Finite Difference - Newton's Series

Newton's Series

The Newton series consists of the terms of the Newton forward difference equation, named after Isaac Newton; in essence, it is the Newton interpolation formula, first published in his Principia Mathematica in 1687, namely the discrete analog of the continuum Taylor expansion,

f(x)=\sum_{k=0}^\infty\frac{\Delta^k (a)}{k!} ~(x-a)_k = \sum_{k=0}^\infty {x-a \choose k}~ \Delta^k (a) ~,

which holds for any polynomial function f and for most (but not all) analytic functions. Here, the expression

is the binomial coefficient, and

is the "falling factorial" or "lower factorial", while the empty product (x)0 is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of x, h=1 of the generalization below.

Note also the formal correspondence of this result to Taylor's theorem. Historically, this, as well as the Chu-Vandermonde identity,

(following from it, and corresponding to the binomial theorem), are included in the observations which matured to the system of the umbral calculus.

To illustrate how one may use Newton's formula in actual practice, consider the first few terms of the Fibonacci sequence f = 2, 2, 4... One can find a polynomial that reproduces these values, by first computing a difference table, and then substituting the differences which correspond to x0 (underlined) into the formula as follows,

\begin{matrix} \begin{array}{|c||c|c|c|} \hline x & f=\Delta^0 & \Delta^1 & \Delta^2 \\ \hline 1&\underline{2}& & \\ & &\underline{0}& \\ 2&2& &\underline{2} \\ & &2& \\ 3&4& & \\ \hline \end{array} & \quad \begin{matrix} f(x)=\Delta^0 \cdot 1 +\Delta^1 \cdot \dfrac{(x-x_0)_1}{1!} + \Delta^2 \cdot \dfrac{(x-x_0)_2}{2!} \quad (x_0=1)\\ \\ =2 \cdot 1 + 0 \cdot \dfrac{x-1}{1} + 2 \cdot \dfrac{(x-1)(x-2)}{2} \\ \\ =2 + (x-1)(x-2) \\ \end{matrix} \end{matrix}

For the case of nonuniform steps in the values of x, Newton computes the divided differences,

the series of products,

and the resulting polynomial is the scalar product, .

In analysis with p-adic numbers, Mahler's theorem states that the assumption that f is a polynomial function can be weakened all the way to the assumption that f is merely continuous.

Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series will not, in general, exist.

The Newton series, together with the Stirling series and the Selberg series, is a special case of the general difference series, all of which are defined in terms of suitably scaled forward differences.

In a compressed and slightly more general form and equidistant nodes the formula reads

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