The forward difference is a finite difference defined by
(1)
|
Higher order differences are obtained by repeated operations of the forward difference operator,
(2)
|
so
(3)
| |||
(4)
| |||
(5)
| |||
(6)
| |||
(7)
|
In general,
(8)
|
where is a binomial coefficient (Sloane and Plouffe 1995, p. 10).
The forward finite difference is implemented in the Wolfram Language as DifferenceDelta[f, i].
Newton's forward difference formula expresses as the sum of the th forward differences
(9)
|
where is the first th difference computed from the difference table. Furthermore, if the differences , , , ..., are known for some fixed value of , then a formula for the th term is given by
(10)
|
(Sloane and Plouffe 1985, p. 10).