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

T' = Tx − xT,

which means that for any polynomial f(x),

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

This concept is named after Salvatore Pincherle (1853—1936).

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

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

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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