Does anyone know of a reference or have any idea for an explicit description of the ring of differential operators for polynomial algebras over the integers? I'm hoping there is something analogous to the case of a polynomial ring over a field $K$, where if the field has characteristic $0$ an explicit description is given by the Weyl algebra. In particular, for $K[x_1,\ldots,x_n]$ this is the noncommutative $K$-algebra spanned by the symbols $x_1,\ldots,x_n$ and $d_1,\ldots,d_n$ where the $x_i$'s commute, the $d_i$'s commute and one has the Leibniz rule: $d_i x_j - x_j d_i = \delta_{i,j}$ where $\delta_{i,j}$ is the Kronecker delta.

If the field has characteristic $p > 0$, then one also has divided powers $\frac{1}{p!}(\frac{d^p}{dx_i^p})$, (Hasse derivatives or hyperdifferential operators) which are not in the Weyl algebra but can be constructed using Frobenius. Of course, for $\mathbf{Z}[x_1,\ldots,x_n]$ the divided powers $\frac{1}{t!}(\frac{d^t}{dx_i^t})$ for integers $t > 1$ are not in the Weyl algebra. Is the full ring of differential operators spanned by the Weyl algebra and these divided powers? Is there a finite generating set?