Pincherle derivative

Summary

The Pincherle derivative of a linear operator $T$ on polynomials is the commutator $T' = Tx - xT$ of $T$ with multiplication by $x$. It behaves like a derivative: it is linear and satisfies the product rule $(ST)' = S'T + ST'$. It turns the differentiation operator into the identity, and each shift operator $f(x) \mapsto f(x + h)$ into $h$ times itself. It is a basic tool in umbral calculus.