Chao Meng, School of Science, Shenyang Aerospace University, P. O. Box 110136, Shenyang, China, e-mail: mengchaosau@163.com
Abstract: 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 $n\geq7$ (or $n\geq10$) 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.
Keywords: entire function; difference polynomial; uniqueness
Classification (MSC 2010): 30D35, 39A05
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