Complex symmetric weighted composition-differentiation operators on Hardy-type spaces

In this note, we completely characterize the complex symmetric weighted composition-differentiation operator on the Hardy space $H^2$ with respect to the conjugation operator $C_{\lambda,\xi}$. The normal and self-adjoint of weighted composition-differentiation operators on the Hardy space $H^2$ are also studied. We also give the results on $S^2_{\alpha,\beta}$ and $\mathcal{H}^2_1(\mathbb{D})$.