Is there a closed form for $(x + D)^n$ where D is the differential operator with respect to x?

Avi's comment in this post helped me understand this expression a bit better, but I'm still curious if you can write it out with sums and binomial coefficients in some clever way.

Edit: I ran into this expression while deriving properties of the Hermite polynomials so I could use them in the context of a harmonic oscillator. This question is purely out of mathematical curiosity.

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    $\begingroup$ Why not explain what you tried and say where you're stuck? $\endgroup$ – Fly by Night Oct 2 '13 at 21:13
  • $\begingroup$ Where did this problem come from? $\endgroup$ – Mhenni Benghorbal Oct 2 '13 at 21:29
  • $\begingroup$ This operator is proportional to the lowering operator for the quantum mechanical harmonic oscillator. As such it has a lot of really nice properties, for example $\exp(-\alpha(x+D))$ is a translation operator in phase space. But I don't know if $(x+D)^n$ has a simple expression. $\endgroup$ – Jas Ter Oct 3 '13 at 6:29
  • $\begingroup$ My gut tells me there isn't. I haven't seen it written out, but I figured I'd ask. I thought maybe there was a fancy series of named numbers that you could use, or something similar to that. $\endgroup$ – gradi3nt Oct 3 '13 at 22:22

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