Some results on zeros and the uniqueness of one certain type of high difference polynomials
© Zhang; licensee Springer 2012
Received: 10 July 2012
Accepted: 29 August 2012
Published: 12 September 2012
In this paper, we investigate the distribution of the zeros of some high differential-difference polynomials and obtain some uniqueness theorems with regard to it.
Keywordsuniqueness entire function difference equation small function
1 Introduction and main results
possibly outside of a set E with finite linear measure, not necessarily the same at each occurrence. A meromorphic function is said to be a small function with respect to if . We say that two meromorphic functions and share the value a IM (ignoring multiplicities) if and have the same zeros. If and have the same zeros with the same multiplicities, then we say that they share the value a CM (counting multiplicities). A polynomial is called a differential-difference polynomial in f if Q is a polynomial in f, its derivatives and shifts with meromorphic coefficients, say , such that for all . We define the difference operators and .
In 2007, Laine and Yang  considered zeros of one certain type of difference polynomials and obtained the following theorem.
Theorem A Let f be a transcendental entire function of finite order and c be a nonzero complex constant. If , then has infinitely many zeros, where .
Liu  considered the case of general difference products of a meromorphic function and made relevant improvements to Theorem A. Recently, a number of papers [5, 6] focusing on the distribution of zeros of some difference polynomials of differential types emerged. In this paper, we consider the general differential difference cases in some sense and obtain some results as follows.
then the differential-difference polynomial has infinitely many zeros.
Theorem 2 Let f be a transcendental entire function of finite order and , and be a small function with respect to . If , then the differential-difference polynomial has infinitely many zeros.
Theorem 3 Let f be a transcendental meromorphic function of finite order, λ be a nonzero constant, and be a small function with respect to . If , then the differential-difference polynomial has infinitely many zeros.
, where , i.e., if m, n are two prime integers, then ;
Recently, Yang and Laine  considered the following interesting differential equation and they proved
admits no transcendental entire solutions. If is a nonzero constant, then equation above admits three distinct transcendental entire solutions, provided .
They also presented some results on difference analogues of the equation in Theorem B and obtained the following theorem.
where is a nonconstant polynomial and are two nonzero constants, has no transcendental entire function of finite order. If is a nonzero constant, then the difference equation above admits three distinct transcendental entire functions of finite order, provided and for a nonzero integer n.
In this paper, we also consider a more general class of difference equation related to Theorem B and Theorem C and obtain the following results, which generalize the above related results.
has no transcendental entire function of finite order.
where , .
2 Some lemmas
To prove our results, we need some lemmas as follows.
Lemma 1 (see )
Lemma 2 (see )
Combing the two equations above, we can get our conclusion immediately. □
The proof of Lemma 4 is completed. □
Lemma 5 (see )
holds possibly outside of an exceptional set of finite logarithmic measure.
Lemma 6 (see )
has no transcendental meromorphic solutions satisfying .
Lemma 7 (see )
are not constants for ;
For , . (, ).
Lemma 8 (see )
Remark denotes the counting function of zeros of f where an m-fold zero is counted m times if and p times if .
Lemma 9 (see )
3 The proof of theorems
Next, we discuss the following two cases separately.
which contradicts our assumption .
which also contradicts our assumption . Therefore, has infinitely many zeros. The proof of Theorem 1 is completed. □
which contradicts our assumption . Thus we proved has infinitely many zeros. The proof of Theorem 2 is completed. □
which contradicts our assumption . So we proved has infinitely many zeros. The proof of Theorem 3 is completed. □
Next, we consider three cases according to Lemma 9.
which is impossible when .
Set , and suppose it is a nonconstant meromorphic function.
which is also impossible when . So .
which is impossible when . Thus, we obtain h is a constant such that . Furthermore, if , then and .
The proof of Theorem 4 is completed. □
which is impossible when . The proof of Theorem 5 is completed. □
where , are two constants. Next, we claim , are two nonzero constants.
By Lemma 7, we obtain , which is impossible. Thus , and we can obtain in a similar way. Thus, our claim holds.
If is a nonconstant polynomial, then we can obtain a contradiction from Equation (22) obviously.
If is a nonzero constant, then we can set according to Equation (22), where , . It is easy to see or −x.
If , then and we obtain from Equation (22), which is a contradiction. So .
The proof of Theorem 6 is completed. □
The author would like to thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This research was supported by the Fundamental Research Funds for the Central Universities (No. 2011QNA25).
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