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

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).