# Pincherle derivative

In mathematics, the Pincherle derivative of a linear operator T on the space of polynomials in x is another linear operator T′ defined by

[itex]T'=Tx-xT,[itex]

which means that for any polynomial f(x),

[itex]T'\left\{f(x)\right\}=T\left\{xf(x)\right\}-xT\left\{f(x)\right\}.[itex]

This is a derivation satisfying the sum and product rules: (T + S)′ = T′ + S′ and (TS)′ = TS + TS′, where TS is the composition of T and S.

If T is shift-equivariant, then so is T′. Every shift-equivariant operator on polynomials is of the form

[itex]\sum_{n=0}^\infty \frac{c_n D^n}{n!}[itex]

where D is differentiation with respect to x. When an operator is written in this form, then it is easy to find its Pincherle derivative in this form, by using the fact that

[itex](D^n)'=nD^{n-1},[itex]

which may be proved by mathematical induction.

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