Omitted-node identities for divided differences
= Omitted-node identities for divided differences
{title2=$D_i-D_j=(x_j-x_i)f[x_0,\ldots,x_n]$}
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 $0<i<n$,
$$
D_i=\frac{x_i-x_0}{x_n-x_0}D_n+\frac{x_n-x_i}{x_n-x_0}D_0.
$$
The identity is algebraic and does not require the nodes to be ordered; the coefficients need not be nonnegative.