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

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.


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