This article is devoted to study the existence conditions of solutions to several complex differential-difference equations. We obtain some Malmquist theorems related to complex differential-difference equations with a more general form than the previous equations given by Zhang, Huang, and others. Moreover, some examples are provided to demonstrate why some restrictive conditions in some of our theorems cannot be removed.
1 Introduction and main results
The purpose of this article is to investigate the properties of the solutions of some differential-difference equations of Malmquist-type. We first assume that the reader is familiar with the basic notions of Nevanlinna value distribution theory (see [1,2, 3,4]). Now, we recall the well-known Malmquist theorem about the existence of meromorphic solutions to a certain type of differential equation (see ) and obtain
(See .) Let
where and are relatively prime polynomials in , and the coefficients and are rational functions. If equation (1.1) admits a transcendental meromorphic solution, then and .
where are rational functions.
We can find that Malmquist did not make use of Nevanlinna value distribution theory to prove Theorems A and B. In 1933, Yosida  proved Malmquist’s theorem by applying the Nevanlinna value distribution theory. In the 1970s, Laine , Yang , and Hille  gave a generalization of Theorems A and B when the coefficients of are meromorphic functions. In 1978, Steinmetz  considered the equation and extended Malmquist’s theorem by relying on a number of auxiliary functions; in 1980, Gackstatter and Laine  obtained a generalized result of Theorems A and B by using the properties of Valiron deficient values. In 2018, Zhang and Liao  further studied the existence of meromorphic solutions when is replaced by the differential polynomial in equation (1.1) and obtained
(See [12, Theorem 1.3].) If the algebraic differential equation
is a differential polynomial in f with meromorphic coefficients and and are meromorphic functions, possesses an admissible meromorphic solution, then is reduced to a polynomial in f of degree , where .
In the past 20 years, there were a lot of results focusing on the growth of order and the existence of the solutions to Malmquist-type difference equations (see [13,14, 15,16,17, 18,19]), by applying the Nevanlinna theory for meromorphic functions. In particular, Heittokangas et al.  discussed the following difference equations:
where and is an irreducible rational function in with meromorphic coefficients such that . They proved that if equations (1.3) and (1.2) admit a transcendental meromorphic solution of finite order. After 4 years, Laine et al.  further studied the existence of solutions to the following complex difference equation:
where is a collection of all subsets of , and are relatively prime polynomials in over the field of rational functions, and the coefficients are rational functions, and obtained that if equation (1.2) admits a transcendental meromorphic solution of finite order. In 2010, Zhang and Liao  further studied the solutions of the difference equation with more general form than (1.2). Laine et al. , also proved the form of the solutions to complex difference equation (1.4) under some theoretical assumptions.
where and are rational functions and is a transcendental entire function satisfying a difference equation of the form either
were and are nonempty disjoint subsets of , , , and .
In view of the aforementioned theorems of Malmquist-type, a natural question is whether similar results hold for more general difference equations of Malmquist-type. For this question, the first aim of this article is to investigate the existence of solutions to the equation, when the left-hand sides of equations (1.1)–(1.4) are replaced by a differential-difference polynomial of . To state our results, we first introduce the following definition.
(See [22, Definition 2.1] or .) A differential-difference polynomial in is a finite sum of difference products of , derivatives of , and derivatives of their shifts, with all coefficients of these monomials being small functions of .
Now, we give a differential-difference polynomial in with the form
where is a finite set of multi-indices , are not equal to 0, simultaneously, for ; , and , are distinct complex constants, and the meromorphic coefficients are of growth . The degree of the monomial is defined by , and the degree of is defined by
The first result of this article related to Malmquist theorem is obtained as follows.
Assume that is a transcendental meromorphic solution of the differential-difference equation:
where P and Q are relatively prime polynomials in f, the coefficients of and are stated as in Theorem A, and the coefficients are of growth . If f has at most finitely many poles and , then we have , and it must take the form
where and are rational functions, is a transcendental entire function, and there exist integers and a constant such that
The following example shows that the condition that has at most finitely many poles is necessary in Theorem 1.1.
Let , , , and . Then satisfies the following equation:
but is not of the form .
The following example shows that the conclusions in Theorem 1.1 can be realized.
Let and . Then satisfies the equation
where , , and . It can be seen that . Thus, it follows that satisfies
Assume that is a transcendental meromorphic solution with finite order of the differential-difference equation (1.6), where P and Q are relatively prime polynomials in f, the coefficients of and are stated as in Theorem 1.1, and the coefficients of are rational functions. If is of finite order with at most finitely many poles, then can be reduced to a polynomial in of degree .
The following example shows that the condition “ has at most finitely many poles” cannot be omitted in Theorem 1.2.
Let , , , and , then satisfies the differential-difference equation:
It is easy to see that and . However, is not a polynomial in .
The following example shows that the restriction “ is finite order” also cannot be omitted in Theorem 1.2.
Let , , and , then satisfies the differential-difference equation
The following example shows that the conclusion “ ” is sharp to some extent in Theorem 1.2.
Let , , and , then satisfies the differential-difference equation
This shows that the equality in can be attained in Theorem 1.2.
The second purpose of this article is to study the existence and growth of solutions for a special differential-difference equation
where , , , are of growth , and and are relatively prime polynomials in with coefficients and such that . Denote
For the growth of solutions of equation (1.7), we obtain the following results.
Assume , is a transcendental meromorphic solution of complex differential-difference equation of Malmquist-type (1.7). Let the coefficients and satisfy and . If , then .
Assume , is a transcendental meromorphic solution of complex differential-difference equation of Malmquist-type (1.7). Let the coefficients and satisfy . If there is a dominant coefficient or satisfying
then . If is of finite regular growth and , then .
2 Proof of Theorem 1.1
 Let be a meromorphic function with order and let be a fixed nonzero complex number, then for each , we have
and if the exponent of convergence of poles , then
(Valiron-Mohon’ko) . Let be a meromorphic function. Then, for all irreducible rational functions in ,
with meromorphic coefficients and , the characteristic function of satisfies that
where and .
. Suppose that ( ) are meromorphic functions, and are entire functions satisfying the following conditions:
are not constants for ;
For , ,
where is the finite linear measure of finite logarithmic measure.
. Let be a meromorphic function and let be given by
where are small meromorphic functions relative to , then either
Proof of Theorem 1.1
Assume that is a transcendental meromorphic solution of equation (1.6). Since the coefficients and are rational functions, in view of [20, p. 80], it follows that and have only finitely many common zeros. Thus, by Lemma 2.3 in , it yields
By Lemma 2.4, it follows that either
where is a rational function, or
Since has at most finitely many poles, and in view of (2.2) and (2.3), we can deduce that , which is a contradiction. Thus, and has at most finitely many poles. In view of (2.1), we can see that has finitely many zeros. As a result, if there exist a rational function and an entire function such that
then we have
where is the th root of and . Set , , , and , where , then it follows that
where , , and , , and are rational functions.
If is a nonconstant polynomial, let
then, for any , it follows that
Thus, equation (1.6) can be represented as
Due to the small functions , the rational functions , , and (2.6), it follows that and for any . Given the assumptions on and , it follows from Lemma 2.2 and (2.7) that . In view of , equation (2.7) can be represented as the form:
for some and , or
for some and . As a result, there exist integers , and a constant such that
Therefore, this completes the proof of Theorem 1.1.□
3 Proof of Theorem 1.2
Proof of Theorem 1.2
Suppose that is a transcendental meromorphic solution of finite order for the differential-difference equation (1.6) and has at most finitely many poles. If , then it follows from Theorem 1.1 that , which is a contradiction with being finite order. Thus, , implying that is reduced to a polynomial. Hence, by Lemma 2.2, it follows that
Since has at most finitely many poles, then it yields . In view of Lemma 3.1, we conclude that
which leads to .
Therefore, this completes the proof of Theorem 1.2.□
4 Proof of Theorem 1.3
We recall some notations from . If has more than poles of a certain type, then we claim that the integrated counting function of these poles is not of type . Thus, we use to denote a pole (zero) of with multiplicity . The following lemma is also from [27, Lemma 3.1].
([27, Lemma 3.1].) Let be a meromorphic function having more than poles and be small meromorphic functions with respect to f. Denote the maximum order of zeros and poles of the functions at by . Then, for any , there are at most points such that
([22, Lemma 2.2].) Let be a transcendental meromorphic solution of finite order of a difference equation of the form
where , and are differential-difference polynomials in f such that the total degree of is . Then, for any ,
possibly outside of an exceptional set of finite logarithmic measure.
Proof of Theorem 1.3
We will use the method in the proof of Proposition 5.4 in . Let
Suppose that is a transcendental meromorphic solution with finite order of equation (1.7). Since , then, in view of Lemma 4.2, it yields that , which implies that has more than poles, counting multiplicity. Thus, we can choose such poles sequence such that and , where is the maximum order of poles or zeros of the coefficients , and in (1.7) at , and is any small constant. Taking the subsequence of poles as our starting point, and let , we may prove that
Analyzing the poles of and considering , we can see that at least one of the points is a pole of with order . We first apply Lemma 4.2 to obtain that there are more than such points with and . Then, we choose only one of these points and denote it by . Thus, we have for each permitted , and a pole satisfying , where
Continuing the above process, we can obtain a sequence satisfying and .
Next, we continue to give the estimate of the counting function . Set and . Thus, by a simple geometric observation, it is easy to see that
where is an open disc of radius centered at . For sufficiently large , it yields , which leads to
implying that . This is a contradiction with the assumption of finite order . Thus, . Therefore, this completes the proof of Theorem 1.3.□
5 Proof of Theorem 1.4
To prove Theorem 1.4, we need the following lemma.
Proof of Theorem 1.4
Without losing generality, assume that
Now, two cases will be discussed below.
Case 1. If , it then follows from (5.1) that
for , , , and . So, in view of Lemma 2.2 and , combining the above equations yields
In addition, by Lemma 5.1, we have
This is a contradiction.
Hence, we can conclude that . This completes the proof of Theorem 1.4.□
The authors are very thankful to the referees for their valuable comments, which helped to improve the presentation of the article.
Funding information: This work was supported by the National Natural Science Foundation of China 12161074 and the Talent Introduction Research Foundation of Suqian University.
Author contributions: H. Y. Xu: conceptualization, writing – original draft preparation, writing – review and editing, funding acquisition; H. Li: writing – review and editing; M. Y. Yu: funding acquisition.
Conflict of interest: The authors declare no conflict of interest.
Data availability statement: No data were used to support this study.
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