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"]
(for the origin of the ad notation, see the article on the adjoint representation) so that 
This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).
Properties.
The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators formula_3 and formula_4 belonging to formula_5
One also has formula_9 where formula_10 is the usual Lie bracket, which follows from the Jacobi identity.
The usual derivative, "D" = "d"/"dx", is an operator on polynomials. By straightforward computation, its Pincherle derivative is
This formula generalizes to
by induction. It proves that the Pincherle derivative of a differential operator