A note on entire functions sharing a finite set with applications to difference equations

Value distribution and uniqueness problems of difference operator of an entire function have been investigated in this article. This research shows that a finite ordered entire function $f$ when sharing a set $\mathcal{S}=\{\alpha (z),\beta (z)\}$ of two entire functions $\alpha$ and $\beta$ with $\max \{\rho (\alpha ),\rho (\beta )\}<\rho (f)$ with its difference ${\mathcal{L}}_c^n(f)={\sum }_{j=0}^n{a}_{j}f(z+jc)$, then ${\mathcal{L}}_c^n(f)\equiv f$, and more importantly certain form of the function $f$ has been found. The results in this paper improve those given by \emph{k. Liu}, \emph{X. M. Li}, \emph{J. Qi, Y. Wang and Y. Gu} etc. Some constructive examples have been exhibited to show the condition $ \max{\rho(\alpha), \rho(\beta)}<\rho(f) $ is sharp in our main result. Examples have been also exhibited to show that if $CM$ sharing is replaced by $IM$ sharing, then conclusion of the main results ceases to hold.