# 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]

$T':=[T,x]=Tx-xT=-\operatorname {ad} (x)T,\,$ (for the origin of the ad notation, see the article on the adjoint representation) so that

$T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad \forall p(x)\in \mathbb {K} [x].$ 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 $S$  and $T$  belonging to $\operatorname {End} \left(\mathbb {K} [x]\right)$

1. ${(T+S)^{\prime }=T^{\prime }+S^{\prime }}$  ;
2. ${(TS)^{\prime }=T^{\prime }\!S+TS^{\prime }}$  where ${TS=T\circ S}$  is the composition of operators ;

One also has ${[T,S]^{\prime }=[T^{\prime },S]+[T,S^{\prime }]}$  where ${[T,S]=TS-ST}$  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

$D'=\left({d \over {dx}}\right)'=\operatorname {Id} _{\mathbb {K} [x]}=1.$

This formula generalizes to

$(D^{n})'=\left({{d^{n}} \over {dx^{n}}}\right)'=nD^{n-1},$

by induction. It proves that the Pincherle derivative of a differential operator

$\partial =\sum a_{n}{{d^{n}} \over {dx^{n}}}=\sum a_{n}D^{n}$

is also a differential operator, so that the Pincherle derivative is a derivation of $\operatorname {Diff} (\mathbb {K} [x])$ .

When $\mathbb {K}$  has characteristic zero, the shift operator

$S_{h}(f)(x)=f(x+h)\,$

can be written as

$S_{h}=\sum _{n\geq 0}{{h^{n}} \over {n!}}D^{n}$

by the Taylor formula. Its Pincherle derivative is then

$S_{h}'=\sum _{n\geq 1}{{h^{n}} \over {(n-1)!}}D^{n-1}=h\cdot S_{h}.$

In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars ${\mathbb {K} }$ .

If T is shift-equivariant, that is, if T commutes with Sh or ${[T,S_{h}]=0}$ , then we also have ${[T',S_{h}]=0}$ , so that $T'$  is also shift-equivariant and for the same shift $h$ .

The "discrete-time delta operator"

$(\delta f)(x)={{f(x+h)-f(x)} \over h}$

is the operator

$\delta ={1 \over h}(S_{h}-1),$

whose Pincherle derivative is the shift operator ${\delta '=S_{h}}$ .