# Talk:Pincherle derivative

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Mathematics rating:
 Start Class
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Field:  Algebra

the Pincherle derivative of a linear operator ${\displaystyle \scriptstyle {T:\mathbb {K} [x]\longrightarrow \mathbb {K} [x]}}$ on the vector space of polynomials in the variable ${\displaystyle \scriptstyle x}$ over a field ${\displaystyle \scriptstyle {\mathbb {K} }}$ is another linear operator ${\displaystyle \scriptstyle {T':\mathbb {K} [x]\longrightarrow \mathbb {K} [x]}}$ defined as
${\displaystyle T'=[T,x]=Tx-xT=-\operatorname {ad} (x)T,\,}$
${\displaystyle T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad \forall p(x)\in \mathbb {K} [x].}$