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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.

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See Combinatorial Models of Creation-Annihilation by Blasiak and Flajolet. http://arxiv.org/abs/1010.0354

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    $\begingroup$ Great resource! Proposition 2 is exactly my problem, but more than that is a database of differential operator manipulations. Thanks. $\endgroup$ – Bobby Ocean Jun 29 '14 at 2:00

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