Newton series

Summary

The Newton series of a function $f$ at a point $a$ is the series $\sum_{k \ge 0} \binom{x-a}{k} \Delta^k f(a)$, built from the forward differences of $f$ at $a$. It is the difference analogue of the Taylor series, with generalized binomial coefficients in place of powers. For a polynomial it is a finite sum equal to the polynomial; for other functions it may or may not converge to $f$.