Pincherle derivative
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In mathematics, the Pincherle derivative of a linear operator T on the space of polynomials in x is another linear operator T′ defined by
- <math>T'=Tx-xT,<math>
which means that for any polynomial f(x),
- <math>T'\left\{f(x)\right\}=T\left\{xf(x)\right\}-xT\left\{f(x)\right\}.<math>
This is a derivation satisfying the sum and product rules: (T + S)′ = T′ + S′ and (TS)′ = T′S + 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
- <math>\sum_{n=0}^\infty \frac{c_n D^n}{n!}<math>
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
- <math>(D^n)'=nD^{n-1},<math>
which may be proved by mathematical induction.