Entire Solutions of the Second-Order Fermat-Type Differential-Difference Equation

Abstract

In this paper, the entire solutions of finite order of the Fermat-type differential-difference equation and the system of equations have been studied. We give the necessary and sufficient conditions of existence of the entire solutions of finite order.

1. Introduction

In this work, we assume that the readers are familiar with general definitions and fundamental theories of Nevanlinna theory [1–3]. is a meromorphic function which means is meromorphic in the finite complex plane . If has no poles, we call is an entire function. We denote by , any function satisfying, , outside of a possible exceptional set of finite logarithmic measure. For a meromorphic function , we define its shift by and its difference operators by

The classic Fermat-type functional equationhas been intensively studied in recent years. If , equation (2) has no transcendental meromorphic solutions [4]. If , equation (2) has no transcendental entire solutions [4]. If , all entire solutions are the forms of and , where is any entire function [5].

Yang [6] studied the following generalized Fermat-type functional equation:where and are positive integers and obtained the following theorem.

Theorem 1. If , then equation (3) has no nonconstant entire solutions and .

For further research solutions of the Fermat-type functional equation, Yang and Li [7] considered the following special Fermat-type functional equation:and they obtained the following theorem.

Theorem 2. The transcendental meromorphic solutions of (4) must satisfy , where and are nonzero constants.

In the following, is a nonzero constant, unless otherwise specified.

Liu [8] investigated the following special Fermat-type functional equation:

He obtained that each transcendental entire solution of (5) with the finite order must satisfy , where , and . Liu et al. [9] proved that the nonconstant finite-order entire solutions of (5) must have order one. Then, Liu et al. [10] considered the entire solutions of the following difference equations:and obtained the following results.

Theorem 3. Equation (6) has no nonconstant entire solution if or ; has no transcendental entire solution with finite order if ; and has the general solutions , where is any entire function periodic with period .

Theorem 4. Equation (7) has no transcendental entire solution with finite order, provided that , where and are positive integers.

Liu et al. [10] obtained that satisfies the equation when ; satisfies the equation when , where ; and satisfies the equation when , where or and is an integer.

Liu et al. [10] also investigated the following special Fermat-type functional equation:and obtained the following theorem.

Theorem 5. Equation (8) has no transcendental entire solutions with finite order, provided that , where and are positive integers.

Liu et al. built exact solutions for equation (8); for , the equation has a transcendental entire solution , where and ; for , the equation admits a transcendental entire solution , where , and , where is odd; and for , the equation admits a transcendental entire solution , where is an integer, and is a constant.

In 2019, Liu et al. [11] researched the entire solutions with finite order of the Fermat-type differential-difference equationand the following system of differential-difference equations:

In 2019, Dang and Chen [12] obtained the meromorphic solutions of the following special Fermat-type functional equation:

In the following, we will consider the entire solutions with finite order of the Fermat-type differential-difference equationand obtain the following result.

Theorem 6. be entire solutions of finite order of differential-difference equation (12) if and only if be the following forms:where , and are constants, , , and ; , where for , for , and for .

Obviously, from Theorem 6, we immediately get the following example.

Example 1. The transcendental entire function solutions with finite order of the differential-difference equationmust satisfywhere , and are constants, , , and ; , where .
Then, we will study the following system of differential-difference equations:and obtain the next result.

Theorem 7. be the transcendental entire solutions of finite order of the system of differential-difference equation (16) if and only if be the following forms:where , , and ;where , , and ; for ; , where for ; , where for ; and for , where , and are constants, .

Obviously, from Theorem 7, we immediately get the following example.

Example 2. Let . Then, the transcendental entire function solutions with finite order of the system of differential-difference equationsmust be in the following forms:where , , and ;where , , and .

2. Lemmas

Lemma 1. (see [1]). Let be a meromorphic function, , being nonconstant, satisfying and . If andwhere and , then .

Lemma 2. (see [1]). Suppose that are meromorphic functions and are entire functions satisfying the following conditions:(1)(2)The orders of are less than those of for Then, .

Lemma 3. (Hadamard’s factorization theorem; see [1]). Let be an entire function of finite order with zeros and -fold zero at the origin. Then,where is the canonical product of formed with nonnull of and is a polynomial of degree less than .

3. Proof of Theorem 6

Suppose that is an entire solution with finite order which satisfies (12). We rewrite (12) as follows:

It follows that and have no zeros. By Lemma 3, we havewhere is a polynomial.

From (25), we get

Case 1. Suppose that is a transcendental entire function. Then, it follows from (26) that is a nonconstant polynomial. Let ; then, . It follows from (26) thatIt follows from (27) thatThen,Combining (28) and (30), we getThis impliesTherefore,If , then for , we haveand by (33) and Lemma 2, we obtain , which is contradicting.
Hence, . Let , where ; then, , , , and . Then, by (33), we haveThis impliesThen,This impliesThen,This impliesThen,Obviously, , and . From (41) and Lemma 2, we obtain thatBy (42), it is easy to get and . It follows from (42) that . Thus, it follows from (26) that

Case 2. Suppose that is a polynomial. Then, it follows from (26) that is a constant, and . If , then . It follows from (12) that . If , then . It follows from (12) that . If , then . It follows from (12) that .Thus, Theorem 6 is proved.

4. Proof of Theorem 7

Suppose that be an entire solution with finite order which satisfies equation (16). We rewrite (16) as follows:

It follows that

By Lemma 3, we havewhere and are polynomials.

From (46), we get

Case 3. Suppose that is a transcendental entire function with finite order. Then, it follows from (47)–(50) that and are two nonconstant polynomials. Let ; then, .
It follows from (47)–(50) thatThen,From (51) and (55), we getFrom (52) and (56), we getIt follows thatThen,If are nonconstant polynomials for any , then we obtain that are nonconstants for any . Assume ; then, does not vanish; then, we multiply on both sides of (61), and we haveThen, by Lemma 1, we have ; this contradicts with .
Hence, there exist such that or , where is a constant.
Suppose that . Obviously, . We assert that . Otherwise, we assume that ; then, for any and for any .
From (63), we haveThen, it follows from (64) and Lemma 2 that . This implies is a constant, and , which is contradicting; therefore, the assertion is proved.
Accordingly, and . Thus, we can assume that ; then,for any .
Combining (61) and (62), we getFurthermore,It follows from (68) and (69) thatBy the binomial formula and (70)–(73), we haveFrom (75) and (77), we get ; hence, . This implies , where is an integer. From (74) and (75) and , we obtain that and , where ; and , where .
It follows from (47) and (49) thatTherefore,where and are constants.
Combining (79) and , we get that , where and are constants.
If , then it follows from (74) and (79) that andIf , then it follows from (74) and (79) that and

Case 4. Suppose that is a polynomial. Then, it follows from (47)–(50) that are two constants. Hence, and , where , and are constants. If , then . It follows from (16) that , and . If , then . It follows from (16) thatTherefore, , or . Therefore,where , andwhere .
If , we have . It follows from (16) that , where is a constant.
Thus, Theorem 7 is proved.

5. Discussion

Consider the differential-difference equationwhere and are positive integers. If or or , it follows easily that (85) has no nonconstant entire solution by Theorem 1. For the case , the entire solutions with finite order have been illustrated in Theorem 6.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the Visiting Scholar Program of Shiing-Shen Chern Institute of Mathematics at Nankai University.