Uniqueness on Meromorphic function concerning their differential-difference operators

TL;DR

This paper investigates the uniqueness of meromorphic functions concerning their differential-difference operators, establishing conditions under which the functions are identical based on shared values, using novel methods to improve previous results.

Contribution

It introduces a new approach to prove the uniqueness of meromorphic functions related to differential-difference operators, extending and improving prior work by Chen-Xu.

Findings

Proves that under certain sharing conditions, a meromorphic function equals its differential-difference derivative.

Uses a novel method to establish uniqueness results for meromorphic functions.

Improves upon previous results by Chen-Xu in the field of complex analysis.

Abstract

In this paper, we study the uniqueness of the differential-difference of meromorphic functions. We prove the following result: Let $f$ be a nonconstant meromorphic function of $\rho_{2}(f)<1$, let $\eta$ be a non-zero complex number, $n\geq1, k\geq0$ two integers and let $a\not\equiv0,\infty$ be a small function of $f$. If $f$ and $(\Delta_{\eta}^{n}f)^{(k)}$ share $0,\infty$ CM and share $a$ IM, then $f\equiv(\Delta_{\eta}^{n}f)^{(k)}$, which use a completely different method to improve some results due to Chen-Xu [1].