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- T' = Tx - xT,
which means that for any polynomial f(x),
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
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
- (Dn)' = nDn - 1,
which may be proved by mathematical induction.
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