#### Abstract

The characteristic functions of differential-difference polynomials are investigated, and the result can be viewed as a differential-difference analogue of the classic Valiron-Mokhon’ko Theorem in some sense and applied to investigate the deficiencies of some homogeneous or nonhomogeneous differential-difference polynomials. Some special differential-difference polynomials are also investigated and these results on the value distribution can be viewed as differential-difference analogues of some classic results of Hayman and Yang. Examples are given to illustrate our results at the end of this paper.

#### 1. Introduction

Throughout this paper, we use standard notations in the Nevanlinna theory (see, e.g., [1–3]). Let be a meromorphic function. Here and in the following the word “meromorphic’’ means being meromorphic in the whole complex plane. We use normal notations , , , , , , and . And we also use to denote the hyperorder of and to denote the Nevanlinna deficiency of with respect to . Moreover, we denote by any real quantity satisfying as outside of a possible exceptional set of finite logarithmic measure.

Recently, with some establishments of difference analogues of the classic Nevanlinna theory (two typical and most important ones can be seen in [4–6]), there has been a renewed interest in the properties of complex difference expressions and meromorphic solutions of complex difference equations (see, e.g., [4–17]). By combining complex differentiates and complex differences, we proceed in this way in this paper.

It is well known that the following Valiron-Mokhon’ko Theorem, due to Valiron [18] and A. Z. Mokhon’ko and V. D. Mokhon’ko [19], is of essential importance in the theory of complex differential equations and functional equations.

Theorem A (see [2, 3]). *Let be a meromorphic function. Then for all irreducible rational functions in ,
**
with meromorphic coefficients , , the characteristic function of satisfies
**
where and .*

Noting that the difference analogue of Theorem A may not hold, we have obtained a result of this type in [16] by adding some additional assumptions as follows.

Theorem B (see [16]). *Suppose that is a difference polynomial of the form**
containing just one monomial of degree , and is a transcendental meromorphic function of finite order. If also satisfies , then we have
*

In this paper, we consider removing the assumption “ contains just one monomial of degree ’’ in Theorem B and obtain a weaker result, which is also generalized into differential-difference case. The concrete result can be seen in Section 2.

Next, we recall a classic result concerning Picard’s values of meromorphic functions and its derivatives, due to Hayman [20].

Theorem C (see [20]). *Let be a transcendental entire function. Then*(a)*for and , assumes all finite values infinitely often;*(b)*for , assumes all finite values except possibly zero infinitely often.*

Corresponding difference analogues of Theorem C can be seen in [12, 17].

Theorem D (see [12, 17]). *Let be a transcendental entire function of finite order, and let be a nonzero complex constant. Then*(a)*for and , assumes all finite complex values infinitely often;*(b)*for , assumes all finite complex values except possibly zero infinitely often.*

After Theorem C, many results have been obtained on the value distribution of differential polynomials. A typical one is as follows.

Theorem E (see [21, 22]). *Let be a transcendental meromorphic function with , and let be a differential polynomial in of the form
**
with no constant term. Furthermore, assume the degree, , of is greater than one and , , for all . Then for all . Moreover, if all the terms of have different degrees at least two, that is, is nonhomogeneous, then for all .*

We also consider deficiencies of difference polynomials of meromorphic functions of finite order in [16], which can be viewed as difference analogues of Theorem E, as well as generalizations of Theorem D.

In this paper, we proceed to investigate deficiencies of differential-difference polynomials of meromorphic functions. The concrete results can be seen in Section 3.

Examples are given in Section 4 to illustrate our results.

#### 2. A Differential-Difference Analogue of Valiron-Mokhon’ko Theorem

In what follows, we will consider differential-difference polynomials. A differential-difference polynomial is a polynomial in , its shifts, its derivatives, and derivatives of its shifts (see [14]), that is, an expression of the form where is a finite set of multi-indices , and () and are distinct complex constants. And we assume that the meromorphic coefficients , of are of growth . We denote the degree of the monomial of by . Then we denote the degree and the lower degree of by respectively. In particular, we call a homogeneous differential-difference polynomial if . Otherwise, is nonhomogeneous.

In the following, we assume and .

We prove a weaker differential-difference version of the classic Valiron-Mokhon’ko Theorem as follows.

Theorem 1. *Suppose that is a transcendental meromorphic function, and is a differential-difference polynomial of the form (6). If also satisfies and
**
then one has
*

*Remark 2. *If is a homogeneous differential-difference polynomial in addition, then

*Remark 3. *Especially, assumption (8) can be replaced by the assumption “’’. In fact, if satisfies , then is of regular growth, and (8) holds consequently.

To prove Theorem 1, we need the following lemmas.

Lemma 4 (see [6]). *Let be a nonconstant meromorphic function, , and . If , then
**
for all outside of a set of finite logarithmic measure.*

Lemma 5 (see [6]). *Let be a nondecreasing continuous function and let . If the hyperorder of is strictly less than one, that is, and , then
**
where runs to infinity outside of a set of finite logarithmic measure.*

It is shown in [23, p.66] and [7, Lemma ] that the inequality holds for and . And from the proof, the above relation is also true for counting function. By combing Lemma 5 and these inequalities, we immediately deduce the following lemma.

Lemma 6. *Let be a nonconstant meromorphic function of , and let be a nonzero complex constant. Then one has
*

Lemma 7. *Let be a transcendental meromorphic function of , and let be a differential-difference polynomial of the form (6); then we one has
**
Furthermore, if also satisfies
**
then one has
*

*Proof. *For , , we define . We also define
Thus,
By the definitions of and , , , we have
It follows by (19) and (20) that
Lemmas 4 and 6 and the logarithmic derivative lemma imply that, for and ,
Then (15) follows by (21) and (22).

It is easy to find that

Then (16), (23), and Lemma 6 yield that

Thus, (17) follows by (15) and (24).

Lemma 8. *Let be a transcendental meromorphic function of , and let be a differential-difference polynomial of the form (6); then one has
*

*Proof. *Similar to (19), we have
where and
By the definition of , we have . Thus, (20), (22), and (26) yield that
that is, (25).

Now, we can finish the proof of Theorem 1 in the end.

*Proof of Theorem 1. *We deduce from (8), (24), and Lemma 8 that
that is,
Then, (9) follows by (17) and (30).

#### 3. Deficiencies of Some Differential-Difference Polynomials

In the following, we assume that is a meromorphic function of growth .

In this section, we will apply Theorem 1 to consider the deficiencies of general homogeneous or nonhomogeneous differential-difference polynomials.

Theorem 9. *Suppose that is a transcendental meromorphic function satisfying and (8), and is a differential-difference polynomial of the form (6).*(a)*If is a homogeneous differential-difference polynomial, then one has
*(b)*If is a nonhomogeneous differential-difference polynomial with , then one has
**Thus, has infinitely many zeros, whether is homogeneous or nonhomogeneous.*

Furthermore, one considers some differential-difference polynomials of special forms, which are generalizations of both differential cases and difference cases, that is, Theorems C–E.

Theorem 10. *Suppose that is a transcendental meromorphic function satisfying and (16), is a differential-difference polynomial of the form (6), and , , is a polynomial of with meromorphic coefficients , of growth . If , , then
**
satisfies
**
Thus, has infinitely many zeros.*

When is of a special form , we can deduce the following result from Theorem 9.

Theorem 11. *Suppose that is a transcendental meromorphic function satisfying and (16), and is a differential-difference polynomial of the form (6). If and , then
**
satisfies , where . Thus, has infinitely many zeros.*

*Remark 12. *On the one hand, we can also apply Theorem 9 to with the assumption “’’ and obtain the same result as Theorem 10. But our present assumption “’’ has no concern with and , so we think Theorem 10 is better to some extent. On the other hand, we can also apply Theorem 10 to with the assumption “,’’ which is stronger than “’’ in Theorem 11, showing Theorem 11 is better to some extent.

Theorem 13. *Suppose that is a transcendental meromorphic function satisfying and (16), is a differential-difference polynomial of the form (6), and , , is a polynomial of with meromorphic coefficients , of growth . If , , then
**
satisfies
**Thus, has infinitely many zeros.*

When , one can consider some special cases as follows.

Theorem 14. *Suppose that is a transcendental meromorphic function satisfying and (16), and is a differential-difference polynomial of the form (6).*(a)*If , then
**satisfies
*(b)*If , , then satisfies
**Especially, it holds for .*(c)*If and also satisfies , then satisfies . Especially, it holds for .**Thus, has infinitely many zeros.*

If we assume that in addition, the following result follows immediately by Theorem 9.

Theorem 15. *Suppose that is a transcendental meromorphic function satisfying and (8), and is a differential-difference polynomial of the form (6). If , then satisfies . Thus, has infinitely many zeros.*

*Remark 16. *Noting that, when , hold, we see that the assumption “’’ in Theorem 14(a) is weaker than the assumption “’’ in Theorem 14(b). And these assumptions in Theorem 14 have no concern with ; thus they are different from the assumption “’’ in Theorem 15.

*Remark 17. *From the proofs behind, it is easy to find that
hold, respectively, in Theorems 9, 10, 13, 14(a) and (b), and 15.

Now, we give the proofs of Theorems 9–15.

*Proof of Theorem 9. *It follows by Theorem 1 that
We deduce from (8), (24), (25), and (42) that
Thus, an application of the second main theorem and (24), (42), and (43) imply that
(a)If , then it follows by (44) that
by which (31) holds.(b)If , then we deduce from (30) and (44) thatthat is,

Since , (32) follows immediately by (47).

*Proof of Theorem 10. *We deduce from (16), (17), and (24) that
hold. Next, we consider . Let be a zero of and distinguish three cases.(i) is not a zero of ; then must be a zero of and
where denotes the order of multiplicity of or zero according as is a pole of or not.(ii) is a zero of but not a pole of . Then
(iii) is a zero of and a pole of . Then
(24) and (50)–(52) yield that
Then (48), (49), (53), and an application of the second main theorem to imply that
consequently,

Moreover, by , (16), (24), (25), and Theorem A, we have
consequently,
On the other hand, the evident relation , where the definition of is given after (50), results in
We deduce from (57) and (58) that
Then (17), (55), and (59) yield thatthat is,
From (48) and (61), we deduce that

*Proof of Theorem 11. *Assume to the contrary that . Denoting
we deduce from (16) and (17) that
On the other hand, (16) and (24) yield that

Differentiating both sides of (63), we obtain
where . Clearly, we deduce from (16) and (24) that
Moreover, (64) and (65) yield that
It follows by (66)–(68) that
Then (16), (69), and the fact imply that the assumptions of Theorem 9(b) are satisfied. Thus, Theorem 9(b) yields that , a contradiction. Therefore, we have .

*Proof of Theorem 13. *We deduce from (16), (17), and (24) that
Denote

Now, we estimate the poles, the zeros, and 1-points of accurately. On the one hand, we see by (71) that the poles of occur at zeros of and poles of which are not simultaneously 1-points of , and those poles of which are zeros of but not simultaneously zeros of also have multiplicities at least . On the other hand, we also see by (71) that the zeros of occur at zeros of and poles of which are not simultaneously 1-points of . Moreover, 1-points of occur at zeros of and occur at the common poles, zeros of and with the same multiplicities. Thus, it follows by (16) and (24) that
Then (17), (72), and the second main theorem result in
that is,

Moreover, Theorem A and (17) imply that
that is,
Then (74) and (76) yield that
From (70) and (77), we deduce that

To prove Theorem 14(c), we also need the following lemma of one of Tumura-Clunie type theorems.

Lemma 18 (see [24]). *Let be a meromorphic function, and suppose that has small meromorphic coefficients , , in the sense of . Moreover, assume that . Then .*

*Proof of Theorem 14. *(a) We deduce from (16), (17), and (24) that
Denote
Differentiating both sides of (80), we obtain
that is,
It follows by (15)–(17), (24), (79), and (82) that
that is,
From (79) and (84), we deduce that

(b) It suffices to note that we may see as ; then Theorem 14(b) follows immediately by Theorem 13.

(c) By using a similar reasoning as [13, Theorem ], we can rearrange the expression for the differential-difference polynomial by collecting together all terms having the same total degree and then writing in the form . Now each of the coefficients is a finite sum of products of functions of the form , with each such product being multiplied by one of the original coefficients . We deduce from the logarithmic derivative lemma and Lemmas 4 and 6 that . Clearly, holds by (8) and Lemma 6. Thus, . Denote
Assume to the contrary that . Thus, Theorem A yields that
Then (8), (86), (87), Lemma 18, and the assumption that imply that ; that is,
Noting the fact that and , we deduce from Theorem A that (88) is a contradiction. Therefore, we have .

#### 4. Examples

*Example 1. *We consider nonhomogeneous differential-difference polynomials
and a homogeneous differential-difference polynomial
where , , , and . Clearly, the function satisfies (8) and . Then we have
This example shows that (9) is best possible.

*Example 2. *Consider again. Then the homogeneous case in Example 1 also illustrates Theorem 9(a). And the nonhomogeneous differential-difference polynomials , , in Example 1 also illustrate Theorem 9(b), where , , and . Next, we consider the nonhomogeneous differential-difference polynomial
where