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Properties of meromorphic solutions of Painlevé III difference equations
Advances in Difference Equations volume 2013, Article number: 256 (2013)
Abstract
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.
MSC:30D35, 39A10.
1 Introduction
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. $\rho (w)$, $\lambda (w)$ and $\lambda (1/w)$ 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 $S(r,w)$ any quantity satisfying $S(r,w)=o(T(r,w))$ 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 $\mathcal{S}(w)$. For example, if a difference equation has only rational coefficients, then all nonrational 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 [7] considered the difference equation
where R is rational in w and meromorphic in z with slow growth coefficients, and the zdependence is supposed by writing $\overline{w}\equiv w(z+1)$ and $\underline{w}\equiv w(z1)$. 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 finiteorder meromorphic solutions of difference Painlevé I and II equations. In fact, they studied the equations
where ${a}_{j}$ ($1\le j\le 6$) are constants with ${a}_{1}{a}_{3}{a}_{4}\ne 0$.
Theorem A ([8])
If w is a transcendental finiteorder meromorphic solution of (1.2) or (1.3), then

(i)
w has at most one nonzero finite Borel exceptional value;

(ii)
$\lambda (1/w)=\lambda (w)=\tau (w)=\rho (w)$.
Furthermore, assume that a rational function $w=\frac{P(z)}{Q(z)}$ is a solution of (1.2), where $P(z)$ and $Q(z)$ are relatively prime polynomials with degrees p and q respectively, then $q=p+1$, while (1.3) has no rational solution.
As for the family including Painlevé III difference equations, we recall the following theorem.
Theorem B ([9])
Assume that the equation
has an admissible meromorphic solution w of hyperorder less than one, where $R(z,w)$ is rational and irreducible in w and meromorphic in z, then either w satisfies the difference Riccati equation
where $\alpha ,\beta ,\gamma \in \mathcal{S}(w)$ are algebroid functions, or equation (1.4) can be transformed to one of the following equations:
In (1.5a), the coefficients satisfy ${\kappa}^{2}\overline{\mu}\underline{\mu}={\mu}^{2}$, $\overline{\lambda}\mu =\kappa \underline{\lambda}\overline{\mu}$, $\kappa \overline{\overline{\lambda}}\underline{\lambda}=\underline{\kappa}\lambda \overline{\lambda}$, and one of the following:
In (1.5b), $\eta \overline{\eta}=1$ and $\overline{\overline{\lambda}}\underline{\lambda}=\lambda \overline{\lambda}$. In (1.5c), the coefficients satisfy one of the following:

(1)
$\eta \equiv 1$ and either $\lambda =\overline{\lambda}\underline{\lambda}$ or ${\overline{\lambda}}^{[3]}{\underline{\lambda}}_{[3]}=\overline{\overline{\lambda}}\underline{\underline{\lambda}}$;

(2)
$\overline{\lambda}\underline{\lambda}=\overline{\overline{\lambda}}\underline{\underline{\lambda}}$, $\overline{\eta}\overline{\lambda}=\overline{\overline{\lambda}}\underline{\eta}$, $\eta \underline{\eta}=\overline{\overline{\eta}}{\underline{\eta}}_{[3]}$;

(3)
$\overline{\overline{\eta}}\underline{\underline{\eta}}=\eta \underline{\eta}$, $\lambda =\underline{\eta}$;

(4)
${\overline{\lambda}}^{[3]}{\underline{\lambda}}_{[3]}=\overline{\overline{\lambda}}\underline{\underline{\lambda}}\lambda $, $\eta \lambda =\overline{\overline{\eta}}\underline{\underline{\eta}}$.
In (1.5d), $h\in \mathcal{S}(w)$ and $m\in \mathbb{Z}$, $m\le 2$.
The first author and Yang [10] 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 $w=\frac{P(z)}{Q(z)}$ be given as in Theorem A. If $w(z)$ is a solution of
where η (≠0), λ are constants, then $p=q$ and ${a}^{2}(a1)=\eta (a\lambda )$, where $a=w(\mathrm{\infty})$.
Theorem D Let $w=\frac{P(z)}{Q(z)}$ be given as in Theorem A. If $w(z)$ is a solution of
where λ (≠1) is a constant, then $p=q$ and ${a}^{2}2a+\lambda =0$, where $a=w(\mathrm{\infty})$.
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 [5] and Chiang and Feng [4] 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 nonzero 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 ([6])
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 $a\in \mathcal{S}(w)$, 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 Equations (1.6) and (1.7)
Theorem 3.1 If w is a transcendental finiteorder meromorphic solution of (1.6), then

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

(ii)
If $\lambda =0$, then w has at most one nonzero Borel exceptional value for $\rho (w)>0$.
Proof Denote $\varphi (z)=w(z)z$. So, $\varphi (z)$ is a transcendental meromorphic function and $T(r,\varphi )=T(r,w)+S(r,w)$. Substituting $w(z)=\varphi (z)+z$ in (1.6), we obtain
Let
We get $P(z,0)={(z1)}^{2}(z+1)\eta (z\lambda )\not\equiv 0$, Lemma 2.3 gives
which means $N(r,\frac{1}{wz})=T(r,w)+S(r,w)$, and thus $\tau (w)=\rho (w)$.
If $\lambda \ne 0$, Lemma 2.3 tells us $m(r,1/w)=S(r,w)$. If $\lambda =0$, we rewrite (1.6) as
Noting that $m(r,w)=S(r,w)$, by applying Lemma 2.2 to (1.6), we deduce from Lemma 2.1 and the above equation that
then $\lambda (w)=\rho (w)$ holds.
Next, we prove the conclusion (ii). Some ideas here come from [8]. Assume to the contrary that w has two nonzero Borel exceptional values a and b (≠a). 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 $w=\frac{bfa}{f1}$. Noting that $\lambda =0$, by (1.6) and (3.4), we have
where
From (3.3), we apply Lemma 2.4 on (3.5), resulting in vanishing of all the coefficients. Since a and b are nonzero constants, we deduce from $A(z)=0$ and $D=0$ that
which means that a and b are distinct zeros of the equation
Thus
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
It follows from (3.7) that
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) yields
which yield $\alpha \beta =1$. Thus $a=b$ by (3.8), a contradiction. □
Theorem 3.2 If w is a transcendental meromorphic solution of (1.7) with finite order $\rho (w)>0$, then

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

(ii)
If $\lambda \ne 0$, then $\lambda (w)=\rho (w)$;

(iii)
w has at most one nonzero 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 nonzero Borel exceptional values a and b (≠a). Let f be given by (3.1). Then we still have (3.2)(3.4). Substituting $w=\frac{bfa}{f1}$ in (1.7), we obtain
where
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 ${A}_{1}(z)=0$ and ${E}_{1}=0$ that a and b are distinct zeros of the equation
Thus
Noting this, from ${B}_{1}(z)=0$, we have $b(b\lambda )ab(b1)=0$ or $\overline{g}{g}_{1}+\underline{g}{g}_{2}=0$. If $b(b\lambda )ab(b1)=0$, then (3.10) gives $\lambda =1$, a contradiction. Therefore, $\overline{g}{g}_{1}+\underline{g}{g}_{2}=0$, i.e.,
We get from $\overline{g}{g}_{1}+\underline{g}{g}_{2}=0$ and ${C}_{1}(z)=0$ that
If $a{b}^{2}{a}^{2}{b}^{2}+a\lambda =0$, we have from (3.10) that $a=b$, which is a contradiction. Then $\overline{f}\underline{f}+{f}^{2}=0$, (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 $w=\frac{P(z)}{Q(z)}$ be given as in Theorem A. If $w(z)$ is a solution of (4.1), then one of the following holds:

(i)
$p=q$, ${a}^{2}{(a1)}^{2}={a}^{2}\lambda a+\mu $, where $a=w(\mathrm{\infty})$;

(ii)
$p<q$, $\lambda =\mu =0$ and $P(z)$ is a constant.
Proof Substituting w by $\frac{P(z)}{Q(z)}$ in (4.1), we get
Let $s=pq$. We discuss the following three cases.
Case 1. $s>0$. Then $\frac{P(z)}{Q(z)}=a{z}^{s}(1+o(1))$ as r tends to infinite, and (4.2) gives
which is a contradiction as r tends to infinite.
Case 2. $s<0$. Now, $\frac{P(z)}{Q(z)}=o(1)$ and $\frac{P(z+1)}{Q(z+1)}=o(1)$. By (4.2), we obtain $\mu =0$. If $\lambda \ne 0$, equation (4.2) turns into
Noting that $\frac{Q(z)}{P(z)}=b{z}^{s}(1+o(1))$ as r tends to infinite, the above equation also leads to a contradiction by the same reasoning as in Case 1.
Then $\lambda =0$. We have
It is easy to find that ${P}^{2}(z)$ and $P(z+1)P(z1)$ have the same zeros, which means that $P(z)$ must be a constant. We get (ii).
Case 3. $s=0$. We suppose that $\frac{P(z)}{Q(z)}=a+o(1)$ as r tends to infinite. The conclusion (i) follows from (4.2). □
Example 4.2 The rational function $w(z)=\frac{1}{{(z+1)}^{2}}$ is a solution of the difference equation $\overline{w}\underline{w}{(w1)}^{2}={w}^{2}$. 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 $\rho (w)$, then

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

(ii)
If $\lambda \mu \ne 0$, then $\lambda (w)=\rho (w)$.
Example 4.4 The function $w(z)={sec}^{2}\frac{\pi z}{2}$ is a solution of the difference equation $\overline{w}\underline{w}{(w1)}^{2}={w}^{2}$. 0 is a Picard exceptional value of w, which shows that $\lambda \mu \ne 0$ is necessary in (ii) of Theorem 4.3.
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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 no. 11201014, 11171013, 11126036 and the YWFZY302854 of Beihang University. This research was also supported by the youth talent program of Beijing.
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Keywords
 meromorphic solution
 difference
 finite order