Compact and order bounded sum of weighted differentiation composition operators

In this paper, we characterize bounded, compact and order bounded sum of weighted differentiation composition operators from Bergman type spaces to weighted Banach spaces of analytic functions, where the sum of weighted differentiation composition operators is defined as $$S_{\vec{u},\tau }^n(f)={\displaystyle \sum _{j=0}^n{D}_{u_j,\tau }^j(f),f\in \mathcal{H}(\mathbb{D})\mathrm{.}}$$ Here $\mathcal{H}(\mathbb{D})$ is the space of all holomorphic functions on $\mathbb{D}$, $\vec{u}=\{u_j{\}}_{j=0}^n$, $u_j\in \mathcal{H}(\mathbb{D})$, $\tau$ a holomorphic self-map of $\mathbb{D}$, $f^{(j)}$ the $j$th derivative of $f$ and weighted differentiation composition operator $D_{u_j,\tau }^j$ is defined as $D_{u_j,\tau }^j(f)=u_j{C}_{\tau }{D}^j(f)=u_j{f}^{(j)}\circ \tau ,f\in \mathcal{H}(\mathbb{D})\mathrm{.}$