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A Newton series uses the binomial-polynomial expansionFor data on nonnegative integers, the value at $x=n$ is always the finite identity $f(n)={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){\Delta }^{k}f(0)$. 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, $x^2=\left(\genfrac{}{}{0ex}{}{x}{1}\right)+2\left(\genfrac{}{}{0ex}{}{x}{2}\right)$.

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.