The forward difference is a finite difference defined by
 |
(1)
|
For a positive integer 
, higher order differences are obtained by repeated operations
of the forward difference operator,
 |
(2)
|
Writing 
,
the second forward difference is
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
For a nonnegative integer 
,
 |
(8)
|
where 
is a binomial coefficient (Sloane and Plouffe
1995, p. 10).
For a step 
,
the first and 
th
forward differences are
 |  |  |
(9)
|
 |  |  |
(10)
|
where 
is a nonnegative integer and 
.
The forward finite difference is implemented in the Wolfram Language as DifferenceDelta[f,
i]. A step 
is specified by DifferenceDelta[f[x],
x,
1, h
],
and the 
th
difference by DifferenceDelta[f[x],
x,
n, h
].
For a nonnegative integer 
, Newton's
forward difference formula gives the finite identity
 |
(11)
|
More generally, if the forward differences 
, 
, 
, ..., are known, then a formula
for a later term is
 |
(12)
|
for nonnegative integers 
and 
(Sloane and Plouffe 1995, p. 10).
