In this paper, we explore the uniqueness problem of differential polynomials $(Q(f))^{(k)}$ and $(Q(g))^{(k)}$ of meromorphic functions $f$ and $g$, respectively with the notion of weighted sharing a set of roots of unity, where $Q$ is a polynomial of one variable. The results of the paper generalize the results due to Sultana and Sahoo[ Mathematica Bohemica, 2024].

In this paper, the meromorphic function means the meromorphic function in the complex plane. We use the standard notations of Nevanlinna theory, which can be found in \cite{ hay, yang, yi}.

A meromorphic function $a \ (\not\equiv 0,\infty)$ is said to be small with respect to $f$ provided $T(r,a)=S(r,f)$ as $r\rightarrow\infty, \ r\not\in E$ where $E$ is a set of positive real numbers of finite Lebesgue measure. We denote by $S(f)$ the collection of all small functions of $f$.

For any two non-constant meromorphic functions $f$ and $g$, and $a\in S(f)\cap S(g)$, we say that $f$ and $g$ share $a$ CM(IM) provided that $f-a$ and $g-a$ have the same zeros counting (ignoring) multiplicities.