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

TL;DR

This paper investigates the value sharing and uniqueness of entire functions with respect to difference operators, establishing conditions under which the function must satisfy a specific functional equation, and improves upon previous results in the field.

Contribution

It introduces new conditions for entire functions sharing a finite set with their difference operators, extending and refining earlier work in value distribution theory.

Findings

If an entire function shares a set with certain entire functions, then it satisfies a specific difference equation.

The condition on the order of the functions is shown to be sharp through examples.

Replacing CM sharing with IM sharing invalidates the main conclusions.

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.