Weighted composition–differentiation operators in the uniformly closed algebra generated by weighted composition operators

Let \(\varphi \) be an analytic self map of the open unit disc \(\mathbb {D}\). Assume that \(\psi \) is an analytic map of \(\mathbb {D}\). Suppose that f is in the Hardy space of the open unit disc \(H^p\). The operator that takes f into \(\psi \cdot f \circ \varphi \) is a weighted composition operator, and is denoted by \(C_{\psi ,\varphi }\). The operator that takes f into \(\psi \cdot f^\prime \circ \varphi \) is a weighted composition-differentiation operator. We prove that some weighted composition-differentiation operators belong to the closed algebra generated by weighted composition operators in the uniform operator topology.