In mathematics, the Pincherle derivative <math>T'</math> of a linear operator <math>T: \mathbb{K}[x] \to \mathbb{K}[x]</math> on the vector space of polynomials in the variable x over a field <math>\mathbb{K}</math> is the commutator of <math>T</math> with the multiplication by x in the algebra of endomorphisms <math>\operatorname{End}(\mathbb{K}[x])</math>. That is, <math>T'</math> is another linear operator <math>T': \mathbb{K}[x] \to \mathbb{K}[x]</math>
:<math>T' := [T,x] = Tx-xT = -\operatorname{ad}(x)T,\,</math>
(for the origin of the <math>\operatorname{ad}</math> notation, see the article on the adjoint representation) so that
:<math>T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad\forall p(x)\in \mathbb{K}[x].</math>
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 <math>S</math> and <math>T</math> belonging to <math>\operatorname{End}\left( \mathbb{K}[x] \right),</math>
- <math>(T + S)^\prime = T^\prime + S^\prime</math>;
- <math>(TS)^\prime = T^\prime\!S + TS^\prime</math> where <math>TS = T \circ S</math> is the composition of operators.
One also has <math>[T,S]^{\prime} = [T^{\prime}, S] + [T, S^{\prime}]</math> where <math>[T,S] = TS - ST</math> 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
: <math>D'= \left({d \over {dx\right)' = \operatorname{Id}_{\mathbb K [x]} = 1.</math>
This formula generalizes to
: <math>(D^n)'= \left(\right)' = nD^{n-1},</math>
by induction. This proves that the Pincherle derivative of a differential operator
: <math>\partial = \sum a_n