Omitted-node identities for divided differences ($D_i-D_j=(x_j-x_i)f[x_0,\dots ,x_n]$)

For distinct interpolation nodes, write $D_i$ for the divided difference with $x_i$ omitted. Symmetry and the divided-difference recurrence give the displayed identity. Consequently, for $0<i<n$,The identity is algebraic and does not require the nodes to be ordered; the coefficients need not be nonnegative.