Uniqueness of Meromorphic Functions Concerning Derivatives and Shifts

In this paper, we mainly prove: Let $$f$$ be a nonconstant meromorphic function of $$\rho_{2}(f)<1$$ , let $$k(\geq 2)$$ be a positive integer, and let $$a,c$$ be two nonzero complex numbers. Suppose that $$f(z),f^{(k)}(z+c)$$ share $$a$$ IM and $$f(z),f(z+c)$$ share $$\infty$$ CM. If $$N\left(r,\frac{1}{f(z)}\right)+N\left(r,\frac{1}{f^{(k)}(z+c)}\right)=S(r,f),\quad\text{then }f(z)\equiv f^{(k)}(z+c).$$ The results obtained in this paper improve and extend the results due to Majumder [Commun. Math. Stat., 2017] and Kaish and Mobdal [Commun. Korean Math. Soc., 2024].