Uniqueness Results for Meromorphic Functions Involving Differential-Difference Polynomials and Shared Values

Abstract:

      Throughout this work, we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial. Under the condition of finite order, we establish three distinct uniqueness results for a meromorphic function $f$ associated with the differential-difference polynomial $L^{n}_{\eta}f=\sum_{k=0}^{n}a_{k}f(z+k\eta)+a_{-1}f'$. These findings lead to a refined characterization of $f(z)\equiv L^{n}_{\eta}f(z)$. Several illustrative examples are provided to demonstrate the sharpness and precision of the results obtained in this study.