# Meromorphic solutions of difference Painlevé IV equations

- Jilong Zhang
^{1}Email author

**2014**:260

https://doi.org/10.1186/1687-1847-2014-260

© Zhang; licensee Springer. 2014

**Received: **14 June 2014

**Accepted: **17 September 2014

**Published: **13 October 2014

## Abstract

In this paper, we investigate the family of difference Painlevé IV equation $(w(z+1)+w(z))(w(z)+w(z-1))=R(z,w)$, where *R* is rational in *w* and meromorphic in *z*. If the equation assumes an admissible meromorphic solution of hyper-order ${\rho}_{2}(w)<1$, we fix the degree of $R(z,w)$, and we give some further discussions.

**MSC:**39A10.

## Keywords

## 1 Introduction

*et al.*[1]. Halburd and Korhonen [2] studied the second order complex difference equation

*w*and meromorphic in

*z*. They showed that if (1.1) has an admissible finite order meromorphic solution, then either

*w*satisfies a difference Riccati equation or (1.1) can be transformed into difference Painlevé I, II equations or a linear difference equation. The work on the family $w(z+1)w(z-1)=R(z,w)$, which includes the difference Painlevé III equations, was initiated in [3], with an additional assumption that the order of the poles of

*w*is bounded in a certain subcase of this family. Ronkainen [4] gave the full classification of $w(z+1)w(z-1)=R(z,w)$ in his dissertation without the assumption of the order of the poles of

*w*, and in [4] his also studied the family $(w(z)w(z+1)-1)(w(z)w(z-1)-1)=R(z,w)$, which includes the difference Painlevé V equations. The family of difference Painlevé IV equations (see [5])

with constant coefficients was researched by Grammaticos *et al.* [6]. Korhonen [7] treated a subcase of (1.2) with $R(z,w)$ rational in *z*. In this paper, we will study (1.2) with $R(z,w)$ meromorphic in *z*.

*e.g.*, [8–10]). In particular, we denote by $S(r,f)$ any quantity satisfying $S(r,f)=o(T(r,f))$ as

*r*tends to infinity outside of an exceptional set

*E*of finite logarithmic measure

A meromorphic function $a(z)$ is called a small function with respect to $f(z)$, if $T(r,a)=S(r,f)$. The family of all meromorphic functions that are small with respect to *f* is denoted by $\mathcal{S}(f)$. A meromorphic solution *w* of a difference equation is called *admissible* if all coefficients of the equation are in $\mathcal{S}(w)$.

We conclude this section by the following expatiation on the coefficients. While we only consider meromorphic solutions of equations with meromorphic coefficients, we might encounter a situation where the coefficients have finitely-sheeted branchings, and we will use the algebroid version of Nevanlinna theory (see for instance [11]), which studies meromorphic functions on a finitely-sheeted Riemann surface. Whenever the coefficients have branchings, $T(r,\cdot )$ denotes the corresponding Nevanlinna characteristic function of a finite-sheeted algebroid function. Since all algebroid functions we need to consider are small functions with respect to the meromorphic solution of (1.2), the change in notation only affects the error term which needs to be redefined in terms of the algebroid characteristic. The ‘algebroid error term’ will still be denoted by $S(r,\cdot )$ and it remains small with respect to the meromorphic solution of (1.2), we can still denote it by $S(r,w)$.

## 2 Some lemmas

The difference analog of the logarithmic derivative lemma, which was obtained independently by Halburd and Korhonen [12] and by Chiang and Feng [13], plays a key role in the value distribution of difference [14–16]. The original version is valid for functions of finite order, and it was generalized to hold for meromorphic functions with hyper-order less than one recently.

**Lemma 2.1** ([17])

*Let*

*w*

*be a nonconstant meromorphic function with hyper*-

*order*${\rho}_{2}(w)={\rho}_{2}<1$, $c\in \mathbb{C}$,

*and*$\delta \in (0,1-{\rho}_{2})$.

*Then*

*for all* *r* *outside of a set of finite logarithmic measure*.

The Valiron-Mohon’ko identity [18, 19] is a useful tool to estimate the characteristic function of a rational function, the proof of which can be found in [[9], Theorem 2.2.5].

**Lemma 2.2**

*Let*

*w*

*be a meromorphic function and*$R(z,w)$

*a function which is rational in*

*w*

*and meromorphic in*

*z*.

*If all the coefficients of*

*R*

*are small compared to*

*w*,

*then*

We also need the following lemma to detect the hyper-order of a meromorphic function to be at least one.

**Lemma 2.3**

*Let*$T:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*be a nondecreasing continuous function*,

*and*$s\in (0,\mathrm{\infty})$.

*If*

*and*$\delta \in (0,1-{\rho}_{2})$,

*then*

*as* *r* *tends to infinity outside of an exceptional set of finite logarithmic measure*.

**Proposition 2.4**

*Let*

*w*,

*c*,

*and*

*δ*

*be as in Lemma*2.1.

*We have*

*Proof*Noting that $N(r,w(z+c))\le N(r+|c|,w(z))$, Lemma 2.3 yields

Then (2.1) follows from the last equation and (2.2). □

Without the order restriction, we have the following.

**Lemma 2.5** ([[1], lemma 1])

*Given*$\u03f5>0$

*and a meromorphic function*

*w*,

*we have*

*for all* $r\ge 1/\u03f5$.

If a meromorphic function *f* has a pole of the order *n* at ${z}_{0}\in \mathbb{C}$, we denote this by $f({z}_{0})={\mathrm{\infty}}^{n}$. Similarly, an *a*-point of the order *n* is denoted by $f({z}_{0})=a+{0}^{n}$.

It could always happen that the coefficients in (1.2) have poles or zeros when *w* has a pole; whenever we meet these cases, we shall use the following result.

**Lemma 2.6** ([2])

*Let*$w(z)$

*be a meromorphic function with more than*$S(r,w)$

*poles*(

*or*

*c*-

*points*, $c\in \mathbb{C}$)

*counting multiplicities*,

*and let*${a}_{1},{a}_{2},\dots ,{a}_{n}\in \mathcal{S}(w)$.

*Assume moreover that none of the functions*${a}_{i}$

*is identically zero*.

*Denote by*${z}_{j}$

*the poles and zeros of the functions*${a}_{i}$ (

*where*

*j*

*is in some index set*),

*and let*

*be the maximal order of zeros and poles of the functions* ${a}_{i}$ *at* ${z}_{j}$. *Then for any* $\u03f5>0$ *there are at most* $S(r,w)$ *points* ${z}_{j}$ *such that* $w({z}_{j})={\mathrm{\infty}}^{{k}_{i}}$ (*or* $w({z}_{j})=c+{0}^{{k}_{j}}$) *where* ${m}_{j}\ge \u03f5{k}_{j}$.

The next result on the Nevanlinna characteristic is essential in the study of the family (1.2), for the proof, see, *e.g.*, [6].

**Lemma 2.7**

*Let*

*f*,

*h*,

*and*

*g*

*be three meromorphic functions*.

*Then*

## 3 Main results

**Definition 3.1** ([7])

*I*be a finite set of multi-indexes $\lambda =({\lambda}_{0},\dots ,{\lambda}_{n})$. A difference polynomial of

*w*is defined as

*P*is defined by

*P* is said to be homogeneous if ${d}_{\lambda}$ of each term in the sum (3.1) is nonzero and the same for all $\lambda \in I$. The order of a zero of $P(z,x,{x}_{1},\dots ,{x}_{n})$, as a function of ${x}_{0}$ at ${x}_{0}=0$, is denoted by ${ord}_{0}(P)$. Let ${\mathrm{\Lambda}}_{0}$ be the maximum power of $w(z)$ in $P(z,w)$, and let ${\mathrm{\Lambda}}_{i}$ be the maximum power of $w(z+{c}_{i})$ in $P(z,w(z))$. Obviously, ${\mathrm{\Lambda}}_{i}\ge 1$ ($i=0,\dots ,n$). Denote $\mathrm{\Lambda}={\sum}_{i=0}^{n}{\mathrm{\Lambda}}_{i}$.

A difference version of the Clunie lemma, developed by Halburd and Korhonen [12] and by Laine and Yang [20], works on such difference polynomials, say $P(z,w)$, having only one term of maximal degree in the sum (3.1). If $P(z,w)$ is homogeneous, the above version does not work. We will establish a similar version of the Clunie lemma, for the other new type of version, see [7].

**Theorem 3.2**

*Suppose that*$w(z)$

*is a meromorphic solution of*

*with hyper*-

*order*${\rho}_{2}(w)<1$,

*where*$P(z,w)$

*is a homogeneous difference polynomial defined as in*(3.1), $H(z,w)$

*and*$Q(z,w)$

*are polynomials in*$w(z)$

*with no common factors*.

*All the coefficients of*$P(z,w)$, $H(z,w)$,

*and*$Q(z,w)$

*are in*$\mathcal{S}(w)$.

*If*

*then* $m(r,w)=S(r,w)$.

**Theorem 3.3**

*Suppose that*$w(z)$

*is an admissible meromorphic solution of*(1.2)

*with hyper*-

*order*${\rho}_{2}(w)<1$,

*where*$R(z,w)=\frac{P(z,w)}{Q(z,w)}$, $P(z,w)$,

*and*$Q(z,w)$

*are polynomials in*

*w*

*with degrees*

*p*

*and*

*q*,

*respectively*.

*Then*:

- (i)
$p\le 4$

*and*$q\le 2$; - (ii)
*If*$q=2$,*then*$p=4$.*The coefficients of the highest degree of**P**and**Q**are identical*; - (iii)
*If*$q=1$,*then*$p\le 3$.

**Theorem 3.4**

*Suppose that*$w(z)$

*is an admissible meromorphic solution of*

*with hyper*-

*order*${\rho}_{2}(w)<1$,

*where*$L(w)=(\overline{w}+w)(w+\underline{w})$,

*neither of*${a}_{i}$

*and*${h}_{j}$

*vanishes identically*.

*Then either*

*w*

*satisfies a difference Riccati equation*

*where*$\alpha ,\beta ,\gamma \in \mathcal{S}(w)$

*are algebroid functions*,

*or one of the following holds*:

- (1)
${b}_{3}={\underline{a}}_{1}+{\overline{a}}_{1}={\underline{a}}_{2}+{\overline{a}}_{2}$;

- (2)
${b}_{3}={\underline{a}}_{1}+{\overline{a}}_{2}={\underline{a}}_{2}+{\overline{a}}_{1}$,

*where* ${b}_{3}={a}_{1}+{a}_{2}-{H}_{3}$ *and* ${H}_{3}={\sum}_{j=1}^{4}{h}_{j}$. *For each* $n=1,2,3,4$, ${h}_{n}+{\overline{h}}_{k}=0$ *or* ${h}_{n}+{\underline{h}}_{k}=0$ *for some* $k\in \{1,2,3,4\}$.

**Remark 1** In the present paper, we use similar notations to [4].

## 4 Proofs of theorems

*Proof of Theorem 3.2*Since $H(z,w)$ and $Q(z,w)$ are polynomials in $w(z)$ with no common factors, Lemma 2.2 gives us

which is $(\mathrm{\Lambda}-{deg}_{w}P)m(r,w)=S(r,w)$, then $m(r,w)=S(r,w)$ from (3.2). □

*Proof of Theorem 3.3* We restrict *p* and *q* duo to the reasoning by Grammaticos *et al.* [6], where $P(z,w)$ and $Q(z,w)$ are polynomials in *w* with constant coefficients.

so ${deg}_{w}K\le 3$. Thus $q\le 3$. If $q=3$, the degree of $P-{w}^{2}Q$ that was denoted by *k* would be 5 since $p\le 4$, a contradiction. Hence, $q\le 2$, $p\le 4$, and $k\le 3$.

If $q=2$, since the degree of $P-{w}^{2}Q$, $k\le 3$, we have $p=4$ and the coefficients of the highest degree of *P* and *Q* are identical. □

*Proof of Theorem 3.4*We get from (3.3)

for $m=1,2$. From Lemma 2.6, given $\u03f5>0$, there are at most $S(r,w)$ points ${z}_{j}$ where $Q({z}_{j},w)={0}^{{k}_{j}}$, but where $L(w)$ has a pole of order greater that $(1+\u03f5){k}_{j}$ or less than $(1-\u03f5){k}_{j}$ due to poles or zeros of $P({z}_{j},w)$. The combined effect of all such points can be included in the error term, and so we only consider the rest of the zeros of *Q* in what follows.

to be the longest possible list of points such that each ${z}_{j}+2n\in L({z}_{j},w)$ is a zero of $w-{a}_{m}$, $m=1,2$, and each ${z}_{j}+2n+1\in L({z}_{j},w)$ is a pole of *w*.

*w*has more than $S(r,w)$ poles that are not contained in any sequence $L({z}_{j},w)$. Let ${N}^{\ast}(r,w)$ be the integrated counting function of such poles; by assumption we have ${N}^{\ast}(r,w)>CT(r,w)$ for some $C>0$ in a set of infinite logarithmic measure. By (3.3), ${N}^{\ast}(r,L(w))=2{N}^{\ast}(r,w)+S(r,w)$, and so we get

which implies that ${\rho}_{2}(w)\ge 1$ by Lemma 2.3. Therefore all except at most $S(r,w)$ poles of *w* are in some sequence $L({z}_{j},w)$.

We will call the total number of zeros of $w-{a}_{m}$ in $L({z}_{j},w)$ divided by the total number of poles of *w* (both counting multiplicities) the ${a}_{m}/\mathrm{pole}$ ratio of the sequence.

- (i)
Both (4.7) and (4.8) hold $\u22d7S(r,w)$.

- (ii)
Both (4.5) and (4.6) hold $\u22d7S(r,w)$, (4.7) and (4.8) hold $\u22d6S(r,w)$.

- (iii)
Relation (4.6) holds $\u22d7S(r,w)$, (4.5), (4.7), and (4.8) hold $\u22d6S(r,w)$.

- (iv)
Relation (4.5) holds $\u22d7S(r,w)$, (4.6)-(4.8) hold $\u22d6S(r,w)$.

where ${b}_{3}={a}_{1}+{a}_{2}-{H}_{3}$ and ${H}_{3}={\sum}_{j=1}^{4}{h}_{j}$.

*w*must be contained in sequences of the form

Next, we will show that *U* is a small function with respect to *w*. From (4.3), we get $m(r,U)=S(r,w)$. From the definition of *U* and the fact that all but at most $S(r,w)$ poles of *w* are in sequences of the above form, it follows that if *U* has more than $S(r,w)$ poles, then there are more than $S(r,w)$ sequences where ${l}_{j\pm}>0$.

*d*such that

*j*, there exists an ${\u03f5}_{j}$ satisfying

*ϵ*is well defined. Thus we conclude that if ${l}_{j-}>0$, then in such sequences the ${a}_{m}/\mathrm{pole}$ ratio is at most some $2d<2$. We will get a contradiction as the above. If ${l}_{j+}>0$, we will get a contradiction similarly. Therefore, $U\in S(w)$, and so (3.3) becomes the Riccati difference equation

The same reasoning works for case (iv) as we exchange the roles of ${a}_{1}$ and ${a}_{2}$.

In the case that ${a}_{1}={a}_{2}$, the proof is similar. The condition (i) is the only possibility. □

*Restriction of*${h}_{n}$. We claim that for each $n=1,2,3,4$,

provided ${h}_{n}$ is not a solution of (3.3), *i.e.*, ${h}_{n}$ does not satisfy ${h}_{n}+{\overline{h}}_{n}=0$. In the following, we may assume that $w-{h}_{n}$ has a large number of zeros, or (4.14) holds as desired.

for all except at most $S(r,w)$ points ${z}_{j}$ and for either or both choices of ±.

We assume the condition (i) or (ii) is true, otherwise, *w* satisfies a Riccati difference equation. We discuss the following three cases:

Combining the above equation with (4.10) or (4.11), we have $-{h}_{n}={\overline{a}}_{m}$ for $m=1$ or $m=2$, which is a contradiction.

holds for $m=1$ or $m=2$.

holds for $m=1$ or $m=2$.

In this case, we have both (4.16) and (4.17).

where ${s}_{j+}\approx {k}_{j}+{m}_{j+}$, ${s}_{j-}\approx {k}_{j}+{m}_{j-}$, ${s}_{j}\approx {t}_{j}+{m}_{j}\approx {l}_{j}+{r}_{j}$, and *φ* is a pole or a finite value but not the zero of $w-{h}_{n}$. In fact, if *φ* is the zero of $w-{h}_{n}$, it will be a starting point of another sequence.

If (4.20) holds for more than $S(r,w)$ points, there are more than $S(r,w)$ sequences $L({z}_{j},w)$ such that $N(r,\frac{1}{Q(z,w)})\le \beta N(r+1,w)$, where $\beta <2$. This is a contradiction. Hence, (4.20) holds at most $S(r,w)$ points. Then almost all the zeros of $w-{h}_{n}$ are in the sequences (4.18) and (4.19).

However, noting that ${s}_{j+}\approx {k}_{j}+{m}_{j+}$ and ${s}_{j-}\approx {k}_{j}+{m}_{j-}$, we have $N(r+2,w)>(1+\u03f5)N(r,\frac{1}{w-{h}_{n}})+S(r,w)=(1+\u03f5)T(r,w)+S(r,w)$, which contradicts with Lemma 2.3.

## Declarations

### Acknowledgements

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, 301556 of Beihang University. This research was also supported by the youth talent program of Beijing No. 29201443.

## Authors’ Affiliations

## References

- Ablowitz MJ, Halburd RG, Herbst B: On the extension of the Painlevé property to difference equations.
*Nonlinearity*2000, 13: 889-905. 10.1088/0951-7715/13/3/321MathSciNetView ArticleGoogle Scholar - Halburd RG, Korhonen RJ: Finite order solutions and the discrete Painlevé equations.
*Proc. Lond. Math. Soc.*2007, 94: 443-474.MathSciNetView ArticleGoogle Scholar - Halburd RG, Korhonen RJ: Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations.
*J. Phys. A, Math. Theor.*2007, 40: R1-R38. 10.1088/1751-8113/40/6/R01MathSciNetView ArticleGoogle Scholar - Ronkainen O: Meromorphic solutions of difference Painlevé equations.
*Ann. Acad. Sci. Fenn., Math. Diss.*2010, 155: 1-59.MathSciNetGoogle Scholar - Ramani A, Grammaticos B, Hietarinta J: Discrete versions of Painlevé equations.
*Phys. Rev. Lett.*1991, 67: 1829-1832. 10.1103/PhysRevLett.67.1829MathSciNetView ArticleGoogle Scholar - Grammaticos B, Tamizhmani T, Ramani A, Tamizhmani KM: Groth and integrability in discrete systems.
*J. Phys. A, Math. Gen.*2001, 34: 3811-3821. 10.1088/0305-4470/34/18/309MathSciNetView ArticleGoogle Scholar - Korhonen, RJ: A new Clunie type theorem for difference polynomials (2009). arXiv:0903.4394v1 [math.CV]Google Scholar
- Hayman WK:
*Meromorphic Functions*. Clarendon, Oxford; 1964.Google Scholar - Laine I:
*Nevanlinna Theory and Complex Differential Equations*. de Gruyter, Berlin; 1993.View ArticleGoogle Scholar - Yang CC, Yi HX:
*Uniqueness Theory of Meromorphic Functions*. Kluwer Academic, Dordrecht; 2003. (Chinese original: Science Press, Beijing (1995))View ArticleGoogle Scholar - Katajamäki K: Algebroid solutions of binomial and linear differential equations.
*Ann. Acad. Sci. Fenn., Math. Diss.*1993, 90: 1-48.Google Scholar - Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations.
*J. Math. Anal. Appl.*2006, 314: 477-487. 10.1016/j.jmaa.2005.04.010MathSciNetView ArticleGoogle Scholar - Chiang YM, Feng SJ: On the Nevanlinna characteristic of $f(z+\eta )$ and difference equations in the complex plane.
*Ramanujan J.*2008, 16: 105-129. 10.1007/s11139-007-9101-1MathSciNetView ArticleGoogle Scholar - Li N, Yang LZ: Value distribution of certain type of difference polynomials.
*Abstr. Appl. Anal.*2014., 2014: Article ID 278786Google Scholar - Li S, Chen BQ: Results on meromorphic solutions of linear difference equations.
*Adv. Differ. Equ.*2012., 2012: Article ID 203Google Scholar - Zhang JL, Yi HX: Properties of meromorphic solutions of Painlevé III difference equations.
*Adv. Differ. Equ.*2013., 2013: Article ID 256Google Scholar - Halburd, RG, Korhonen, RJ, Tohge, K: Holomorphic curves with shift-invariant hyperplane preimages. Preprint (2009). arXiv:0903.3236Google Scholar
- Valiron G: Sur la dérivée des fonctions algébroïdes.
*Bull. Soc. Math. Fr.*1931, 59: 17-39.MathSciNetGoogle Scholar - Mohon’ko AZ: The Nevanlinna characteristics of certain meromorphic functions.
*Teor. Funkc. Funkc. Anal. Ih Prilozh.*1971, 14: 83-87. (Russian)MathSciNetGoogle Scholar - Laine I, Yang CC: Clunie theorems for difference and
*q*-difference polynomials.*J. Lond. Math. Soc.*2007, 76: 556-566. 10.1112/jlms/jdm073MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.