Pincherle derivative
In mathematics, the Pincherle derivative T ′ {\displaystyle T'} of a linear operator T : K [ x ] → K [ x ] {\displaystyle T:\mathbb {K} [x]\to \mathbb {K} [x]} on the vector space of polynomials in the variable x over a field K {\displaystyle \mathbb {K} } is the commutator of T {\displaystyle T} with the multiplication by x in the algebra of endomorphisms End ( K [ x ] ) {\displaystyle \operatorname {End} (\mathbb {K} [x])} . That is, T ′ {\displaystyle T'} is another linear operator T ′ : K [ x ] → K [ x ] {\displaystyle T':\mathbb {K} [x]\to \mathbb {K} [x]} T ′ := [ T , x ] = T x − x T = − ad ( x ) T , {\displaystyle T':=[T,x]=Tx-xT=-\operatorname {ad} (x)T,\,} (for the origin of the ad {\displaystyle \operatorname {ad} } notation, see the article on the adjoint representation) so that T ′ { p ( x ) } = T { x p ( x ) } − x T { p ( x ) } ∀ p ( x ) ∈ K [ x ] .