# Closed form for a binomial containing a differential operator

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.

• Why not explain what you tried and say where you're stuck? – Fly by Night Oct 2 '13 at 21:13
• Where did this problem come from? – Mhenni Benghorbal Oct 2 '13 at 21:29
• 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. – Jas Ter Oct 3 '13 at 6:29
• 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. – gradi3nt Oct 3 '13 at 22:22