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.


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

  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.