On Hasse-Schmidt higher derivations
Let k be a field of characteristic zero and let A be a commutative k- algebra. A higher derivation D of A over k is a sequence Ḏ\(=\{D_ 0,D_ 1,D_ 2,...\}\) of additive k-endomorphisms \(D_ i's\) such that \(D_ 0\) is the identity map of A and \(D_ n(ab)=\sum^{n}_{m=0}D_ m(a)D_{n-m}(b)\quad for\) every a,b\(\in A\). The set of ordered partitions of n into the r-positive integers is denoted by \(P_{n,r}.\) Theorem: There is a one to one correspondence between the set of ordered sequences of k-derivations on A and the set of higher derivations on A in such a way that if \(\{\delta_ n: n\geq 0,\quad \delta_ 0\quad identity,\quad \delta_ n\quad is\quad a\quad k-derivation\}\) and the higher derivation \(D=\{D_ n: n\geq 0\}\) correspond, then \(D_ n=\sum^{n}_{r=1}(r!)^{-1}\sum_{(n_ 1,...,n_ r)\in P_{n,r}}\delta_{n_ 1}\delta_{n_ 2}...\delta_{n_ r}\quad and\delta_ n=\sum^{n}_{r=1}(-1)^{r+1}\cdot r^{-1}\sum_{(n_ 1,...,n_ r)\in P_{n,r}}D_{n_ 1}D_{n_ 2}...D_{n_ r}\quad for\) every \(n\geq 1\) and \(D_ 0=\delta_ 0\).
scientific article; zbMATH DE number 3985389