Research Article | Open Access

# Zeros, Poles, and Fixed Points of Meromorphic Solutions of Difference Painlevé Equations

**Academic Editor:**Zhi-Bo Huang

#### Abstract

In this paper, we mainly study the properties of transcendental meromorphic solutions of difference Painlevé equations and and obtain precise estimations of the exponents of convergence of zeros, poles of and , and of fixed points of for any .

#### 1. Introduction and Main Results

At the beginning of last century, Painlevé, Gambier, and Fuchs classified a large number of second-order differential equations in terms of a characteristic which is now known as the Painlevé property [1–4]. Ablowitz et al. [5] considered discrete equations as delay equations in the complex plane which enabled them to utilize complex analytic methods. They looked at, for instance, difference equations of the type where is rational in both of its arguments. It is shown that if (1) has at least one nonrational finite-order meromorphic solution, then .

In this paper, we use the basic notions of Nevanlinna's theory (see [6, 7]). In addition, we use the notations to denote the order of growth of the meromorphic function ; and , respectively, to denote the exponents of convergence of zeros and poles of ; to denote the exponent of convergence of fixed points of .

The quantity is called the deficiency of the value to . Furthermore, we denote by any quantity satisfying for all outside of a set with finite logarithmic measure and by
the field of small functions with respect to . A meromorphic solution of a difference (or differential) equation is called* admissible* if all coefficients of the equation are in .

Recently, Halburd and Korhonen [8] considered (1) and got the following theorem.

Theorem A. *Let be rational in both of its arguments such that its denominator has at least two distinct roots. If the second-order difference equation (1) admits a nonrational meromorphic solution of finite order such that there is a finite real constant , such that for sufficiently large ,
**
holds, then (1) is a difference Painlevé II equation
**
where , , and are constants.*

*Remark 1. *If has a pole at , we say the singularity at is of type if and of type if . We denote by the number of type poles (ignoring multiplicities) in the disc . Similarly, the function counts poles of type .

In 2010, Chen and Shon [9] researched the properties of finite-order meromorphic solutions of difference Painlevé I and II equations. They mainly discussed the existence and the forms of rational solutions and value distribution of transcendental meromorphic solutions.

For difference Painlevé III equations, we recall the following.

Theorem B (see [10]). *Assume that equation
**
has an admissible meromorphic solution of hyperorder less than one, where is rational and irreducible in and meromorphic in , then either satisfies a difference Riccati equation
**
where , , are algebroid functions or (5) can be transformed to one of the following equations:**In (7a), the coefficients satisfy , , , and one of the following: *(1)*, , ;*(2)*, .**In (7b), and . In (7c), the coefficients satisfy one of the following:*(1)* and either or ;*(2)*, , and ;*(3)*, ;*(4)*, .**In (7d), and , .*

Zhang and Yang [11] investigated the difference Painlevé III equations (7a)–(7d) with constant coefficients and obtained the following results.

Theorem C. *If is a nonconstant meromorphic solution of difference equation (7d), where and is a nonzero constant. Then*(i)* cannot be a rational function;*(ii)*.*

Lan and Chen [12] studied some difference Painlevé III equations and proved the following.

Theorem D. *Suppose that is a nonconstant rational function. Suppose is a transcendental meromorphic solution with finite order of (7d), where . Set . Then *(i)* has no Nevanlinna exceptional value;*(ii)*, .*

In general, , where is a nonzero constant. For example, has no fixed points, but has infinitely many fixed points and . Combining Theorems C and D, we continue to study properties (including fixed points) of transcendental meromorphic solutions of difference Painlevé III equations (7b) and (7c), and obtain the following.

Theorem 2. *Suppose that and are nonconstant polynomials. Suppose is a transcendental meromorphic solution with finite order of difference Painlevé III equation
**
Then*(i)* has at most one Nevanlinna exceptional value;*(ii)*for any , has infinitely many fixed points, and ;*(iii)* and ;*(iv)*if there exists some nonconstant rational function such that , then .*

Theorem 3. *Suppose that is a nonconstant polynomial. Suppose is a transcendental meromorphic solution with finite order of difference Painlevé III equation
**
Then*(i)* has no Nevanlinna exceptional value;*(ii)*for any , has infinitely many fixed points, and ;*(iii)* and ;*(iv)*if there exists some rational function such that , then .*

#### 2. Lemmas for the Proofs of Theorems

Lemma 4 (see [13]). *Let be a meromorphic function of finite order and let be a nonzero complex number. Then for each , one has
*

Lemma 5 (see [13]). *Let be a meromorphic function with order , , and let be a fixed nonzero complex number, then for each , one has
*

Lemmas 4 and 5 show the following.

Lemma 6. *Let be a nonzero constant and be a meromorphic function with finite order . Then for each , one has
*

Lemma 7 (see [14, 15]). *Let be a transcendental meromorphic solution of finite order of difference equation
**
where is a difference polynomial in . If for a meromorphic function , then
*

Lemma 8 (see [15]). *Let be a transcendental meromorphic solution of finite order of difference equation of the form
**
where , , and are difference polynomials such that the total degree in and its shifts, and . If contains just one term of maximal total degree in and its shifts, then for each ,
**
possibly outside of an exceptional set of finite logarithmic measure.*

#### 3. Proofs of Theorems

*Proof of Theorem 2. *(i) By (8), we have
Applying Lemma 8 to (17), we have
which yields , that is, .

By (17), we have
By (18), (19), and Lemma 4, we obtain
Thus,
which means that .

Set
Assume that has two Nevanlinna exceptional values . By and , we see that . By Lemma 7, we have and . That is,
Hence,
Since , then is a constant. This contradicts the fact that is a nonconstant polynomial. So, has at most one Nevanlinna exceptional value.

(ii) For any , substituting for in (8), we obtain

Set . Thus, (25) can be written as
Set
Since and are polynomials,
By and Lemma 7, we have , which follows . By and Lemma 5, we have . Thus, .

(iii) By (8), we have
Applying Valiron-Mohon'ko theorem and Lemma 5 to (29), we obtain
Thus,
By (31) and Lemma 4, we have
Therefore, , that is, .

Substituting , into (8), we see
that is,
Let be a zero of , by (33), is a zero of or . Since , then must be a zero of or . Thus, by (21) and Lemma 6, we obtain
Hence, , that is, .

Applying Valiron-Mohon'ko Theorem and Lemma 5 to (34), we deduce
Thus,
By (18), (37), and Lemma 4, we have
Hence, , that is, .

(iv) Suppose that , where is some nonconstant rational function. Now we prove that . Set
By (34), (39), and , we have

Since is a nonconstant rational function, then . By (22) and , we know
Similarly, we obtain . By and Lemma 7, we have
By (18), (20), (40) and (42), we obtain

By (39), we obtain
It sees from Lemma 4 that
Hence,
From (18), (43), (44), (46) and Lemma 4, we deduce that
which yields
By (18) and (48) we have
and by (31),
Then . So, .

*Proof of Theorem 3. *(i) By (9), we have
Applying Lemma 8 to (51), we have
Thus, , which yields .

Again by (9), we have
From (52), (53), and Lemma 4, we deduce that
which follows
Thus, .

Set
For any , since is a nonconstant polynomial, then . By and Lemma 7, we know that , which means that . Hence, . Combining , , we see that has no Nevanlinna exceptional value.

(ii) For any , substituting for in (9), we see that

Set . Thus, (57) can be written as
Set
Since is a polynomial, then
By this and Lemma 7, we have , which follows . By and Lemma 5, we see that . Thus, .

(iii) Substituting into (9), we have

If is a zero of , by (61), must be a zero of or . Thus, by (55) and Lemma 6, we have
Hence, , that is, .

By (61), we have
Applying Valiron-Mohon'ko theorem and Lemma 5 to (63), we deduce
Hence,
Combining (65) with (52) and Lemma 4, we have
which yields . So, .

By (9), we have
Applying Valiron-Mohon'ko theorem and Lemma 5 to (67), we obtain
Thus,
By (69) and Lemma 4, we see that
Hence, , that is, .

(iv) Suppose that , where is some rational function. Now we prove that . Set
By (63), (71), and , we have

Since is a nonconstant polynomial, is a nonconstant rational function. Then . By (56) and , we know
Similarly, we obtain . By , , and Lemma 7, we have
'By (52), (54), (74), and (72), we obtain

Using the same method as in the proof of (iv) in Theorem 2, we may obtain . By this and (52), we have
and by (69),
Then . So, .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The project was supported by the National Natural Science Foundation of China (11171119).

#### References

- L. Fuchs, “Sur quelques équations différentielles linéares du second ordre,”
*Comptes Rendus de l'Académie des Sciences*, vol. 141, pp. 555–558, 1905. View at: Google Scholar | Zentralblatt MATH - B. Gambier, “Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes,”
*Acta Mathematica*, vol. 33, no. 1, pp. 1–55, 1910. View at: Publisher Site | Google Scholar | MathSciNet - P. Painlevé, “Mémoire sur les équations différentielles dont l'intégrale générale est uniforme,”
*Bulletin de la Société Mathématique de France*, vol. 28, pp. 201–261, 1900. View at: Google Scholar | MathSciNet - P. Painlevé, “Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme,”
*Acta Mathematica*, vol. 25, no. 1, pp. 1–85, 1902. View at: Publisher Site | Google Scholar | MathSciNet - M. J. Ablowitz, R. Halburd, and B. Herbst, “On the extension of the Painlevé property to difference equations,”
*Nonlinearity*, vol. 13, no. 3, pp. 889–905, 2000. View at: Publisher Site | Google Scholar | MathSciNet - W. K. Hayman,
*Meromorphic Functions*, Clarendon Press, Oxford, UK, 1964. View at: MathSciNet - L. Yang,
*Value Distribution Theory and Its New Research*, Science Press, Beijing, China, 1982, (Chinese). - R. G. Halburd and R. J. Korhonen, “Existence of finite-order meromorphic solutions as a detector of integrability in difference equations,”
*Physica D: Nonlinear Phenomena*, vol. 218, no. 2, pp. 191–203, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z.-X. Chen and K. H. Shon, “Value distribution of meromorphic solutions of certain difference Painlevé equations,”
*Journal of Mathematical Analysis and Applications*, vol. 364, no. 2, pp. 556–566, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - O. Ronkainen, “Meromorphic solutions of difference Painlevé equations,”
*Annales Academiae Scientiarum Fennicae*, vol. 155, p. 59, 2010. View at: Google Scholar | Zentralblatt MATH - J. L. Zhang and L. Z. Yang, “Meromorphic solutions of Painlevé III difference equations,”
*Acta Mathematica Sinica*, vol. 57, no. 1, pp. 181–188, 2014 (Chinese). View at: Google Scholar - S.-T. Lan and Z.-X. Chen, “On properties of meromorphic solutions of certain difference painlevé III equations,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 208701, 9 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of $f(z+\eta )$ and difference equations in the complex plane,”
*Ramanujan Journal*, vol. 16, no. 1, pp. 105–129, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 314, no. 2, pp. 477–487, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Laine and C.-C. Yang, “Clunie theorems for difference and $q$-difference polynomials,”
*Journal of the London Mathematical Society*, vol. 76, no. 3, pp. 556–566, 2007. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Shuang-Ting Lan and Zong-Xuan Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.