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What is the relationship between higher-order derivations (in the sense of Hasse-Schmidt) and differential operators?
Let $A$ and $B$ be $R$-algebras. A Hasse-Schmidt $m$-derivation $D : A \to B$ is a tuple $(D_0, D_1, \dots, D_m)$ of $R$-linear maps $A \to B$ satisfying the generalized Leibniz law,
$$ D_k(xy) = \...
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