# Differential Operator Simplification

Does anyone know the explicit formulation for the $q_k$'s in, $$(x+D)^n=\sum_{k=0}^n q_k(x)D^k\ \ \ \ ?$$

I know that $e^{-x^2/2+x}$ is a fixed point of $(x+D)$. I also, know that $$(x+D)H_n(x)e^{-x^2/2} = 2n H_{n-1}(x)e^{-x^2/2},$$ where $H_n(x)$ are the Hermite polynomials. Hence, $$(x+D)^n H_k(x)e^{-x^2/2} = 0$$ for all $k<n$. However, this knowledge hasn't proven useful yet.