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 $\mathcal{L}^n_c(f)=\sum_{j=0}^{n}a_jf(z+jc)$, then $\mathcal{L}^n_c(f)\equiv f$, 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.