Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum\_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some functions, this series does indeed converge to the derivative of $f$ (e.g., polynomials, and apparently even functions of the form $f(x) = x^p$ for non-natural $p$). Alas, however, this formula cannot hold in complete generality (there's no reason the local data of $f'(x)$ for a particular $x$ should be constrained at all by the global data of $f(x), f(x + 1), f(x + 2), ...$).


So one might ask (and I do!): is there a nice characterization of conditions under which this identity does hold?



Derivative in terms of finite differences
This is a special case of well-known general formula for the n-th derivative via differences and Stirling numbers. Cf. good books on finite differences, e.g. of Gelfond or Jordan.


