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The purpose of this paper is twofold. Therst is to establish a uniqueness theorem for entire function sharing two polynomials with its k-th derivative, by using the theory of normal families. Meanwhile, the theorem generalizes some related results of Rubel and Yang and of Li and Yi. Several examples are provided to show the conditions are necessary. The second is to generalize the Br uck conjecture with the idea of sharing polynomial.

THE BRÜCK CONJECTURE AND ENTIRE FUNCTIONS SHARING POLYNOMIALS WITH THEIR κ-TH DERIVATIVES