Abstract
In this paper, we prove the difference equation does not have meromorphic solution of finite order over the complex plane . We also discuss an application to the unique range set problem.
1. Introduction and Main Result
This kind of questions is from the analogues to the Fermat diophantine equations
for a positive integer . When , Gross [1] and Baker [2] proved that (1) does not nonconstant meromorphic solution in the complex plane .
Gross [1] and Baker [2] also showed for there are no entire solutions. For the case , Gross [1] and Baker [2] also got the meromorphic solutions such as
and
where is a Weierstrass -function.
Now the equation defines an algebraic function whose Riemann surface has genus 1, and there is accordingly a uniformization by Weierstrass elliptic functions. Weierstrass elliptic function is a meromorphic function with double periods and defined as
which is even and satisfies, after appropriately choosing and , to the form
In the same paper, Gross [1] conjectured all meromorphic solutions of are necessarily elliptic functions of entire functions. The conjecture was proved by Baker [2]. He proved.
Theorem 1. Any function , which are meromophic in the complex plane and satisfyhave the formwhere is an entire function and is a cube-root of unity.
We assume that the reader is familiar with the standard notations and results such as the proximity function , counting function , characteristic function , the first and second main theorems, lemma on the logarithmic derivatives etc. of Nevanlinna theory, (see [3, 4]). Given a meromorphic function , we shall call a meromorphic function a small function of if , where is used to denote any quantity that satisfies as , possibly outside of a set of finite logarithmic measure. Here, the order is defined by
Now, the estimate the order of , Bank and Langley [5] indicates that
Nevanlinna value distribution theory of meromorphic functions has been extensively applied to resolve growth and solvability of meromorphic solutions of linear and nonlinear differential and difference equations (see [4, 6–10]).
Below, we list some well-known facts concerning the order of composite meromorphic functions that can be found in Edrei and Fuchs [11], and Bergweiler [12].
Theorem 2. Let be meromorphic and be entire in . When , then and is transcendental, then .
In recent years, Nevanlinna characteristic of (), the value distribution theory for difference polynomials, Nevanlinna theory of the difference operator and the difference analogue of the lemma on the logarithmic derivative had been established, see e.g., [13–22]. Due to these theories, there has been a recent study on whether the derivative or can be replaced by the difference operator or in the above question.
We may want to study all meromorphic solutions of the following difference equation
for a fixed nonzero constant [23, 24]. Shimomura [25] proved that the (10) has an entire solution of infinite order for the ,. Later, Liu et al. [26] proved that Eq. (10) has no transcendental entire solutions of finite order when . If , Liu et al. [27] proved that the solutions of (10) are periodic functions of period , and proved that (10) has transcendental entire solutions of finite order for .
For a meromorphic functions , we define its difference operators by
Recently, Lü and Han [28] described a property of meromorphic solutions to equation in (6) with , for , as the following result.
Theorem 3. The difference equation does not have meromorphic solution of finite order.
It is natural to ask whether the shift can be replaced by in above Theorem 3. In this paper, based on the ideas of [28], we mainly consider this problem and prove the following results.
Theorem 4. The difference equation does not have meromorphic solution of finite order.
Proof. It follows from (2) and (3) that Then we haveRewrite it asWe knowAssume . Then, combining (13), (14) with (5), one hasFinally, we getSoand . Note that is a of finite order and (9) and Theorem 2 combined force to be a polynomial.
Notice when , then . By (5) Now, write all the zeros of by and as . Note that is a polynomial, then that all the zeros of are simple zeros of for , where is a positive integer, and for . Now, we divide two steps to prove our result.
Step 1. If there exists a subsequence of (with , , which we still denote it by ) such that . Then we have and , so we also have is a simple zero of .
Differential (14), we haveSubstitute (for enough large ) into the above equation and by , , we haveNoting that and . Without of loss generality, together with (14), we also assume there exists a sub-sequence of (here we still denote it by ) such that four cases:
Case 1. If and , we getCase 2. If and , we getCase 3. If and , we getCase 4. If and , we getNoting that are polynomials and infinite many (with when ), we would have to getEither or , and by comparing the highest coefficient of the both polynomials of the above equations, These are simply impossible.
So there exists a positive integer satisfying for .
Step 2. In view of and , we have that for .
Thus, we get the proposition that the zeros of are the poles of except for finitely many points. This is to saywhere is a positive number. Noting that is transcendental function, so and By and , we havewhich implies that and . By the equation , we deduce that all zeros of , and are with multiplicities at least 3.
Then, we have which implies that . It leads to .
Note the form of , we have . Furthermore, we haveThen, all the above discussions yield that a contradiction.
Thus, we finish the proof of Theorem 4.
Remark 1. It is easy to get the solution of the equation is only for constant. For , the solution of the equation , Liu (see Proposition 5.3 in [19]) proved there is no nonconstant finite order entire solution.
2. An Application
For a meromorphic function and a set , we define
We say that and share a set CM, provided that .
In 1976, Gross [29] posed the following question:
Question 1 (see [29, Question 6]). Can one find two (or possibly even one) finite sets such that any two entire functions and satisfying must be identical?
If the answer to Question 1 is affirmative, it would be interesting to know how large both sets would have to be. Many authors have been considering about it, and got a lot of related results. Some of them are due to Yi [30], Mues and Reinders [31], and Frank and Reinders [32].
Recently, value distribution in difference analogues of meromorphic functions has become a subject of great interest [33, 34]. Zhang [35] obtained the following results.
Theorem B. Let and , where and is a positive integer, let . Suppose that is a nonconstant meromorphic function of finite order such that . If , then , where .
Recently L and Han [28] proved that
Theorem C. Under the conditions of Theorem B and if , then the conclusion of Theorem B still holds.
In this paper, we mainly consider the and share the set which has three elements and get the following result.
Theorem 5. Let , for , and . Take . Let be a meromorphic function of finite order satisfying for . Then, either or .
Proof. By Theorem C, we set . Obviously, and share CM. Then we can assume thatwhere is an entire function. Note that is of finite order, we have is a polynomial. Obviously, .
Rewrite (33) asSimilarly as above, we get just has zeros with multiplicities at least . Furthermore, we have just has zeros with multiplicities at least . Suppose that are distinct from each other. Then, by the second main theorem, we have a contradiction. Thus, there are at lest two functions equal in the setWe consider two cases.
Case 1. or and then , then must be a constant with , so , and then , at last .
Then, we have . It follows from (33) that the assertion holds.
Case 2. . That is .
Noting that is a polynomial, we deduce that is a constant by the assumption of Case 2. Furthermore, and . By (33), we get . Then, it follows from Theorem 1 that the case cannot occur.
Thus, we finish the proof of Theorem 5.
Example 1. Consider . Then for , and . Notice that , share , CM, but .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors typed, read, and approved the final manuscript.
Acknowledgments
The work presented in this paper is supported by the Plateau Disciplines in Shanghai. Also this work was supported by Leading Academic Discipline Project of Shanghai Dianji University (16JCXK02) and Philosophy and Social Sciences Planning Project of the Ministry of Education (Grant No. 18YJC630120).