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. 



Differential operator simplification
(Updated Jan. 2 and 3, 2022)


I had forgotten this question by the time I wrote up last year in OEIS [A344678](https://oeis.org/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](https://www.researchgate.net/publication/226801341_Evolution_operator_equations_Integration_with_algebraic_and_finitedifference_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)](https://mathoverflow.net/questions/107159/pochhammer-symbol-of-a-differential-and-hypergeometric-polynomials/107191#107191); 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](https://oeis.org/A094638)).


There are numerous other OEIS entries linked to normal reordering, several related to this [MSE-Q](https://math.stackexchange.com/questions/116633/is-there-a-formula-similar-to-fxa-ea-fracddxfx-to-express-f-al). A hub OEIS entry for iterated Lie derivatives $(g(x)D\_x)^n$ is [A145271](https://oeis.org/A145271) with several refs.
[This answer](https://mathoverflow.net/questions/97512/in-splendid-isolation/257205#257205) and [this one](https://mathoverflow.net/questions/97512/in-splendid-isolation/98213#98213) 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](https://arxiv.org/abs/1010.0354)" by Blasiak and Flajolet.


