Questions regarding the Newton series expansion of functions or other finite-difference based series expansions. Issues including convergence, calculation of coefficients, and bounds on remainders.
A Newton series (so named because Isaac Newton used it extensively) is an expansion of a univariate function $$ f(x) = \sum c_k \binom{x}{k}. $$ Any polynomial can be represented exactly as a terminating series in this way. In general, $$ f(x) = \Delta^d f(0) \binom{x}{d} + \Delta^{d-1} f(0) \binom{x}{d-1} + \cdots + \Delta f(0) \binom{x}{1} + f(0)\binom{x}{0}. $$ Non-terminating Newton series, of the form $$ f(x) = \sum{n=0}^\infty \Delta^n[f](0) \binom{x}{n} $$ where $x$ need not be an integer, are closely analagous to Taylor series, and can be written as $$ f(a+x) = \frac{f(a)}{0!} x^{\underline{0}}+\frac{\Delta f(a)}{1!}x^{\underline{1}} + \frac{\Delta^2 f(a)}{2!}x^{\underline{2}}+\frac{\Delta^3 f(a)}{3!}x^{\underline{3}}+\cdots $$ where $x^{\underline{n}}$ is the "falling power" notation, e.g., $x^{\underline{4}} = x(x-1)(x-2)(x-3)$.
Multivariate function expansions, in terms of multinomial coefficients and multivariate differences, are also possible.
