Pincherle derivative

In mathematics, the Pincherle derivative[1] 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).


The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators   and   belonging to  

  1.   ;
  2.   where   is the composition of operators ;

One also has   where   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


is also a differential operator, so that the Pincherle derivative is a derivation of  .

When   has characteristic zero, the shift operator


can be written as


by the Taylor formula. Its Pincherle derivative is then


In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars  .

If T is shift-equivariant, that is, if T commutes with Sh or  , then we also have  , so that   is also shift-equivariant and for the same shift  .

The "discrete-time delta operator"


is the operator


whose Pincherle derivative is the shift operator  .

See alsoEdit


  1. ^ Rota, Gian-Carlo; Mullin, Ronald (1970). Graph Theory and Its Applications. Academic Press. pp. 192. ISBN 0123268508.

External linksEdit