Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 + ...$ for some $r_i\in R$. As far as I know, the dual Hopf algebra $B$ of ${\mathcal O}(\text{Aut}(\hat{\mathbb A}^1))$ is called the Landweber-Novikov algebra. Apart from this algebro-geometric and also the original topological definition in terms of $MU$, it can also be described as a particular algebra of differential operators acting on the infinite polynomial ring ${\mathbb Z}[x_1,x_2,...]$ with all $x_i$ in degree $1$.

Q: Is there a way to see the action of $B$ on ${\mathbb Z}[x_1,x_2,...]$ form the above algebro-geometric definition?

$\text{Aut}(\hat{\mathbb A}^1)$ acts on the functor of $1$-dimensional formal group laws, but while this indeed results in an action of $B$ on an infinite polynomial ring, the Lazard ring, it can't be the one I'm looking for since the generators are not in the right degrees. Still, the Lazard ring naturally embeds into ${\mathbb Z}[x_1,x_2,...]$ through the map classifying the formal group law with the "universal logarithm", but I don't know if and how the action of $B$ can be extended along this emedding, or if it's even reasonable to expect this.

Any help and any comment about what I said is highly appreciated -- since I'm quite unfamiliar with the subject, it might well be that I already mixed up some things in what I said above.

Thank you!