 Derivatives of Bernstein polynomials

Differentiating the Bernstein basis $p_{n,j}(x)=\left(\genfrac{}{}{0ex}{}{n}{j}\right)x^j(1-x{)}^{n-j}$ gives $p_{n,j}^{\prime }=n(p_{n-1,j-1}-p_{n-1,j})$, with out-of-range terms zero. Telescoping coefficient differences and induction give the displayed formula, where $(n{)}_r=n!/(n-r)!$ and $0\le r\le n$. For $n>r$, set $g_{n,r}(t)=(n{)}_r{\Delta }_{1/n}^{r}f((n-r)t/n)$; then $B_n^{(r)}f=B_{n-r}{g}_{n,r}$. The rescaled argument compensates for the different sampling meshes.