Entire solutions for several general quadratic trinomial differential difference equations

Abstract: This paper is devoted to exploring the existence and the forms of entire solutions of several quadratic trinomial differential difference equations with more general forms. Some results about the forms of entire solutions for these equations are some extensions and generalizations of the previous theorems given by Liu, Yang and Cao. We also give a series of examples to explain the existence of the finite order transcendental entire solutions of such equations.


Introduction
The main aim of this paper is to investigate the transcendental entire solutions with finite order of the quadratic trinomial difference equation (1) and the quadratic trinomial differential difference equation (2) where ( ) ≠ α 0, 1 2 , c are constants and ( ) g z is a polynomial. When = α 0 and ( ) = g z 0, the above equations become the Fermat-type difference equation ( ) ( ) + + = f z c f z 1 2 2 and differential difference equations ( ) ( ) + + ′ = f z c f z 1 2 2 , which are discussed by Liu and his colleagues (see [1][2][3]). They pointed out that the transcendental entire solution with finite order of the latter must satisfy ( ) , where B is a constant and = c kπ 2 or ( ) = + c k π 2 1 , k is an integer. For the general Fermat-type functional equation Gross [4] had discussed the existence of solutions of equation (3) and showed that the entire solutions are , where ( ) a z is an entire function. In recent years, with the development of Nevanlinna theory and difference Nevanlinn theory of meromorphic function [5][6][7][8], many scholars obtained lots of results about the solutions of Fermat-type functional equations [1][2][3][9][10][11][12][13][14][15][16][17].
In fact, when = ± α 1, it is easy to get the entire solution of equations ( ) ( ) ( ) ± + = ± f z f z c e g z 1 2 and ( ) ( ) ( ) ′ ± + = ± f z f z c e g z 1 2 , , Liu and Yang [9] in 2016 studied the existence and the form of solutions of some quadratic trinomial functional equations and obtained the following results in equations (1) and (2).
has no transcendental meromorphic solutions.
Theorem B. (see [9,Theorem 1.4]) If ≠ ± α 1, 0, then the finite order transcendental entire functions of equation must be of order equal to one.
In recent years, Han and Lü [18] gave the description of meromorphic solutions for the functional equation (3) when ( ) ( ) = ′ g z f z and 1 is replaced by , and obtained the following results.
The meromorphic solutions f of the following differential equation Here, They also proved that all the trivial meromorphic solutions of ( ) ( )

Results and some examples
Motivated by the above question, this article is concerned with the entire solutions for the difference equation (1) and the differential difference equation (2). The main tools used in this paper are the Nevanlinna theory and the difference Nevanlinna theory. Our principal results obtained generalize the previous theorems given by Liu, Cao, and Yang [1][2][3]9]. Here and below, let ≠ α 0, 1 2 , and Entire solutions for several differential difference equations  1019 The first main theorem is about the existence and the forms of the solutions for the quadratic trinomial difference equation (1).
and ( ) g z be a polynomial. If the difference equation (1) admits a transcendental entire solution ( ) f z of finite order, then ( ) g z must be of the form , . Furthermore, ( ) f z must satisfy one of the following cases: and a c A A η , , , , The following examples show that the forms of solutions are precise to some extent. Thus, ( ) f z is a solution of (1) with ( ) = g z z and = c πi 2 .
When (1), we obtain the second theorem as follows.
and ( ) g z be a polynomial, and if the differential equation admits a transcendental entire solution ( ) f z of finite order, then ( ) g z must be of the form , .
The following example shows that the forms of solutions are precise to some extent.
Then it is easy to get that the function From Theorem 2.2, it is easy to get the following corollary.
. Then the following partial differential difference equation admits no transcendental entire solution with finite order.
For the differential difference counterpart of Theorem 2.2, we have and ( ) g z be a nonconstant polynomial. If the differential difference equation . Furthermore, ( ) f z must satisfy one of the following cases: and a c A A η , , , , The following examples explain the existence of transcendental entire solutions with finite order of (2).  then ( ) f z is a transcendental entire solution with finite order of equation (2) with ( ) ( ) = + g z a a z 1 2 1 and = c πi.
From Theorem 2.3, we obtain the following corollary.
, . Then the differential difference equation (2) has no transcendental entire solution with finite order.
Entire solutions for several differential difference equations  1021

Some lemmas
The following lemmas play the key role in proving our results.
Lemma 2.1. [19] If g and h are entire functions on the complex plane 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.

Proof of Theorem 2.1
Suppose that ( ) f z is a transcendental entire solution with finite order of equation (1). Let where u v , are entire functions. Thus, equation (1) can be written as Since f is a finite order transcendental entire function and g is a polynomial, there thus exists a polynomial ( ) p z such that Denote By combining with (12) where A A , 1 2 are defined in (7). Thus, in view of (14) and (15), it follows that We will discuss two cases below. where . This is a contradiction, which implies that ( ) p z is a constant. Let = η e p . Substituting this into (14) and (15) In view of where a b , are constants satisfying is not a constant, and by applying Lemma 2.2 for (16), it follows that Entire solutions for several differential difference equations  1023  Therefore, this completes the proof of Theorem 2.1. □

Proof of Theorem 2.2
Suppose that ( ) f z is a transcendental entire solution with finite order of equation (8). By using the same argument as in the proof of Theorem 2.1, we have (14) and which leads to By combining with (13) and (24), we have If ( ) p z is not a constant, then it follows from (25) that Since ( ) ( ) p z g z , are polynomials, the left of equation (26) is transcendental, but the right of equation (26) is a polynomial. Thus, a contradiction can be obtained from (26). Hence, it follows that . Let = ξ e p 2 , in view of (25), it follows that which leads to Thus, we have ( ) . Hence, ( ) g z must be of the form ( ) = + g z az b.
Therefore, this completes the proof of Theorem 2.2. □

Proof of Theorem 2.3
Suppose that ( ) f z is a transcendental entire solution with finite order of equation (2). By using the same argument as in the proof of Theorem 2.1, we have (23) and where ( ) p z is a polynomial and ( ) ( ) γ z γ z , 1 2 are stated as in (13). In view of (23) and (27), it follows that  Thus, in view of (32) and (33), this completes the proof of Theorem 2.3(i).
is not a constant, it follows from (13) that ( ) p z is not a constant. Then we have that ′ γ 1 and ′ γ 2 cannot be equal to 0 at the same time. Otherwise, it yields that ( ) where ε 1 is a constant. In view of (28), it thus follows that ( ) ( ) ( ) ′ ≡ − + γ z e 1 γ z γ z c 1 1 2 , this means where ε 2 is a constant. In view of (36) and (37) is not a constant. Thus, we get the conclusions of Theorem 2.3(ii) from Case 2.
Therefore, this completes the proof of Theorem 2.3. □