# 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]

$T' := [T,x] = Tx-xT = -\operatorname{ad}(x)T,\,$

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 $\scriptstyle S$ and $\scriptstyle T$ belonging to $\scriptstyle \operatorname{End} \left( \mathbb K[x] \right)$

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

One also has $\scriptstyle{ [T,S]^\prime = [T^\prime , S] + [T, S^\prime ] }$ where $\scriptstyle{ [T,S] = TS - ST}$ 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 $\scriptstyle \operatorname{Diff}(\mathbb K [x])$.

The shift operator

$S_h(f)(x) = f(x+h) \,$

can be written as

$S_h = \sum_{n=0} {{h^n} \over {n!} }D^n$

by the Taylor formula. Its Pincherle derivative is then

$S_h' = \sum_{n=1} {{h^n} \over {(n-1)!} }D^{n-1} = h \cdot S_h.$

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

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

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 $\scriptstyle{ \delta ' = S_h }$.