Forward Difference

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).

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 877, 1972.

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, p. 10, 1995.