Hermite–Genocchi formula

Remarks

The point $t_0 x_0 + \cdots + t_n x_n$ runs over the convex hull of the nodes, so the integrand is defined. For $n = 1$ the formula reads $\frac{f(x_1) - f(x_0)}{x_1 - x_0} = \int_0^1 f'(x_0 + t(x_1 - x_0)) \, dt$, which is the fundamental theorem of calculus. The general case follows by induction on $n$: integrating over the last variable and using the case $n - 1$ reproduces the recursion that defines divided differences.

Since $\Sigma_n$ is a simplex of volume $1/n!$, the formula gives $f[x_0, \dots, x_n] = f^{(n)}(\xi)/n!$ for some $\xi$ between the smallest and largest node. This is a form of the mean value theorem for divided differences. The right side is a continuous function of $(x_0, \dots, x_n)$ even when nodes coincide. So it serves as the definition of divided differences with repeated nodes, which is what Hermite interpolation uses.

Going deeper

For a student

The formula shows at once that $f[x_0, \dots, x_n]$ is symmetric in the nodes and depends continuously on them. Letting all nodes tend to one point $x$ gives $f[x, \dots, x] = f^{(n)}(x)/n!$, the Taylor coefficient. It is the bridge between Newton interpolation and Taylor polynomials.