## 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.

### 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
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