The backward difference is a finite difference defined by
(1)

Higher order differences are obtained by repeated operations of the backward difference operator, so
(2)
 
(3)
 
(4)

In general,
(5)

where is a binomial coefficient.
The backward finite difference are implemented in the Wolfram Language as DifferenceDelta[f, i].
Newton's backward difference formula expresses as the sum of the th backward differences
(6)

where is the first th difference computed from the difference table.