Shared values and Borel exceptional values for high order difference operators
Bull. Korean Math. Soc. 2016 Vol. 53, No. 1, 49-60
https://doi.org/10.4134/BKMS.2016.53.1.49
Printed January 31, 2016
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
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