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