Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2016; 53(1): 49-60

Printed January 31, 2016

https://doi.org/10.4134/BKMS.2016.53.1.49

Copyright © The Korean Mathematical Society.

Shared values and Borel exceptional values for high order difference operators

Liangwen Liao and Jie Zhang

Nanjing University, China University of Mining and Technology

Abstract

In this paper, we investigate the high order difference counterpart of Br\"uck's conjecture, and we prove one result that for a transcendental entire function $f$ of finite order, which has a Borel exceptional function $a$ whose order is less than one, if $\Delta ^nf$ and $f$ share one small function $d$ other than $a$ CM, then $f$ must be form of $f(z)=a+ce^{\beta z},$ where $c$ and $\beta$ are two nonzero constants such that $\frac{d-\Delta^na}{d-a}= {(e^{\beta}-1)}^n.$ This result extends Chen's result from the case of $\sigma(d)<1$ to the general case of $\sigma(d)<\sigma(f)$.

Keywords: uniqueness, entire function, difference equation, order

MSC numbers: Primary 30D35, 34M10