Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables

1.   Introduction

As is known to all, Nevanlinna theory is an important tool in studying the value distribution of meromorphic solutions on complex differential equations ^[18]. In recent, with the development of difference analogues of Nevanlinna theory in $\mathbb{C}$, many scholars paid consideration attention to considering the properties on complex difference equations, by using the difference analogue of the logarithmic derivative lemma given by Chiang and Feng ^[3], Halburd and Korhonen ^[8], respectively. In particular, Liu et al. ^[22,23,24] investigated the existence of entire solutions with finite order of the Fermat type differential-difference equations

They proved that the transcendental entire solutions with finite order of Eq (1.1) must satisfy $f(z) = \sin(z\pm Bi)$, where $B$ is a constant and $c = 2k\pi$ or $c = (2k+1)\pi$, $k$ is an integer, and the transcendental entire solutions with finite order of Eq (1.2) must satisfy $f(z) = 1 2\sin(2z+Bi)$, where $c = (2k+1)\pi$, $k$ is an integer, and $B$ is a constant.

The study of complex differential-difference equations in $\mathbb{C}$ can be traced back to Naftalevich's research ^[28,29]. He used operator theory and iteration method to consider the meromorphic solutions on complex differential-difference equations. But recently, by using Nevanlinna theory, a number of results on complex differential-difference equations in $\mathbb{C}$ are rapidly obtained until now, readers can refer to ^[25,31,32].

Corresponding to Eq (1.1), Gao ^[5] in 2016 discussed the form of solutions for a class of system of differential-difference equation

and obtained

Theorem A (see [5,Theorem 1.1]). Suppose that $(f_1, f_2)$ is a pair of finite order transcendental entire solutions for the system of differential-difference Eq (1.3). Then $(f_1, f_2)$ satisfies

where $b, b_1$ are constants, and $c = k\pi$, $k$ is a integer.

Here these conclusions are stated in several complex variables as follows. In many previous articles ^{[13,16,19,25,26,35]} about Fermat-type partial differential equations with several complex variables, G. Khavinson ^[16] pointed out that any entire solutions of the partial differential equations $\left(\frac{\partial f}{\partial z_1}\right)^2+\left(\frac{\partial f}{\partial z_2}\right)^2 = 1$ in $\mathbb{C}^2$ are necessarily linear. This partial differential equations in real variable case occur in the study of characteristic surfaces and wave propagation theory, and it is the two dimensional eiconal equation, one of the main equations of geometric optics (see ^[4,6]). Later, Li ^[20,21] further discussed a series of partial differential equations with more general forms including $\left(\frac{\partial f}{\partial z_1}\right)^2+\left(\frac{\partial f}{\partial z_2}\right)^2 = e^{g}$, $\left(\frac{\partial f}{\partial z_1}\right)^2+\left(\frac{\partial f}{\partial z_2}\right)^2 = p$, etc., where $g, p$ are polynomials in $\mathbb{C}^2$, and gave a number of important and interesting results about the existence and the forms of solutions for these partial differential equations.

In 2012, Korhonen [17,Theorem 3.1] gave a logarithmic difference lemma for meromorphic functions in several variables of hyper order strictly less that 2/3. In 2016, Cao and Korhonen ^[2] improved it to the case for meromorphic functions with hyper order $< 1$ in several variables. In 2018, Xu and Cao ^[39] investigated the existence of the entire and meromorphic solutions for some Fermat-type partial differential-difference equations by utilizing the Nevanlinna theory and difference Nevanlinna theory of several complex variables ^[2,17], and obtained:

Theorem B (see [39,Theorem 1.1]). Let $c = (c_1, c_2)\in\mathbb{C}^2$. Then the Fermat-type partial differential-difference equation

doesn't have any transcendental entire solution with finite order, where $m$ and $n$ are two distinct positive integers.

Theorem C (see [39,Theorem 1.2]). Let $c = (c_1, c_2)\in\mathbb{C}^2$. Then any transcendental entire solutions with finite order of the partial differential-difference equation

has the form of $f(z_1, z_2) = \sin(Az_1+B)$, where $A$ is a constant on $\mathbb{C}$ satisfying $Ae^{iAc_1} = 1$, and $B$ is a constant on $\mathbb{C}$; in the special case whenever $c_1 = 0$, we have $f(z_1, z_2) = \sin(z_1+B)$.

Inspired by the above theorems, the authors ^[37] in 2020 extended the results of Theorems A, B from the complex Fermat types partial differential difference equations to the Fermat types system of partial differential-difference equations and obtained:

Theorem D (see [37,Theorem 1.1]). Let $c = (c_1, c_2)\in\mathbb{C}^2$, and $m_j, n_j$ $(j = 1, 2)$ be positive integers. If the following system of Fermat-type partial differential-difference equations

satisfies one of the conditions

$(i)$ $m_1m_2 > n_1n_2$;

$(ii)$ $m_j > \frac{n_j}{n_j-1}$ for $n_j\geq 2$, $j = 1, 2$.

Then system (1.4) does not have any pair of transcendental entire solution with finite order.

Theorem E (see [37,Theorem 1.3]). Let $c = (c_1, c_2)\in\mathbb{C}^2$. Then any pair of transcendental entire solutions with finite order for the system of Fermat-type partial differential-difference equations

have the following forms

where $L(z) = a_1z_1+a_2z_2$, $B_1$ is a constant in $\mathbb{C}$, and $a_1, c, A_{21}, A_{22}$ satisfy one of the following cases

$(i)$ $A_{21} = -i$, $A_{22} = i$, and $a_1 = i$, $L(c) = (2k+\frac{1}{2})\pi i$, or $a_1 = -i$, $L(c) = (2k-\frac{1}{2})\pi i$;

$(ii)$ $A_{21} = i$, $A_{22} = -i$, and $a_1 = i$, $L(c) = (2k-\frac{1}{2})\pi i$, or $a_1 = -i$, $L(c) = (2k+\frac{1}{2})\pi i$;

$(iii)$ $A_{21} = 1$, $A_{22} = 1$, and $a_1 = i$, $L(c) = 2k\pi i$, or $a_1 = -i$, $L(c) = (2k+1)\pi i$;

$(iv)$ $A_{21} = -1$, $A_{22} = -1$, and $a_1 = i$, $L(c) = (2k+1)\pi i$, or $a_1 = -i$, $L(c) = 2k\pi i$.

From Theorems D and E, we can see that there only contains the partial differentiation of the first variable $z_1$ of the unknown functions $f_1, f_2$ in those systems of partial differential difference equations. Naturally, a question arises: What will happen when the system of the partial differential-difference equations include both the difference $f_j(z+c)$ and $\frac{\partial f_j(z_1, z_2)}{\partial z_1}, \frac{\partial f_j(z_1, z_2)}{\partial z_2}, (j = 1, 2)$? In the past two decades, in spite of a number of important and meaningful results about the complex difference equation of single variable and the complex Fermat difference equation were obtained (can be found in ^{[9,10,11,24,31,33]}), but as far as we know, there are few results concerning the complex differential and complex difference equation in several complex variables. Further more, it appears that the study of systems of this Fermat type equations in several complex variables has been less addressed in the literature before.

The main purpose of this paper is concerned with the description of the transcendental entire solutions for some Fermat-type equations systems which include both difference operator and two kinds of partial differentials by utilizing the Nevanlinna theory and difference Nevanlinna theory of several complex variables ^[2,17]. We obtained some results about the existence and the forms of the transcendental entire solutions of some Fermat type systems of partial differential difference equations in $\mathbb{C}^2$, which improve the previous results given by Xu and Cao, Xu, Liu and Li, Gao ^{[5,37,38,39,40]}.

2.   Results and examples

Here and below, let $z+w = (z_1+w_1, z_2+w_2)$ for any $z = (z_1, z_2)$ and $w = (w_1, w_2)$. Now, our main results of this paper are stated below.

Theorem 2.1. Let $c = (c_1, c_2)\in\mathbb{C}^2$, and $m_j, n_j\; (j = 1, 2)$ be positive integers. If the following system of Fermat-type partial differential-difference equations

satisfies one of the conditions

$(i)$ $n_1n_2 > m_1m_2$;

$(ii)$ $m_j > \frac{n_j}{n_j-1}$ for $n_j\geq 2$, $j = 1, 2$.

Then system (2.1) does not admit any pair of transcendental entire solution with finite order.

Remark 2.1. Here, $(f, g)$ is called as a pair of finite order transcendental entire solutions for system

if $f, g$ are transcendental entire functions and $\rho(f, g) = \max\{\rho(f), \rho(g)\} < \infty$.

The following examples show system (2.1) admits a transcendental entire solution of finite order when $m_1 = m_2 = 2$ and $n_1 = n_2 = 1$.

Example 2.1. Let

Then $\rho(f_1, f_2) = 1$ and $(f_1, f_2)$ satisfies system (2.1) with $(c_1, c_2) = (1, 2)$, $m_1 = m_2 = 2$ and $n_1 = n_2 = 1$.

Theorem 2.2. Let $c = (c_1, c_2)\in\mathbb{C}^2$. If $(f_1, f_2)$ is a pair of transcendental entire solutions with finite order for the system of Fermat-type difference equations

Then $(f_1, f_2)$ is of the following forms

where $\phi(z) = L(z)+H(c_2z_1-c_1z_2)$, $L(z) = a_1z_1+a_2z_2$, $a_1, a_2, B_0$ is a constant in $\mathbb{C}$, $H$ is a polynomial in $\mathbb{C}$, and

and $c, A_{21}, A_{22}$ satisfy one of the following cases

$(i)$ $L(c) = 2k\pi i$, here and below, $k\in \mathbb{Z}$, $A_{21} = -i$ and $A_{22} = i$;

$(ii)$ $L(c) = (2k+1)\pi i$, $A_{21} = i$ and $A_{22} = -i$;

$(iii)$ $L(c) = (2k+\frac{1}{2})\pi i$, $A_{21} = -1$ and $A_{22} = -1$;

$(iv)$ $L(c) = (2k-\frac{1}{2})\pi i$, $A_{21} = 1$ and $A_{22} = 1$.

Here, we only list the following examples to explain the existence of transcendental entire solutions with finite order for system (2.2).

Example 2.2. Let $a = (a_1, a_2) = (2i, -i)$, $H = 4\pi^2(z_1-z_2)^2$, $A_{21} = -i, A_{22} = i$ and $B_0 = 0$. That is

where

Thus, $\rho(f_1, f_2) = 2$ and $(f_1, f_2)$ satisfies system (2.2) with $(c_1, c_2) = (2\pi, 2\pi)$.

Example 2.3. Let $a = (a_1, a_2) = (2i, -i)$, $H = \pi^n(z_1-z_2)^n, n\in \mathbb{Z}_+$, $A_{21} = i, A_{22} = -i$ and $B_0 = 0$. That is

where

Thus, $\rho(f_1, f_2) = n$ and $(f_1, f_2)$ satisfies system (2.2) with $(c_1, c_2) = (\pi, \pi)$.

Remark 2.2. From the conclusions of Theorems C and E, there only exists finite order transcendental entire solutions with growth order $\rho(f_1, f_2) = 1(\rho(f) = 1)$. However, in view of Theorem 2.2, we can see that there exists transcendental entire solution of system (2.2) with growth order $\rho(f_1, f_2) > 1$, for example, $\rho(f_1, f_2) = 2$ in Example 2.2 and $\rho(f_1, f_2) = n, n\in \mathbb{Z}_+$ in Example 2.3, these properties are quite different from the previous results. Hence, our results are some improvements of the previous theorems given by Xu, Liu and Li ^[37], Xu and Cao^[39], Liu Cao and Cao ^[23].

To state our last result, throughout this paper, let $s_1 = z_2-z_1$, $G_1(s_1)$, $G_2(s_1)$ be the finite order entire period functions in $s_1$ with period $2s_0 = 2(c_2-c_1)$, where $c_1\neq c_2$, $G_1(s_1)$, $G_2(s_1)$ can not be the same at every time occurrence.

Theorem 2.3. Let $c = (c_1, c_2)\in \mathbb{C}^2$ and $c_1\neq c_2$. Then any pair of transcendental entire solutions with finite order for the system of Fermat-type partial differential-difference equations

are one of the following forms

(i)

where $A_0 = \frac{\xi_2+\xi_1}{2(c_2-c_1)}$, $G_2\left(s_1+s_0\right) = G_1(s_1)+\frac{\xi_2-\xi_1}{2},$ and $\xi_1^2 = \xi_2^2 = 1$;

(ii)

where $A_0 = \frac{\xi_1+\xi_2-2c_1\xi_3}{2(c_2-c_1)}$, $G_2\left(s_1+s_0\right) = G_1(s_1)+\frac{\xi_1-\xi_2}{2}$, $\xi_1^2+\xi_3^2 = 1$, and $\xi_1 = \pm\xi_2;$

(iii)

where

$(iv)$

where

and $G_3(s_1), G_4(s_1), D_0$ satisfying one of the following cases:

$(iv_1)$ $L(c) = 2k\pi i$, $D_0 = 1$,

and

$(iv_2)$ $L(c) = (2k+1)\pi i$, $D_0 = -1$,

and

where $G_2(s+s_0) = G_1(s)$.

Some examples are listed to exhibit the existence of solutions for system (2.3).

Example 2.4. Let $G_1(s_1) = -G_2(s_1) = e^{-\pi is_1}$, and $\xi_1 = -1$, $\xi_2 = 1$. Then it follows that $A_0 = 0$ and

Thus, $\rho(f_1, f_2) = 1$ and $(f_1, f_2)$ satisfies the system (2.3) with $(c_1, c_2) = (1, 2)$.

Example 2.5. Let $\xi_1 = 0$, $\xi_2 = 0$, $\xi_3 = 1$ and $G_1(s_1) = G_2(s_1) = e^{2\pi i s_1}$. Then it follows that $A_0 = -1$ and

Thus, $\rho(f_1, f_2) = 1$ and $(f_1, f_2)$ satisfies system (2.3) with $(c_1, c_2) = (1, 2)$.

Example 2.6. Let $L(z) = -i(z_1+z_2)$ and $G_1(s_1) = -G_2(s_1) = e^{\frac{i}{4}s_1}$. That is

Thus, $\rho(f) = 1$ and $(f_1, f_2)$ satisfies system (2.3) with $(c_1, c_2) = (-\pi, 3\pi)$.

Example 2.7. Let $a_1 = i$, $a_2 = -i$, $L(z) = i(z_1-z_2)$, $G_1(s_1) = G_2(s_1) = e^{\frac{i}{4}s_1}$ and $B_1 = 0$. Then, it follows that

Thus, $\rho(f_1, f_2) = 1$ and $(f_1, f_2)$ satisfies system (2.3) with $(c_1, c_2) = (-2\pi, 2\pi)$.

Example 2.8. Let $a_1 = 1$, $a_2 = -1$, $L(z) = z_1-z_2$, $G_1(s_1) = -G_2(s_1) = e^{s_1}$ and $B_1 = 0$. Then, it follows that

Thus, $(f_1, f_2)$ satisfies system (2.3) with $(c_1, c_2) = (-\frac{1}{2}\pi i, \frac{1}{2}\pi i)$.

3.   Conclusions and discussion

From Theorems 2.1–2.3, we can see that our results are some extension of the previous results given by Xu and Cao ^[39] from the equations to the systems, and some supplements of the results given by Xu, Liu and Li ^[37]. More importantly, Examples 2.2 and 2.3 show that system (2.2) can admit the transcendental entire solutions of any positive integer order. However, the conclusions of Theorem C and Theorem E showed that the order of the transcendental entire solutions of the equations must be equal to 1. In fact, this is a very significant difference. Finally, one can find that we only focus on the finite-order transcendental entire solutions of systems (1.4)–(2.2) in this article; thus, the following question can be raised naturally:

Question 3.1. How should the meromorphic solutions of systems (2.2) and (2.3) be characterized?

4.   Proofs of Theorems 2.1–2.3

Similar to the argument as in the proof of Theorems 1.1 and 1.3 in Ref. ^[37], one can obtain the conclusions of Theorems 2.1 and 2.2 easily. Thus, we only give the proof of Theorem 2.3 as follow. However, the following lemmas play the key roles in proving Theorem 2.3.

Lemma 4.1. (^[34,36]) For an entire function $F$ on $\mathbb{C}^n$, $F(0)\neq0$ and put $\rho(n_F) = \rho < \infty$. Then there exist a canonical function $f_F$ and a function $g_F\in \mathbb{C}^n$ such that $F(z) = f_F (z)e^{g_F (z)}$. For the special case $n = 1$, $f_F$ is the canonical product of Weierstrass.

Remark 4.1. Here, denote $\rho(n_F)$ to be the order of the counting function of zeros of $F$.

Lemma 4.2. (^[30]) If $g$ and $h$ are entire functions on the complex plane $\mathbb{C}$ and $g(h)$ is an entire function of finite order, then there are only two possible cases: either

(a) the internal function $h$ is a polynomial and the external function $g$ is of finite order; or else

(b) the internal function $h$ is not a polynomial but a function of finite order, and the external function $g$ is of zero order.

Lemma 4.3. (^[15] or [14,Lemma 3.1]) Let $f_j(\not\equiv0), j = 1, 2, 3$, be meromorphic functions on $\mathbb{C}^m$ such that $f_1$ is not constant. If $f_1+f_2+f_3 = 1$, and if

for all $r$ outside possibly a set with finite logarithmic measure, where $\lambda < 1$ is a positive number, then either $f_2 = 1$ or $f_3 = 1$.

Remark 4.2. Here, $N_2(r, \frac{1}{f})$ is the counting function of the zeros of $f$ in $|z|\leq r$, where the simple zero is counted once, and the multiple zero is counted twice.

Lemma 4.4. Let $c = (c_1, c_2)$ be a constant in $\mathbb{C}^2$, $c_1\neq0, c_2\neq0$ and $c_2\neq c_1$. Let $p(z), \; q(z)$ be two polynomial solutions of the equation

and $q(z+c)-p(z) = \zeta_1, p(z+c)-q(z) = \zeta_2$, where $\zeta_1, \zeta_2, \gamma_0 \in \mathbb{C}$, then $p(z) = L(z)+B_1$, $q(z) = L(z)+B_2$, where $L(z) = a_1z_2+a_2z_2$, $a_1, a_2, B_1, B_2 \in \mathbb{C}$.

Proof. The characteristic equations of the Eq (4.1) are

Using the initial conditions: $z_1 = 0, z_2 = s_1$, and $h = h(0, s_1): = h_0(s_1)$ with a parameter. Then $z_1 = t$, $z_2 = t+s_1$, and $h = \int_0^t \gamma_0dt+h_0(s_1) = \gamma_0 t+h_0(s)$, where $s_1 = z_2-z_1$, $h_0(s_1)$ is a function in $s_1$. Since $p(z), q(z)$ are the solutions of (4.1), then it yields that

where $h_1(s_1), h_2(s_1)$ are two polynomials in $s_1$. Substituting (4.2) into $q(z+c)-p(z) = \zeta_1, q(z)-p(z+c) = \zeta_2$, it leads to

Hence, we have

where $\varepsilon_0 = \zeta_1+\zeta_2-2\gamma_0 c_1$. The fact that $h_1(s_1), h_2(s_1)$ are polynomials leads to

where $\gamma_1 = \frac{\varepsilon_0}{2s_0}$, $B_1, B_2\in \mathbb{C}$. By combining with (4.2) and (4.5), it follows that

where $a_1 = \gamma_0+\gamma_1, \; a_2 = -\gamma_1$.

Therefore, Lemma 4.4 is proved.

The Proof of Theorem 2.3: Let $(f_1, f_2)$ be a pair of transcendental entire functions with finite order satisfying system (2.3). In view of ^[7,27], we know that the entire solutions of the Fermat type functional equation $f^2+g^2 = 1$ are $f = \cos a(z), g = \sin a(z)$, where $a(z)$ is an entire function. Hence, we only consider the following cases.

(i) Suppose that $\frac{\partial f_1(z_1, z_2)}{\partial z_1}+\frac{\partial f_1(z_1, z_2)}{\partial z_2} = 0$. Then it follows from (2.3) that

and

In view of (4.6), (4.7) and (2.3), it yields that

By solving the equations $\frac{\partial f_j}{\partial z_1}+\frac{\partial f_j}{\partial z_2} = 0$ $(j = 1, 2)$, we have

where $g_1, g_2$ are transcendental entire functions of finite order. Substituting (4.9) into (4.6), (4.8), it yields that

which implies

Thus, in view of (4.10) and (4.11), we conclude that

where $A_0 = B_0 = \frac{\xi_1+\xi_2}{2(c_2-c_1)}$, $G_1(s_1), G_2(s_1)$ are the finite order entire period functions with period $2s_0$, and

Thus, this proves the conclusions of Theorem 2.3 (i).

(ii) Suppose that

In view of (2.3), we conclude that

The characteristic equations of Eq (4.12) are

Using the initial conditions: $z_1 = 0, z_2 = s_1$, and $f_1 = f_1(0, s_1): = g_1(s_1)$ with a parameter. Thus, we have $z_1 = t$, $z_2 = t+s_1$ and $f_1(t, s_1) = \int_0^t \xi_3dt+g_1(s_1) = \xi_3t+g_1(s_1)$, where $s_1 = z_2-z_1$, $g_1(s_1)$ is a transcendental entire function of finite order. Hence, it yields that

Similar to the same argument for Eq (4.13), we have

where $g_2(s_1)$ is a transcendental entire function of finite order.

Substituting (4.16), (4.17) into (4.14), (4.15), we conclude that

which lead to

Since $g_1(s_1), g_2(s_1)$ are the transcendental entire functions of finite order. then we have

where $A_0 = B_0 = \frac{\xi_1+\xi_2-2c_1\xi_3}{2(c_2-c_1)}$, $G_1(s_1), G_2(s_1)$ are the finite order entire period functions with period $2s_0$, and

Thus, this proves the conclusions of Theorem 2.3 (ii).

(iii) If $\frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2}$ is transcendental, then $f_2(z_1+c_1, z_2+c_2)-f_1(z_1, z_2)$ is transcendental. Here, we can deduce that $f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2)$ and $\frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2}$ are transcendental.

Suppose that $f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2)$ is not transcendental. Since $\frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2}$ is transcendental, then $\frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2}$ is transcendental. In view of (2.3), it yields that $f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2)$ is transcendental, a contradiction.

Suppose that $\frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2}$ is not transcendental. In view of (2.3), it follows that $f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2)$ is not transcendental. Thus, it yields that $\frac{\partial f_1(z+c)}{\partial z_1}+\frac{\partial f_1(z+c)}{\partial z_2}$ is not transcendental. This is a contradiction with $\frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2}$ is transcendental.

Hence, if $\frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2}$ is transcendental, then $f_2(z_1+c_1, z_2+c_2)-f_1(z_1, z_2)$, $f_1(z_1+c_1, z_2+c_2)-f_2(z_1, z_2)$ and $\frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2}$ are transcendental. Thus, system (2.3) can be rewritten as

Since $f_1, f_2$ are transcendental entire functions with finite order, then by Lemmas 4.1 and 4.2, there exist two nonconstant polynomials $p(z), q(z)$ such that

Thus, it follows from (4.21) that

which implies

Obviously, $\frac{\partial p}{\partial z_1}+\frac{\partial p}{\partial z_2}\neq -i$. Otherwise, it follows that $-e^{2q(z+c)}\equiv1$, this is impossible because $q(z)$ is a nonconstant polynomial. Similarly, $\frac{\partial q}{\partial z_1}+\frac{\partial q}{\partial z_2}\neq -i$. Thus, by Lemma 4.3, and in view of (4.23), (4.24), we conclude that

and

Next, we consider the following four cases.

Case 1.

Since $p(z), q(z)$ are polynomials, then from (4.25), we know that $\frac{\partial p}{\partial z_1}+\frac{\partial p}{\partial z_2}$ and $\frac{\partial q}{\partial z_1}+\frac{\partial q}{\partial z_2}$ are constants in $\mathbb{C}$. Otherwise, we obtain a contradiction from the fact that the right of the above equations is not transcendental, but the left is transcendental. In addition, we get that $q(z+c)-p(z)\equiv C_1$ and $p(z+c)-q(z)\equiv C_2$, that is, $p(z+2c)-p(z)\equiv C_1+C_2$ and $q(z+2c)-q(z)\equiv C_1+C_2$. In view of $c_1\neq c_2$, then by Lemma 4.4, it means that $p(z) = L(z)+B_1, q(z) = L(z)+B_2$, where $L$ is a linear function as the form $L(z) = a_1z_1+a_2z_2$, $a_1, a_2, B_1, B_2$ are constants.

By combining with (4.23)–(4.25), we have

This means

Thus, it follows that $a_1+a_2 = -2i$ or $a_1+a_2 = 0$.

If $a_1+a_2 = -2i$. Thus, $e^{2L(c)} = 1, e^{B_1-B_2} = -e^{L(c)}$. The characteristic equations of the first equation in (4.22) are

Using the initial conditions: $z_1 = 0, z_2 = s_1$, and $f_1 = f_1(0, s_1): = g_0(s_1)$ with a parameter. Thus, it follows that $z_1 = t$, $z_2 = t+s_1$ and

where $g_1(s_1)$ is a finite order entire function, and

In view of $z_1 = t$, $z_2 = t+s_1$, we have

Similar to the same argument for the third equation in (4.22), we have

where $g_2(s_1)$ is a finite order entire function.

Substituting (4.28), (4.29) into (4.22), and applying (4.27), it yields that

Thus, from (4.30), we can deduce that

where $G_1(s_1), G_2(s_1)$ are the finite order entire period functions with period $2s_0$, and

If $a_1+a_2 = 0$, then it follows that $e^{2L(c)} = 1, e^{B_1-B_2} = e^{L(c)}$. In view of $z_1 = t$, $z_2 = t+s_1$, it leads to $L(z) = a_1z_1+a_2z_2 = a_2s_1$. The characteristic equations of the first equation in (4.22) are

Using the initial conditions: $z_1 = 0, z_2 = s_1$, and $f_1 = f_1(0, s): = G_3(s)$ with a parameter. This leads to

that is

where $G_3(s_1)$ is an entire function of finite order. Similarly, we have

where $G_4(s_1)$ is an entire function of finite order.

If $e^{L(c)} = 1$, that is, $L(c) = 2k \pi i$, which leads to $e^{B_1-B_2} = 1$. Thus, it follows that

Substituting (4.31), (4.32) into the second and fourth equations in (4.22), we have

that is,

Thus, we can deduce that

where $G_1(s_1), G_2(s_1)$ are the finite order entire period functions with period $2s_0$, and $G_2(s_1+s_0) = G_1(s_1).$

If $e^{L(c)} = -1$, that is $L(c) = (2k+1)\pi i$, then it follows that $e^{B_1-B_2} = -1$, which means

Substituting (4.31), (4.33) into the second and fourth equations in (4.22), we have

Thus, it yields

This leads to

where $G_1(s_1), G_2(s_1)$ are the finite order entire period functions with period $2s_0$, and $G_2(s+s_0) = G_1(s).$

Case 2.

In view of (4.34), the fact that $p(z), q(z)$ are polynomials leads to $q(z+c)-p(z)\equiv C_1$ and $q(z)+p(z+c)\equiv C_2$. This means $q(z+2c)+q(z)\equiv C_1+C_2$, this is a contradiction with the assumption of $q(z)$ being a nonconstant polynomial.

Case 3.

In view of (4.35), the fact that $p(z), q(z)$ are polynomials leads to $p(z)+q(z+c)\equiv C_1$ and $p(z+c)-q(z)\equiv C_2$. This means $p(z+2c)+p(z)\equiv C_1+C_2$, this is a contradiction with the assumption of $p_2(z)$ being a nonconstant polynomial.

Case 4.

In view of (4.36), the fact that $p_1(z), p_2(z)$ are polynomials leads to $p(z)+q(z+c)\equiv C_1$ and $q(z)+p(z+c)\equiv C_2$, that is, $p(z+2c)-p(z)\equiv C_1+C_2$ and $q(z+2c)-q(z)\equiv C_2+C_1$. Similar to the same argument in Case 1 of Theorem 2.3, we obtain that $p(z) = L(z)+B_1, q(z) = -L(z)+B_2$, where $L$ is a linear function as the form $L(z) = a_1z_1+a_2z_2$, $a_1, a_2, B_1, B_2$ are constants. In view of (4.23), (4.24) and (4.36), we have

which implies $\left(a_1+a_2+i\right)^2 = \left(a_1+a_2-i\right)^2$, that is, $a_1+a_2 = 0$. Thus, we conclude from (4.37) that

Similar to the argument as in Case 1 of Theorem 2.3, we get the conclusions of Theorem 2.3 (iv).

Therefore, from Cases 1–4, we complete the proof of Theorem 2.3.

Acknowledgments

The first author is supported by the Key Project of Jiangxi Province Education Science Planning Project in China (20ZD062), the Key Project of Jiangxi Province Culture Planning Project in China (YG2018149I), the Science and Technology Research Project of Jiangxi Provincial Department of Education (GJJ181548, GJJ180767), the 2020 Annual Ganzhou Science and Technology Planning Project in China. The third author was supported by the National Natural Science Foundation of China (11561033), the Natural Science Foundation of Jiangxi Province in China (20181BAB201001) and the Foundation of Education Department of Jiangxi (GJJ190876, GJJ202303, GJJ201813, GJJ191042) of China.

Conflict of interest

The authors declare that none of the authors have any competing interests in the manuscript.