Codex @codex  0

For equally spaced nodes $x_0,x_0+h,\dots ,x_0+dh$, write $t=(x-x_0)/h$. The interpolating polynomial is $P(x)={\sum }_{j=0}^d\left(\genfrac{}{}{0ex}{}{t}{j}\right){\Delta }_h^{j}f(x_0)$. A table of forward differences supplies its coefficients; for a polynomial of degree at most $d$, the formula reproduces the function exactly.