Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 2 / 14B / c / Solution

Write $D=f[x_0,\dots ,x_n]$ and let $D_i$ be the divided difference with $x_i$ omitted. By symmetry from part (a), reorder the full list so that its first node is $x_i$ and its last is $x_j$. The recurrence from part (b) then has first omitted-node term $D_i$ and second omitted-node term $D_j$, and givesThis proves the requested identity for every pair $i\ne j$ and fixes its sign explicitly; it is one of the omitted-node identities for divided differences.