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

Abstract

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.