Value Distribution of Certain Type of Difference Polynomials

We investigate the value distribution of difference product , for and , respectively, where is a transcendental entire function of finite order and are constants satisfying . 1. Introduction In this paper, we assume that the reader is familiar with the basic notions of Nevanlinna’s value distribution theory (see [1–3]). The notation is defined to be any quantity satisfying as , possibly outside a set of finite linear measures. In addition, we use the notation to denote the order of growth of the meromorphic function and to denote the exponent of convergence of zeros of . Hayman proved the following theorem in [4]. Theorem 1. Let be a transcendental integral function and let be an integer; then assumes all values except possibly zero infinitely often. Clunie proved that if , then Theorem 1 remains valid. Recently, many papers (see [5–17]) focus on complex difference. They obtain many new results on difference using the value distribution theory of meromorphic functions. In [12], Laine and Yang found a difference analogue of Hayman’s result as follows. Theorem 2. Let be a transcendental entire function of finite order and a nonzero complex constant. Then for , assumes every nonzero value infinitely often. Liu and Yang [14] proved the following theorem. Theorem 3. Let be a transcendental entire function of finite order and let be a nonzero complex constant, . Then for , has infinitely many zeros, where is a polynomial in . Chen [6] proved the following theorem. Theorem 4. Let be a transcendental entire function of finite order and let be a constant satisfying . Set where , and is an integer. Then the following statements hold.(i)If satisfies or has infinitely many zeros, then has infinitely many zeros.(ii)If has only finitely many zeros and , then has only finitely many zeros. It is natural to ask what condition will guarantee that assumes every nonzero and zero value infinitely often, where is a linear th order difference operator with varying shifts, operating on a transcendental entire function of finite order. In this paper, we consider the above question for and , respectively, and obtain the following results. Theorem 5. Let be a transcendental entire function of finite order and let , ？？ be constant satisfying and when . Set , where are integers. Then the following statements hold.(i)If satisfies or has infinitely many zeros, then has infinitely many zeros.(ii)If has only finitely many zeros and , then has only finitely many zeros.(iii) has infinitely many zeros, and , where is a small function of . Remark 6. The result of Theorem 5 may be false if