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SHARED VALUES AND BOREL EXCEPTIONAL VALUES FOR HIGH ORDER DIFFERENCE OPERATORS

  • Liao, Liangwen (Department of Mathematics Nanjing University) ;
  • Zhang, Jie (College of Science China University of Mining and Technology)
  • Received : 2014.11.29
  • Published : 2016.01.31

Abstract

In this paper, we investigate the high order difference counterpart of $Br{\ddot{u}}ck^{\prime}s$ 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

Acknowledgement

Supported by : Central Universities, NSF of China

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