Some Properties on Complex Functional Difference Equations

Abstract

We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form  , where is a finite set of multi-indexes , are distinct complex constants, is a polynomial, and , , and are small meromorphic functions relative to . We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero and pole.

1. Introduction and Main Results

In this paper, a meromorphic function means meromorphic in the whole complex plane . For a meromorphic function , let be the order of growth and the lower order of . Further, let (resp., ) be the exponent of convergence of the zeros (resp., poles) of . We also assume that the reader is familiar with the fundamental results and the standard notations of Nevanlinna theory of meromorphic functions (see, e.g., [1]). Given a meromorphic function , we call a meromorphic function a small function relative to if as , possibly outside of an exceptional set of finite logarithmic measure. Moreover, if is rational in with small functions relative to as its coefficients, we use the notation for the degree of with respect to . In what follows, we always assume that is irreducible in .

Meromorphic solutions of complex difference equations have recently gained increasing interest, due to the problem of integrability of difference equations. This is related to the activity concerning Painlevé differential equations and their discrete counterparts in the last decades. Ablowitz et al. [2] considered discrete equations to be delay equations in the complex plane. This allowed them to analyze these equations with the methods from complex analysis. In regard to related papers concerning a more general class of complex difference equations, we may refer to [3–5]. These papers mainly dealt with equations of the form where is a collection of all nonempty subsets of , are distinct complex constants, is a transcendental meromorphic function, () are small functions relative to , and is a rational function in with small meromorphic coefficients. Moreover, if the right-hand side of (1) is essentially like the composite function of and a rational function , Laine et al. reversed the order of composition; that is, they considered the composite function of and a rational function , which resulted in a complex functional difference equation. The following theorem [5, Theorem 2.8] gives an example.

Theorem A (see [5, Theorem 2.8]). Suppose that is a transcendental meromorphic solution of equation where is a polynomial of degree . Moreover, one assumes that the coefficients are small functions relative to and that . Then where .
At this point, we briefly introduce some notations used in this paper. A difference monomial of a meromorphic function is defined as where , are distinct constants, and    are natural numbers. A difference polynomial of a meromorphic function , a finite sum of difference monomials, is defined as where is a finite set of multi-indexes , are small functions relative to . The degree and the weight of the difference polynomial (5), respectively, are defined as Consequently, . For instance, the degree and the weight of the difference polynomial , respectively, are four and three. Moreover, a difference polynomial (5) is said to be homogeneous with respect to if the degree of each monomial in the sum of (5) is nonzero and the same for all .
In the following, we proceed to prove generalizations of Theorem A and investigate some new results for the first time. We permit more general expressions on both sides of (1).

Theorem 1. Let be a transcendental meromorphic solution of equation where is defined as (5), is a polynomial with constant coefficients and of the degree , and and are small meromorphic functions relative to such that . Set . If , then where .

Similar to the proof of Theorem 1, we easily obtain the following result, which is a generation of Theorem A.

Theorem 2. Let be distinct constants and be a transcendental meromorphic solution of equation where is a polynomial with constant coefficients and of the degree and and are small functions relative to such that . If , then where .

We then proceed to consider the distribution of zeros and poles of meromorphic solutions of (7). The following result indicates that solutions having Borel exceptional zeros and poles appear only in special situations.

Theorem 3. Let , let be distinct constants, and let be a finite order transcendental meromorphic solution of equation where is a polynomial with constant coefficients and of the degree and and are small meromorphic functions relative to such that . If then (11) is either of the form or where is some constant.

Example 4. solves difference equation Here . Clearly, . This example shows that condition (12) is necessary and cannot be replaced by
Moreover, we obtain a result parallel to Theorem 5.4 in [6] for the difference case.

Theorem 5. Suppose that the equation has a meromorphic solution of finite order, where , are distinct constants, and is a nontrivial meromorphic function. If has only finitely many poles, then , where is a rational function and is a polynomial, if and only if , where is a rational function and is a polynomial.

Example 6. Difference equation of the type (17) is solved by . Here, and satisfy Theorem 5.
As an application of Theorem 3, we obtain the following.

Theorem 7. Let and let be a finite order transcendental meromorphic solution of equation where is a polynomial with constant coefficients and of the degree and and are small meromorphic functions relative to such that . Then has at most one Borel exceptional value.

If the degree of polynomial is in Theorem 7, the result does not hold. For example, we have the following.

Example 8. solves difference equation of the type (19). Obviously, has two Borel exceptional values .
If we remove the assumption used in Theorem 3, we obtain a result similar to Theorem 12 in [4].

Theorem 9. Let be a transcendental meromorphic solution of equation where is defined as (5) and and are small meromorphic functions relative to such that . If , then .
In fact, the following examples show that the assertion of Theorem 9 does not remain valid identically if .

Example 10. solves the difference equation Clearly, and .

Example 11. satisfies the difference equation Obviously, and .

Example 12 (see [7, pages 103–106] and [8, page 8]). The following difference equation, derives from a well-known discrete logistic model in biology. It has been proved that all other meromorphic solutions are of infinite order, apart from the constant solutions and . For instance, (24) has one-parameter families of entire solutions of infinite order: Here, .

Example 13. solves the difference equation We get and .
If the difference polynomial in the left-hand side of (21) is homogeneous, we further obtain the following theorem.

Theorem 14. Let be a transcendental meromorphic solution of (21), where is defined as (5) and and are small meromorphic functions relative to such that . Suppose that is homogeneous and has at least one difference monomial of type If , then .

2. Proof of Theorem 1

We need some preliminaries to prove Theorem 1.

Lemma 15 (see [9, Lemma 4]). Let be a transcendental meromorphic function and let be a polynomial of degree . Given , denote and . Then, given and , one has, for all ,

Lemma 16 (see [10, Theorem  ]). Given distinct meromorphic functions , let denote the collection of all nonempty subsets of , and suppose that for each . Then

By denoting below, it is an easy exercise to prove the following result from Lemma 16.

Lemma 17. Let be a meromorphic function, let be a finite set of multi-indexes , and let be small functions relative to for all . Then the characteristic function of the difference polynomial (5) satisfies where .

Lemma 18 (see [11, Lemma 5]). Let and be monotone nondecreasing functions on such that for all , where is a set of finite logarithmic measure. Let be a given constant. Then there exists an such that for all .

Lemma 19 (see [12, Lemma 3]). Let be a function of , positive and bounded in every finite interval.(i)Suppose that , where , , , and are constants. Then with , unless and ; and if and , then for any .(ii)Suppose that (with the notation of (i)) . Then for all sufficiently large values of , with for some positive constant .

Proof of Theorem 1. For any , we may apply Valiron-Mohon’ko lemma, Lemmas 15 and 17, and (5) and (7) to conclude that holds for all sufficiently large , possibly outside of an exceptional set of finite logarithmic measure, where and is defined as Lemma 15. Now, we may apply Lemma 18 to deal with the exceptional set and conclude that, for every , there exists an such that holds for all . Denote . Then (32) can be written in the form Since , we get for all . Thus, we now apply Lemma 19(i) to conclude that The proof of Theorem 1 is completed.

3. Proof of Theorems 3 and 5

We again need some preliminaries.

Lemma 20 (see [13, Theorem 1.5]). Suppose that are meromorphic functions and are entire functions satisfying the following conditions.(1).(2) are not constants for .(3)For , , where is of finite linear measure or finite logarithmic measure.
Then .

Lemma 21 (see [14, Theorem 4]). Let , be polynomials such that and then every finite order transcendental meromorphic solution of equation satisfies .

Remark 22. Replacing by , where are distinct nonzero complex constants, Lemma 21 remains valid.

Proof of Theorem 3. Let be the multiplicity of pole of at the origin, and let be a canonical product formed with nonzero poles of . Since , then is an entire function such that and is a transcendental entire function with
If is a polynomial, we obtain quickly that . Otherwise, we conclude from the last assertion of Lemma 15, (37), and (38) that Therefore, Now, substituting into (11), we conclude that Obviously, it follows from (37)–(40) and Lemma 15 that Denoting , we get from (42) that Since zeros and poles are Borel exceptional values of by (12), we may apply a result due to Whittaker; see [15, Satz 13.4], to deduce that is of regular growth. Thus, we use Lemma 15 and (12) again to get Similarly, if we set , we also deduce from the lemma of the logarithmic derivative, Lemma 15, (12), (38), and (43) that Denoting , Therefore, we deduce from Lemma 15 and (42) that the coefficients of and are small functions relative to . Thus, (41) can be written in the form Denoting we get from (45) and (47). We also conclude from the lemma of logarithmic derivative, Lemma 15, and (12) that where is defined as Lemma 15.
Since we conclude that Now, writing in (51), regarding then (51) as an algebraic equation in with coefficients of growth , and comparing the leading coefficients, we deduce that By integrating both sides of the last equality above, we conclude that for some . Therefore, by combining the representations of with (53), we conclude that If , we deduce from (11) and (54) that From this, we get that is not irreducible in , a contradiction. Thus, or . Therefore, we deduce from (54) that or The proof of Theorem 3 is completed.

Proof of Theorem 5. Assume first that , where is a rational function and is a polynomial. One can see from (17) that where is rational and is a polynomial.
Suppose next that , where is a rational function and is a polynomial. Since has only finitely many poles, we conclude from (17) that Thus, has only finitely many zeros and poles, and , where is rational and is an entire function. In the following, we only prove is a polynomial. Now, substituting and into (17), we get Thus, we deduce from Lemma 20 that two exponents in (61) cancel each other to a constant such that that is, Suppose that is not a polynomial. If is a transcendental entire function of finite order, we get from Lemma 21, Remark 22, and (63) that . Otherwise, is a transcendental entire function of infinite order. These both show that , contradicting the assumption that is finite order. Thus, is a polynomial. The proof of Theorem 5 is completed.

4. Proof of Theorem 7

Lemmas 23 and 25 reveal some properties of the maximal module of the polynomial in composite function with a meromorphic function and a polynomial , which are useful for proving the existence of Borel exceptional value of finite order meromorphic solutions of functional difference equation of type (19).

Lemma 23. Let be a nonconstant entire function of order . Suppose that are small meromorphic functions relative to . Then there exists a set of lower logarithmic density 1 such that hold simultaneously for all as , where the lower logarithmic density of set is defined by

Remark 24. The proof of Lemma 23 is similar to the proof of Lemma 2.4 and Remark 2.5 in [16]. Here, we omit it.

Lemma 25. Let be a finite order transcendental meromorphic function satisfying (12), and is a polynomial with constant coefficients and of the degree . Suppose that is a polynomial in , where is a positive integer and , are small meromorphic functions relative to . Then there exists a set of lower logarithmic density 1 such that for all as , where . Hence, .

Proof of Lemma 25. Let be the multiplicity of pole of at the origin, and let be a canonical product formed with the nonzero poles of . Since satisfies (12), then is an entire function. Thus, is entire, and (37), (38), and (40) also hold.
Now, substituting into (66), we conclude that
We note from Lemma 15 and (40) that Therefore, we deduce from Lemma 23 that there exists a set of lower logarithmic density 1 such that Moreover, according to the choosing of in the proof of Lemma 23, we know that for have no zeros and poles for all . Thus, we conclude from (68) and (70) that, for any , and so for all and .
Therefore, we deduce from Lemma 15 and (38) that for all , where . It is obvious that has lower logarithmic density 1. The proof of Lemma 25 is completed.

Proof of Theorem 7. Suppose that has two finite Borel exceptional values and . For the case where one of and is infinite, we can use a similar method to prove it. Set Then and It follows from (74) that Now, substituting (76) into (19), we conclude that Since and are irreducible in , we conclude that at least one of the following three inequalities holds; that is, Thus, we deduce from Theorem 3 that where is meromorphic function satisfying and . Clearly, and is of regular growth from (75); see [15, Staz 13.4]. Therefore, .
If , we conclude from (77) and (79) that Thus, we deduce from Lemma 25 that (80) is a contradiction. If , we use the same method as above to get another contradiction. Therefore, has at most one Borel exceptional value. The proof of Theorem 7 is completed.

5. Proof of Theorems 9 and 14

We first recall two lemmas.

Lemma 26 (see [17, Lemma 2.1]). Let be a nonconstant meromorphic function, ,  , and the set of all such that If the logarithmic measure of is infinite, that is, , then is of infinite order of growth.

Lemma 27 (see [18, Corollary 2.6] and [19, Corollary 2.2]). Let be a meromorphic function of finite order, and let . Then for all outside of a possible exceptional set of finite logarithmic measure.

Proof of Theorem 9. For any , we may apply Valiron-Mohon'ko lemma, Lemma 17, (5), and (21) to conclude that for all outside of a possible exceptional set of finite logarithmic measure.
Denote Then since and . Thus, holds for all in a set with infinite logarithmic measure. Therefore, we deduce from Lemma 26 and (85) that . The proof of Theorem 9 is completed.

Proof of Theorem 14. Assume, contrary to the assertion, that is meromorphic of finite order. Taking into account the assumption that is homogeneous, we deduce from Lemma 27 that for all outside of a possible exceptional set of finite logarithmic measure.
Denote . Since is homogeneous and has at least one difference monomial of type , we immediately conclude that, by looking at pole multiplicities, summing over , and integrating logarithmically, Therefore, for all outside of a possible exceptional set of finite logarithmic measure. The remainder can be proven by a similar method in Theorem 9. The proof of Theorem 14 is completed.

Conflict of Interests

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

Acknowledgments

The authors are grateful to the referees for their helpful suggestions to improve this paper. The first author also thanks Professor Ilpo laine and Professor Risto Korhonen for their valuable suggestion to the present paper. Research is supported by National Natural Science Foundation of China (nos. 11171119 and 11171121) and Guangdong National Natural Science Foundation (no. S2012040006865).