Exponential Polynomials and Nonlinear Differential-Difference Equations

Extensive application of Nevanlinna theory has prompted scholars to acquire a number of results on differential equations, difference equations, and differential-difference equations. In this paper, we assume readers are familiar with the standard notations and fundamental results, see [1–4]. Given a meromorphic function f and a constant c. We take c � 1 for simplicity. Δf(z) � f(z + 1) − f(z) and Δf(z) � Δ(Δ 1f(z)) are the first-order difference operator and n-th order difference operator of f, respectively. We adopt the notations ρ(f) and λ(f) to denote the order and the exponent of convergence of zeros of f, respectively. Recall the definition of exponential polynomial of the form


Introduction and Main Result
Extensive application of Nevanlinna theory has prompted scholars to acquire a number of results on differential equations, difference equations, and differential-difference equations. In this paper, we assume readers are familiar with the standard notations and fundamental results, see [1][2][3][4].
Given a meromorphic function f and a constant c. We take c � 1 for simplicity. Δf(z) � f(z + 1) − f(z) and Δ n f(z) � Δ(Δ n− 1 f(z)) are the first-order difference operator and n-th order difference operator of f, respectively. We adopt the notations ρ(f) and λ(f) to denote the order and the exponent of convergence of zeros of f, respectively.
In 2012, Wen et al. [17] classified finite-order entire solutions of the following nonlinear difference equation: where n ≥ 2 is an integer and q(z), Q(z), and P(z) are polynomials such that q(z) is not identically zero and Q(z) is not a constant. ey obtained the following result.
Theorem 1 (see [17]). Let n ≥ 2 be an integer, c ∈ C, and q(z), Q(z), and P(z) be polynomials such that q(z) is not identically zero and Q(z) is not a constant. en, the finiteorder entire solutions f of equation (3) (1), In 2016, Liu [9] investigated finite-order transcendental entire solutions of the following nonlinear differential-difference equation: where n ≥ 2 and k ≥ 1 are integers and q(z), Q(z), and P(z) are polynomials such that q(z) is not identically zero and Q(z) is not a constant. He obtained a result which is similar to eorem 1.
In 2019, Chen et al. [6] considered solutions of equation (3), where P(z) is replaced by p 1 e λz + p 2 e − λz . ey obtained the following result.
Theorem 2 (see [6]). Let n ≥ 3 be an integer and c, λ, p 1 , and p 2 be nonzero constants. Suppose q(z) and Q(z) are polynomials such that q(z) is nonvanishing and Q(z) is not a constant. If f is an entire solution of finite order of then the following conclusions hold: If a solution f belongs to Γ 0 , then q(z) must be a constant and one of the following two relation groups holds: where both b and B are constants. Remark 1. Chen et al. [6] gave an example: f(z) � e z is an entire solution of finite order of the following difference equation: From the example, they conjectured that the conclusions of eorem 2 are still valid if n � 2.
We consider the conjecture and prove a more generalized result. Moreover, we solve Chen, Gao, and Zhang's conjecture when f(z) is an exponential polynomial of form (1). Theorem 3. Let k ≥ 0 be an integer and c, α 1 , α 2 , p 1 , and p 2 be nonzero constants such that α 1 ≠ α 2 . Suppose q(z) is a nonvanishing polynomial and Q(z) is a nonconstant polynomial. If the differential-difference equation has a transcendental entire solution f, then (2) If f is an exponential polynomial of form (1), then (3) If f belongs to Γ 0 ′ , then one of the following two relation groups holds: where both b and B are constants and g(z) is a polynomial.

Corollary 1.
Let k ≥ 0 be an integer and c, α 1 , α 2 , p 1 , and p 2 be nonzero constants such that α 1 ≠ α 2 . Suppose q(z) is a nonvanishing polynomial and Q(z) is a nonconstant polynomial. If the differential-difference equation (7) has solutions f satisfying f ∈ Γ 0 , then ρ(f) � deg Q � 1 and q(z) must be a constant and one of the following two relation groups holds: and q(α 1 /2) k e (α 2 c/2)+b+B � p 2 , where both b and B are constants.
Next, we give two examples to illustrate equation (7).
In 2015, Zhang et al. [18] studied the existence of entire solutions of the following nonlinear difference equation: (10) ey obtained the following result.
Theorem 5. Suppose that p 1 , p 2 , and λ are nonzero constants and that a 1 (z) and a 2 (z) are nonzero polynomials. If f is a nontrivial exponential polynomial solution of then f has solutions of finite order of the following form: where Moreover, one of the following conclusions holds: , and a 1 (z) and a 2 (z) satisfy Especially, if a 1 and a 2 are constants, then then c 0 (z) � a 1 (z)/2 and a 1 (z) and a 2 (z) satisfy Especially, if a 1 is a constant, then c 2 0 � − 2c 1 c 2 . e following examples show the existences of solution of equation (13).

Example 4.
An entire solution f(z) � 2 + e (πi/2)z − 2e − (πi/2)z solves the following difference equation: is paper is organized as follows. In Section 2, we introduce the background of exponential polynomials and some indispensable lemmas. Sections 3 and 4 contain the detailed proofs on eorems 3 and 5. In Section 5, we will discuss the methods of the main results obtained in the paper.

Preliminaries
We recollect a basic result on exponential polynomials. Let For exponential polynomials f(z) of form (1), Wen et al. [17] followed the reasoning in [19] and acquired some instrumental tools.
Suppose the polynomials Q j (z) in (1) are pairwise different and normalized by Q j (0) � 0. en, representation (1) is uniquely determined and the functions P j (z)e Q j (z) are linearly independent. Let and let w 1 , w 2 , . . . , w m be pairwise different leading coefficients of the polynomials Q j (z) of maximum degree q. us, (1) can be written in the following normalized form: where H i (z)(0 ≤ i ≤ m) are either exponential polynomials of degree less than q or ordinary polynomials in z.
A convex hull of a set W ⊂ C, denoted by co(W), is the intersection of all convex sets containing W. If W contains only finitely many elements, then co(W) is obtained as an intersection of finitely many closed half-planes. Hence, co(W) is either a compact polygon (with a nonempty interior) or a line segment. We denote the perimeter of co(W) by C(co(W)). If co(W) is a line segment, then C(co(W)) equals to twice the length of this line segment. We fix the notation for Theorem 6 (see [19], Satz 1]). Let f be given by (23). en, Next, we can find the following consequence from the result of Steinmetz ([20], Satz 1), i.e., holds for an exponential polynomial f(z) in form (23) (also see [21], Section 3). Some auxiliary results are necessary. e first one is a difference analogue of logarithmic derivative lemma given by Chiang and Feng.
Lemma 1 (see [22], Corollary 2.5). Let f(z) be a meromorphic function with finite order ρ(f). Suppose c is a fixed nonzero complex constant. en, for each ε > 0, we have e following lemma is a useful tool to solve differentialdifference equations and difference equations.
Halburd and Korhonen proved a difference analogue of Clunie lemma under the condition finite order.
Lemma 3 (see [23]). Let f(z) be a nonconstant finite-order meromorphic solution of where P(z, f) and Q(z, f) are difference polynomials in f with small meromorphic coefficients. Suppose c ∈ C and δ < 1.
If the total degree of Q(z, f) is a polynomial in f and its shifts are less than or equal to n, then for all r outside of a possible exceptional set with finite logarithmic measure.
Remark 2. Similar to Lemma 3, if f is a transcendental exponential polynomial in form (23), P(z, f) and Q(z, f) are differential-difference polynomials in f and the coefficients of P(z, f) and Q(z, f) are polynomials a i (z)(i � 1, 2, . . . , n), for each ε > 0, then an obtained result is where r is sufficiently large. Chen and Yang proved the following lemma.
Lemma 4 (see [24]). Let λ be a nonzero constant and H(z) be a nonvanishing polynomial. en, the differential equation has a special solution c 0 (z) which is a nonzero polynomial.
In addition, the following lemma is similar to Lemma 5.3 of [17] and Lemma 2.7 of [9]. e proof can be given word by word.
Lemma 5. Let f be given by (23), where q ≥ 2. If f is a solution of equation (7), then m � 1.

Proof of Theorem 3
Proof of Conclusion 1. Suppose that f(z) is a finite-order entire solution of equation (7). Applying the lemma on the logarithmic derivative and Lemma 1 to equation (7), we obtain 4 Journal of Function Spaces us,

Proof of Conclusion 2.
Since f is an exponential polynomial in form (1), we can consider its equivalent form (23). Suppose q ≥ 2, by Lemma 5 we know m � 1. at is, we have f(z) � H 0 (z) + H 1 (z)e w 1 z q . Substituting the expression of f(z) into equation (7) yields where Q 0 (z) � Q(z) − b q z q and P 1 (z) � w 1 (z + c) q − w 1 z q . In addition, H T 1 (z + c) is a differential polynomial in H 1 (z + c), w 1 (z + c) q , and their derivatives. We see that Q 0 (z) and P 1 (z) are two polynomials with degree less than or equal to q − 1. We discuss two cases b q ≠ w 1 and b q � w 1 : Taking b q � − w 1 , b q � 2w 1 and b q ∉ ±w 1 , 2w 1 , respectively, we apply Lemma 2 to equation (35) to obtain H 1 (z) ≡ 0, which is a contradiction. Case 2. b q � w 1 . Equation (35) can be rewritten as We utilize Lemma 2 again to obtain Assume that z 0 is a zero of the above equation. Obviously, z 0 is a simple zero of p 1 e α 1 z + p 2 e α 2 z , but z 0 is the multiple zero of (H 0 (z)) 2 . is is a contradiction. We have where a and A are nonzero constants, b and B are constants, and g(z) is a nonvanishing polynomial. It follows from formula (38) that Substituting formulas (38)-(40) into equation (7), we have Journal of Function Spaces We consider the following four cases: Using Lemma 2, it follows from equation (41) that p 1 � 0. It is a contradiction. (41) and Lemma 2. A contradiction occurs. Now, we consider that only two of 2A − α 2 , A + a − α 2 , and α 1 − α 2 coincide. Without loss of generality, assuming 2A − α 2 � A + a − α 2 ≠ α 1 − α 2 , we see that equation (41) is represented as From the above equation, using Lemma 2, we have p 1 � p 2 � 0, which implies a contradiction.

Proof of Theorem 5
Assume that the difference equation (13) has a transcendental entire solution f of finite order.

Substituting equation (54) into equation (56) yields
where T 3 (f) � 4f 2 Q 1 (f) + R 3 (f) is a differential-difference polynomial in f and its total degree is not greater than three. Now, we discuss two cases.
e general entire solution f(z) of the above equation is where c 1 and c 2 are constants satisfying c 1 c 2 ≡ 0. We obtain

Conclusions
In this study, we mainly consider the solution of two equations when the solution is an exponential polynomial. First, we consider the nonlinear differential-difference equation (7) proposed by Chen et al. [6]. ey conjecture that the conclusions of eorem 2 are still valid. We consider the conjecture in eorem 3. In the first step, we proved that ρ(f) � deg Q. From this, it seems plausible that f is an exponential polynomial of form (1). In the second step, we confirmed that ρ(f) � deg Q � 1 when f is an exponential polynomial. In the last step, we give the solution when f belongs to Γ 0 ′ by Conclusion 2.
Second, we consider a difference equation which is similar to (10), where f 3 (z) is also replaced by f 2 (z). Since we cannot prove that 4f ″ − λ 2 f is a polynomial if f has no restriction, a new Clunie Lemma is given in Remark 2 where f is an exponential polynomial. We obtain the expression of the solution of equation (13) if the solution is an exponential polynomial by the special Clunie Lemma.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.