On Properties of Meromorphic Solutions of Certain Difference Painlevé III Equations

Abstract

We mainly study the exponents of convergence of zeros and poles of difference and divided difference of transcendental meromorphic solutions for certain difference Painlevé III equations.

1. Introduction and Main Results

In this paper, we use the basic notions of Nevanlinna's theory (see [1, 2]). 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 . 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 .

At the beginning of the 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 [3–6]. They are proven to be integrable by using inverse scattering transform technique, for instance [7].

Recently, a number of papers (such as [8–12]) focus on complex difference equations and difference analogues of Nevanlinna's theory. Ablowitz et al. [13] considered discrete equations as delay equations in the complex plane which enabled them to utilize complex analytic methods. They looked at difference equations of the type where is rational in both of its arguments. It is shown that if (2) has at least one nonrational finite order meromorphic solution, then .

Recently, Halburd and Korhonen [14] considered (2), where the coefficients of are in and got Theorem A.

Theorem A. If (2) has an admissible meromorphic solution of finite order, where is rational and irreducible in and meromorphic in , then either satisfies a difference Riccati equation where , or (2) can be transformed to one of the following equations: where are arbitrary finite order periodic functions with period .
Equations (4a), (4c), and (4d) are known as difference Painlevé I equations, while (4f) is often viewed as difference Painlevé II equation. Equations (4b) and (4e) are slight variations of (4a) and (4f), respectively.

In 2010, Chen and Shon [15] 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 [16]). 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) ;(3);(4).
In (7d), and , .

Zhang and Yang [17] investigated 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), where denotes the exponent of convergence of fixed points of .

Theorem D. If is a nonconstant meromorphic solution of difference equation (7d), where and is a nonzero constant, then(i) has no nonzero Nevanlinna exceptional value;(ii) cannot be a rational function;(iii).

In Theorems C and D, is defined as a nonzero constant. A natural question to ask is what can we say on meromorphic solutions of (7a)–(7d) if is a nonconstant meromorphic function? In this paper, we answer this question. In the following theorems, we study the properties of difference and divide difference of transcendental meromorphic solutions of (7a)–(7d).

Theorem 1. Suppose that is a nonconstant rational function. If is a transcendental meromorphic solution with finite order of equation set . Then(i) has no Nevanlinna exceptional value;(ii).

Example 2. The function is a meromorphic solution of difference equation where . By calculation, this solution satisfies Thus,

Theorem 3. Suppose that is a nonconstant rational function. If is a transcendental meromorphic solution with finite order of equation then(i) has no Nevanlinna exceptional value;(ii).

From the following proof of Theorem 3, we have the following.

Remark 4. If is an admissible meromorphic solution with finite order of (12), then .

Example 5. The function is a meromorphic solution of difference equation where . By calculation, this solution satisfies Thus,

Theorem 6. Suppose that is a nonconstant rational function. If is a transcendental meromorphic solution with finite order of equation then(i) has no Nevanlinna exceptional value;(ii).

Theorem 7. Suppose that is a nonconstant rational function. If is a transcendental meromorphic solution with finite order of equation then(i) has no Nevanlinna exceptional value;(ii).

From the following proof of Theorem 7, we see the following.

Remark 8. If is an admissible meromorphic solution with finite order of (17), then .

Example 9. The function is a meromorphic solution of difference equation where . By calculation, this solution satisfies Thus,

From the following proofs of Theorems 1–7, we point out the following.

Remark 10. Suppose that is a meromorphic function satisfying . If is an admissible meromorphic solution with finite order of (7d), where , then Theorems 1–7 still hold.
Equations (7a)–(7c) and can be discussed similarly; we omit it in the present paper.

2. Lemmas for the Proofs of Theorems

Lemma 11 (see [9]). Let be a meromorphic function of finite order and let be a nonzero complex constant. Then

Lemma 12 (see [9]). Let be a meromorphic function with order , and let be a fixed nonzero complex number, then for each , we have

Lemma 13 (see [9]). Let be a meromorphic function with exponent of convergence of poles , and let   be fixed. Then for each ,

Lemmas 11 and 12 show the following.

Lemma 14. Let be a nonzero constant and let be a finite order meromorphic function. Then

Lemma 15 (Valiron-Mohon'ko [18]). Let be a meromorphic function. Then for all irreducible rational functions in , with meromorphic coefficients being small with respect to , the characteristic function of satisfies

Lemma 16 (see [10, 11]). Let be a transcendental meromorphic solution with finite order of difference equation where is a difference polynomial in . If   for a meromorphic function , then

Lemma 17 (see [11]). Let be a transcendental meromorphic solution with finite order of a 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 ,

3. Proofs of Theorems

Proof of Theorem 1. (i) Set . Since is a nonconstant rational function, for any , we know . Lemma 16 gives , which follows . Thus, .
From (8), we have that Applying Lemma 17 to (31), we know which implies . Thus, .
Therefore, for any . So, has no Nevanlinna exceptional value.
(ii) First, we prove that . By (8) and Lemma 12, we obtain Hence, From (34) and Lemmas 11 and 12, we deduce that Thus, , that is, .
By (8) and (31), we know
Set Thus, (36) can be written as . Set . Since is a nonconstant rational function, cannot be a periodic function. Then . Since , by (37) and Lemmas 12 and 16, we have Thus, By (34) and (39), we have Then, , that is, .
Next, we prove . By (8), Applying Lemmas 12 and 15 to (41), we have Hence,
Obviously, it follows from (32) and Lemma 11 that Together with (43), we have which yields . That is, .
Set in (i). By (39), we obtain Combining this with (43), we have Then , that is, .

Proof of Theorem 3. (i) By (12) and Lemma 11, we see that Hence, So, .
Set Since is a nonconstant rational function, for any , we have . Lemma 16 gives , which follows . Thus, . Combining with , we know has no Nevanlinna exceptional value.
(ii) First, we prove . Since , , by (12), we have that is,
Let be a zero of , not pole of . From (52), is a zero of or . Since , then must be a zero of or . Thus, by (50) and Lemma 14, we obtain Hence, , that is, .
If is a pole of with multiplicity , not pole of , then is a pole of with multiplicity . From (53), one of and must have the pole with multiplicity not less than . Thus, by (49) and Lemma 13, we get Hence, , that is, .
Next, we prove that . By (12), we have From (56) and Lemmas 11 and 12, we deduce that Thus, .
Since is a nonconstant rational function, cannot be a periodic function. Thus, by (51), . Lemma 16 gives , which follows By (56), if is a common zero of and , then must be a zero of . Thus, by (56), (58), and Lemma 14, we have Hence, , that is, .

Proof of Theorem 6. (i) Set . Since is a nonconstant rational function, for any , we have . Lemma 16 shows , which yields . Thus, .
We see from (16) and Lemma 17 that which follows ; thus, .
Therefore, for any . So, has no Nevanlinna exceptional value.
(ii) First, we prove . By (16) and Lemma 12, we have Thus, We deduce from (62) and Lemmas 11 and 12 that Then . So, .
By (62), we obtain By (60), (64), and Lemma 11, we have Then , that is, .
Next, we prove that . By (16), we know By this and (16), we have
Set Substituting (68) into (67), we have . Set . Since is a nonconstant rational function, cannot be a periodic function. Thus, . By this and by (68) and Lemmas 12 and 16, we obtain That is, By (62) and (70), we have Thus, , that is, .
Set in (i). By (70), we have Thus, by (64), Hence, , that is, .

Proof of Theorem 7. The proof of (i) is similar to the proof of (i) in Theorem 6; we omit it here.
(ii) We conclude from (17) and Lemmas 12 and 15 that Thus, By (75) and Lemma 11, we know Therefore, .
By (17), we know By this and (17), we have
Set Then (78) can be written as . Set . Since is a nonconstant rational function, cannot be a periodic function. Thus, . Since , by Lemmas 12 and 16, we have thus, By this and (75), we have Then .

We see from (76) that

We deduce from (82) that The last two inequalities show and , respectively. Thus, .

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).