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.
(Updated Jan. 2 and 3, 2022)
I had forgotten this question by the time I wrote up last year in OEIS A344678 a fairly complete characterization of the coefficients of the normal-ordering of the powers of $R = x+D$, the raising op for a family of Hermite polynomials.
For those interested in disentangling operators in general, see the ref in my initial answer below and also "Evolution operator equations: integration with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory" by Dattoli, Ottaviabi, Torre, and Vasquez.
Re-ordering of iterated differential ops in terms of summands of the form $x^pD_x^q$ or $x^q(xD)^p$ gives rise to many classical special functions/sequences of polynomials important in combinatorics, analysis, and mathematical physics, e.g., the confluent hypergeometric polynomials, encompassing Hermite, Laguerre, Lah, and generalized Laguerre polynomials, associated with $(D_xx)^n$, $(xD_xx)^n = x^nD_x^nx^n$, $x^{-\alpha}(xD_xx)^nx^{\alpha}$ and $\binom{xD_x +\alpha+\beta}{\beta}$ (see this MO-Q); the Bell/Touchard/exponential/Stirling polynomials of the second kind, with $(xD_x)^n$; and the Stirling polynomials of the first kind and the generalized 'factorial' or generalized Witt polynomials related to $(x^rD_x)^n$ (see A094638).
There are numerous other OEIS entries linked to normal reordering, several related to this MSE-Q. A hub OEIS entry for iterated Lie derivatives $(g(x)D_x)^n$ is A145271 with several refs. This answer and this one to the MO-Q "In 'Splendid Isolation'" allude to related work, neglected in most accounts, by Charles Graves (and by Pincherle) on the commutator/operator derivatives $[f(L),R] = f'(L)$ and $[L,f(R)] = f'(R)$, where $L$ and $R$ are lowering/annihilation/destruction and raising/creation ladder ops, and the flow equation $\exp[tg(x)D_x] W(x) = W[f^{(-1)}(f(x)+t)]$ where $g(x) = 1/f'(x)$.
Initial post (Jun, 2014):
See "Combinatorial models of creation-annihilation" by Blasiak and Flajolet.
