A study on entire functions of hyper-order sharing a finite set with their high-order difference operators

Abstract

In this paper, due to the Borel lemma and Clunie lemma, we will deduce the relationship between an entire function $f$ of hyper-order less than 1 and its $n$-th difference operator $\Delta^{n}_{c}f(z)$ if they share a finite set and $f$ has a Borel exceptional value 0, where the set consists of two entire functions of smaller orders. Moreover, the exact form of $f$ is given and an example is provided to show the sharpness of the condition.