Divided differences

Remarks

The value $f[x_0, \dots, x_n]$ is a symmetric function of the points: it does not depend on their order. It has the closed form $$f[x_0, \dots, x_n] = \sum_{j=0}^{n} \frac{f(x_j)}{\prod_{k \ne j} (x_j - x_k)}.$$ It is the coefficient of $x^n$ in the interpolation polynomial of $f$ at $x_0, \dots, x_n$. For a real function $f$ with enough derivatives, the definition extends to repeated points by continuity; for example $f[x, x] = f'(x)$.

Example: for $f(x) = x^2$ and the points $0, 1, 3$: $f[0, 1] = 1$, $f[1, 3] = \frac{9 - 1}{2} = 4$, and $f[0, 1, 3] = \frac{4 - 1}{3 - 0} = 1$. This is the leading coefficient of $x^2$, as expected.

Non-example: the recursion cannot be applied to the list $1, 1$ of repeated points, because the denominator $x_1 - x_0$ is $0$.