The forward difference is a finite difference defined by
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(1)
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For a positive integer 
, higher order differences are obtained by repeated
operations of the forward difference operator,
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(2)
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Writing 
, the second forward difference is
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(3)
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(4)
|
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(5)
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(6)
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(7)
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For a nonnegative integer 
,
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(8)
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where 
is a binomial coefficient (Sloane and Plouffe
1995, p. 10).
For a step 
, the first and 
th forward differences are
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(9)
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(10)
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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
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(11)
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More generally, if the forward differences 
, 
, 
, ..., are known, then
a formula for a later term is
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(12)
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for nonnegative integers 
and 
(Sloane and Plouffe 1995, p. 10).
