{2}$, and G are the same as before. We apply Lemma 2.4 on the last equation, resulting in all the coefficients vanish, i.e.,
from which we get ${G}_{1}={G}_{2}=G$. Then $\overline{f}=\underline{f}=f$ by (3.2) and (3.4). Noting (3.14) and $w=\frac{2}{1f}$, it follows that
and
□
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Acknowledgements
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 YWF14SXXY008, YWFZY302854 of Beihang University. This research was also supported by the youth talent program of Beijing No. 29201443.
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Zhang, J., Yi, H. Borel exceptional values of meromorphic solutions of Painlevé III difference equations. Adv Differ Equ 2014, 144 (2014). https://doi.org/10.1186/168718472014144
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Keywords
 Borel exceptional values
 meromorphic solution
 difference