Uniqueness on Meromorphic function concerning their differential-difference operators

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].