Uniqueness of entire functions concerning difference polynomials. (English) Zbl 1340.30133

Meng, Chao

Math. Bohem. 139, No. 1, 89-97 (2014).

Summary: In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let \(f(z)\) and \(g(z)\) be two transcendental entire functions of finite order, and \(\alpha (z)\) a small function with respect to both \(f(z)\) and \(g(z)\). Suppose that \(c\) is a non-zero complex constant and \(n\geq 7\) (or \(n\geq 10\)) is an integer. If \(f^{n}(z)(f(z)-1)f(z+c)\) and \(g^{n}(z)(g(z)-1)g(z+c)\) share “\((\alpha (z),2)\)” (or \((\alpha (z),2)^{*}\)), then \(f(z)\equiv g(z)\). Our results extend and generalize some well known previous results.

MSC:

30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39A05	General theory of difference equations

Keywords:

entire function; difference polynomial; uniqueness