 Divided difference

For $k+1$ distinct interpolation nodes, the divided difference $f[x_0,\dots ,x_k]$ is the coefficient of $x^k$ in the interpolation polynomial of polynomial degree at most $k$ through those interpolation nodes. It is the leading coefficient when that polynomial degree is exactly $k$, and zero when it is smaller. Explicitly,starting with $f[x_i]=f(x_i)$. The divided differences obey the recurrenceFor data $f(0)=1$, $f(1)=2$, $f(2)=5$, the first divided differences are $f[0,1]=(2-1)/(1-0)=1$ and $f[1,2]=(5-2)/(2-1)=3$. The second divided difference is $f[0,1,2]=(3-1)/(2-0)=1$, the leading coefficient of $x^2+1$. The first divided difference is the slope of a secant line; higher divided differences supply successive coefficients for the Newton polynomial.