Unicity on entire function concerning its differential-difference operators

TL;DR

This paper investigates the uniqueness of entire functions concerning their differential-difference operators in several complex variables, establishing conditions under which two functions must be identical or have specific exponential forms.

Contribution

It introduces new uniqueness theorems for entire functions involving differential-difference polynomials in multiple complex variables, extending previous results to higher dimensions.

Findings

If functions share two small meromorphic functions under certain conditions, they are either identical or have specific exponential forms.

The paper characterizes the form of entire functions when related through differential-difference operators.

Special case results include the identity of a function with its difference operator raised to a power.

Abstract

In this paper, we study the uniqueness of the differential-difference polynomials of entire functions on $\mathbb{C}^{n}$. We prove the following result: Let $f(z)$ be a transcendental entire function on $\mathbb{C}^{n}$ of hyper-order less than $1$ and $g(z)=b_{-1}+\sum_{i=0}^{n}b_{i}f^{(k_{i})}(z+\eta_{i})$, where $b_{-1}$ and $b_{i} (i=0\ldots,n)$ are small meromorphic functions of $f$ on $\mathbb{C}^{n}$, $k_{i}\geq0 (i=0\ldots,n)$ are integers, and $\eta_{i} (i=0\ldots,n)$ are finite values. Let $a_{1}(z)\not\equiv\infty, a_{2}(z)\not\equiv\infty$ be two distinct small meromorphic functions of $f(z)$ on $\mathbb{C}^{n}$. If $f(z)$ and $g(z)$ share $a_{1}(z)$ CM, and $a_{2}(z)$ IM. Then either $f(z)\equiv g(z)$ or $a_{1}=2a_{2}=2$, $$f(z)\equiv e^{2p}-2e^{p}+2,$$ and $$g(z)\equiv e^{p},$$ where $p(z)$ is a non-constant entire function on $\mathbb{C}^{n}$. Especially, in the case of $g(z)=(\Delta_{\eta}^{n}f(z))^{k}$, we obtain $f(z)\equiv (\Delta_{\eta}^{n}f(z))^{k}$.