Uniqueness of exponential polynomials

: In this article, we study the uniqueness of exponential polynomials and mainly prove: Let n be a positive integer, let = p z i n 1, 2,…, i ( ) ( ) be nonzero polynomials, and let ≠ = c i n

Abstract: In this article, we study the uniqueness of exponential polynomials and mainly prove: Let n be a positive integer, let ( ) ( ) ( ) ( ) .If f z ( ) and g z ( ) share a and b CM (counting multiplicities), where a and b are two distinct finite complex numbers, then one of the following cases must occur:

Introduction
In this article, an exponential polynomial is an entire function of the form where n is a positive integer, the coefficients In the following, we assume that the reader is familiar with the basic notions of Nevanlinna's value distribution theory [1][2][3].A meromorphic function always means meromorphic in the whole complex plane.
By S r f , ( ), we denote any quantity satisfying = S r f o T r f , , ( ) ( ( )) as → ∞ r possible outside of an exceptional set E with a finite logarithmic measure ∫ ∕ < ∞ r r d A meromorphic function a is said to be a small function of f if it satisfies = T r a S r f , , ( ) ( ).Let f be a nonconstant meromorphic function.The order of f is defined by and the difference operators are defined as follows: where ≥ n 2 ( ) is a positive integer and c is a nonzero constant.Let f and g be two meromorphic functions, and let a be a constant.If − f a and − g a have the same zeros counting multiplicities ignoring multiplicities (IM), we say that f and g share a CM (IM).N r a , ( ) is a counting function of common zeros of − f a and − g a with the same multiplicities, and the multiplicities are counted.
Recently, many articles studied the value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory [4][5][6].In addition, many articles studied the roots of exponential polynomials [7,8].In 2019, Su et al. [9] proved that if two nonconstant exponential polynomials with constant coefficients share four distinct values CM that lie in an angular domain of opening strictly larger than π, they must be identical.In this article, we also study the uniqueness of exponential polynomials.
This was proved in 1974 by Rubel and Yang [10].
Theorem A. Let f be an entire function.If f z ( ) and z sin share 0 and 1 CM, then In 1981, Czubiak and Gundersen [11] proved that Theorem A remains valid if f z ( ) and z sin share 0 and 1 IM.In this article, we extended Theorem A and obtained the following results.
Theorem 1.Let n be a positive integer, let ) be polynomials, and let share a and b CM, where a and b are two distinct finite complex numbers, then one of the following cases must occur: Remark 1.All the cases in Theorem 1 must occur, then we provide two examples to show that the cases of Theorem 1 occurs.
Example 1.Let p 1 and c 1 be two nonzero complex numbers, and let a be a nonzero distinct finite complex

( )
. Obviously, f z ( ) and g z ( ) share a and 0 CM, and Example 2. Let p 1 and c 1 be two nonzero complex numbers, and let a and b be two nonzero distinct finite

Some lemmas
In order to prove our results, we need the following lemmas.
, then there exists an entire function α z Lemma 2. [12] Let f z ( ) and g z ( ) be two nonconstant entire functions, and let a be a nonzero constant.If f z ( ) and g z ( ) share a CM almost and Lemma 3.
[13] Let ≥ n 3 be a positive integer, and let = f z j n 1, …, j ( )( ) be meromorphic functions which are not constants except for f z n ( ).Furthermore, let ∑ ≡ = f z [14] Let f be a nonconstant entire function of finite order, let k be a positive integer, and let a be a nonzero constant.If f and f k ( ) share a CM, then for some nonzero constant C.
Lemma 5. [15] Let α be a meromorphic function, let n be a positive integer, and let c be a nonzero finite complex number.If or α is a polynomial with ≤ − α n deg 1.

Proof of Theorem 1
Since f z ( ) and g z ( ) share a and b CM, then by Lemma 1, we obtain Uniqueness of exponential polynomials  3 where α z ( ) and β z ( ) are two entire functions.By Nevanlinna's first and second fundamental theorems, we have Similarly, we obtain By equations (3.2) and (3.3) and the definition of order, we deduce that From equation (3.1) and above, we have ( ) ( ), y 1 , and y 2 are constants.Now, we consider two cases.
We claim that there exists . Since ≠ b 0, then by Nevanlinna's second fundamental theorem, we obtain In the following, we consider two subcases.
Obviously, there does not exist z 0 such that . Thus, by equation (3.4) and (3.5), we deduce ≡ f g.
, then by ≠ a b, we have ≠ x x 1 2 .From equation (3.9), we obtain + = y c 0.  that , then by Nevanlinna's second fundamental theorem, we obtain a contradiction. Case In the following, we consider two subcases.Hence, ≡ f g.

If ≠
x x Now, we consider two subcases.Case b.1.
By Lemma 3 and mathematical induction, we obtain a contradiction.

Case b.1.2. There exists
x p z e x p z e bx e ax e b a x p z .
If ≠ b 0, then by mathematical induction, we obtain a contradiction.If = b 0, we obtain ≠ a 0, then by equation (3.33), we have Similarly, we obtain a contradiction.

Case b.2. There exists
Using the same argument as used in Case b.1.2,we obtain a contradiction.

Case b.2.2. There exists r
x p z e x p z e bx e ax e b a x p z x p z .
By mathematical induction, we obtain a contradiction.Hence, by equation (3.19), we obtain a contradiction.This completes the proof of Theorem 1.
By Nevanlinna's second fundamental theorem and Lemma 3, we obtain a contradiction.Thus, Theorem 2 is proved.

Proof of Theorem 3
Since f and Δ f c m share a CM, then by Lemma 1, we have where α z ( ) is an entire function.By the definition of order, we obtain . By equation (5.4), we obtain . Then, we obtain ≡ H z p z i i ( ) ( ).Otherwise, by Nevanlinna's second fundamental theorem and Lemma 3, we obtain a contradiction.Hence, we have From equation (5.6), we deduce If m is an even number, then by equation (5.6), we have Since m is an even number, then by equation (5.7), we obtain ).It follows from equations (5.8) and (5.9) that    Similar to Theorem 1, we obtain a contradiction.Thus, Theorem 3 is proved.
of exponential polynomials, and their quotient being z tan .

y 0 1 .
2. = a 0 and ≠ b 0. Using the same argument as used in Case 1.1, we obtain that either ≡ f g or ≡ fg b 2 .Case 1.3.≠ a 0 and ≠ b 0. Next, we consider three subcases.Case 1.3.1.= By equation (3.4), we have

2 .
Nevanlinna's second fundamental theorem and Lemma 3, we obtain a contradiction.f and g share b CM, we have = Using the same argument as above, we obtain ≡ f g.Case 2

. 2 .
.28) Using the same argument as used in Case a.1.1,we deduce that − a and b are distinct, without loss of generality, we assume that ≠ a 0, and then, by equation (3.28) we have Using the same argument as used in Case a.2, we obtain a contradiction.Therefore, from equation (3.19), we obtain a contradiction when = n Suppose that the fact is valid when ≤ n k.When = n k, from equation (3.19), we have i
and b are distinct, without loss of generality, we assume that ≠ a 0, then by equation(3 y are two constants.It follows from equations (5.1) and (5.2) that − = − Δ f a xe f a .
. If m is an odd number, then by equation (5.6), we obtain m is an odd number, then by equations (5.7) and (5.18), we have follows from equations (5.5) and (5.21) that Using the same argument as used in Case 1.3.1,we obtain a contradiction.