Uniqueness of differential polynomial of meromorphic function with its $q$-shift

In the paper, we apply the concept of weighted sharing to study the uniqueness problems of differential polynomial of meromorphic function of zero order with its $q$-shift. The results of the paper improve and extend some recent results due to H. P. Waghamore and M. M. Manakame [Int. J. Open Problems Compt. Math., 18 (2025), 22-34]. A typical theorem obtained in the paper is as follows: Let $P$ be a polynomial, $f(z)$ be a non-constant meromorphic function of zero-order. Suppose that $q$ is a non-zero complex constant, $\eta \in \mathbb{C}$ and $n$ is an integer satisfying $n\geq m+3\tau+3\Omega +6$, where $m=\deg P$, $\tau =\sum_{j=1}^{s}\mu _{j}$ and $\Omega =\sum_{j=1}^{s}j\mu _{j}.$ If $f^{n}(z)P(f(z))\prod _{j=1}^{s}f^{(j)}(z)^{\mu _{j}}$ and $f^{n}(qz+\eta )P(f(qz+\eta ))\prod _{j=1}^{s}f^{(j)}(qz+\eta )^{\mu _{j}}$ share $(1,2)$ and $(\infty,\infty)$, then $\displaystyle f^{n}(z)P(f(z))\prod _{j=1}^{s}f^{(j)}(z)^{\mu _{j}}\equiv f^{n}(qz+\eta )P(f(qz+\eta ))\prod _{j=1}^{s}f^{(j)}(qz+\eta )^{\mu _{j}}.$ Three other similar theorems are also obtained in the paper.