Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2011; 48(3): 499-512

Printed May 1, 2011

https://doi.org/10.4134/JKMS.2011.48.3.499

Copyright © The Korean Mathematical Society.

The Br\"{u}ck conjecture and entire functions sharing polynomials with their $k$-th derivatives

Feng L\"{u} and Hongxun Yi

China University of Petroleum, Shandong University

Abstract

The purpose of this paper is twofold. The first 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\"{u}ck conjecture with the idea of sharing polynomial.

Keywords: Vandermonde determinant, entire functions, Nevanlinna theory, uniqueness, normal family, differential equation

MSC numbers: 30D35, 30D45