= Newton series
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A <Newton series> uses the binomial-polynomial expansion
$$
\sum_{k\geq0}\binom{x}{k}\Delta^k f(0),\qquad\Delta f(n)=f(n+1)-f(n).
$$
For data on nonnegative <integers>, the value at $x=n$ is always the finite identity $f(n)=\sum_{k=0}^n\binom nk\Delta^k f(0)$. For a <polynomial> of degree $d$, higher <finite differences> vanish, and the expansion terminates and equals that polynomial for every real or complex $x$. For example, $x^2=\binom{x}{1}+2\binom{x}{2}$.
For a general nonpolynomial <function>, convergence and recovery away from integer sites need additional hypotheses. Integer samples do not determine an arbitrary continuous <function> or even an <entire function>: $\sin(\pi x)$ vanishes at every integer, so all coefficients computed from its integer samples are zero, although the function is not zero. A formal <Newton series> must therefore be distinguished from an asserted expansion valid on a continuous domain.
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