Pincherle derivative

In mathematics, the Pincherle derivative T’ of a linear operator T:K[x] → K[x] on the vector space of polynomials in the variable x over a field K is the commutator of T with the multiplication by x in the algebra of endomorphisms End(K[x]). That is, T’ is another linear operator T’:K[x] → K[x]

so that

This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).

The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators ${\displaystyle \scriptstyle S}$ and ${\displaystyle \scriptstyle T}$ belonging to ${\displaystyle \scriptstyle \operatorname {End} \left(\mathbb {K} [x]\right)}$

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