Over a field $ k $ of characteristic zero, actions of $ \widehat{\mathbb{G}_{a}} $ on a variety $ \operatorname{Spec}(A) $ are obtained from $ A $-derivations $ \delta $ of $ A $ over $ k $ via the co-action below:
\begin{equation*}
a \mapsto \sum_{j=0}^{\infty} \frac{\delta^{j}(a)t^{j}}{j!}.
\end{equation*}
The sheaf $ \Omega_{A/k} $ parameterizes $ A $-derivations of $ A $ over $ k $.

If $ k $ is a field of positive characteristic then actions of $ \widehat{\mathbb{G}_{a}} $ are parameterized by iterative, higher derivations.  A collection of $ k $-homomorphisms $ \phi_{i}: A \to A $ is a higher derivation if $ \phi_{0} = \operatorname{id}_{A} $ and $ \phi_{n}(a_{1}a_{2}) = \sum_{(i,j) \in \mathbb{N}_{0}^{2}} \phi_{i}(a_{1}) \phi_{j}(a_{2}) $.  A higher derivation is called iterative if $ \phi_{i} \circ \phi_{j} = \binom{i+j}{i} \phi_{i+j} $.  One obtains an action of $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spec}(A) $ from an iterative, higher derivation $ \{\phi_{i}\}_{i \in \mathbb{N}_{0}} $ via the co-action below:
\begin{equation*}
a \mapsto \sum_{j=0}^{\infty} \phi_{j}(a)t^{j}.
\end{equation*}

One could generalize the notion of a $ B $-derivation of $ M $ over $ A $ as follows.  Let $ J_{B/A} $ be the kernel of the ring homomorphism $ B \otimes_{A} B \to B $ which sends $ b_{1} \otimes b_{2} $ to $ b_{1}b_{2} $.  Define $ \delta_{i}(b) $ to be the image of $ b \otimes_{A} 1 $ in $ J_{B/A}^{i}/J_{B/A}^{i+1} $.  A higher $ B $-derivation of $ M $ over $ A $ is a homomorphism $ \phi: B \to \operatorname{gr}_{J_{B/A}}(B \otimes_{A} M) $ such that if $ \phi_{i}(b) \in J_{B/A}^{i}(B \otimes_{A} M)/J_{B/A}^{i+1} $, then $ \phi_{n}(b_{1}b_{2}) = \sum_{(i,j) \in \mathbb{N}_{0}^{2}} \delta_{i}(b_{1})\phi_{j}(b_{2}) $.

Has anyone tried to generalize the notion of higher differentials so that one could create a sheaf $ \Xi_{A/k} $ with a similar universal property to that of $ \Omega_{A/k} $ for derivations?  Specifically are there any references in the literature of such a generalization and sheaf $ \Xi_{A/k} $?