Newton's forward difference formula is a finite difference identity for a polynomial 
of degree at most 
,
where 
is a nonnegative integer. Write 
for unit-spaced samples. The formula uses 
and the powers of the forward
difference operator 
. For ![a in [0,1]](/images/equations/NewtonsForwardDifferenceFormula/Inline7.svg)
,
 |
(1)
|
Here the terms of order greater than 
vanish. When written in the form
 |
(2)
|
with 
the falling factorial, the formula looks suspiciously
like a finite analog of a Taylor series expansion.
This correspondence was one of the motivating forces for the development of umbral
calculus.
An alternate form of this equation using binomial coefficients is
 |
(3)
|
where the binomial coefficient 
represents a polynomial
of degree 
in 
.
For general sampled functions, the finite sum gives a polynomial agreeing with the first 
sampled values, which need not agree with the function
between those points (Abramowitz and Stegun 1972, p. 880). An infinite version
requires additional conditions ensuring that the interpolation remainder tends to
zero. The convergence of the sum alone does not guarantee
equality with 
.
The derivative of Newton's forward difference formula gives Markoff's formulas.
