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Forward Difference

The forward difference is a finite difference defined by

Deltaa_n=a_(n+1)-a_n.
(1)

Higher order differences are obtained by repeated operations of the forward difference operator,

Delta^ka_n=Delta^(k-1)a_(n+1)-Delta^(k-1)a_n,
(2)

so

Delta^2a_n=Delta_n^2
(3)
=Delta(Delta_n)
(4)
=Delta(a_(n+1)-a_n)
(5)
=Delta_(n+1)-Delta_n
(6)
=a_(n+2)-2a_(n+1)+a_n.
(7)

In general,

Delta_n^k=Delta^ka_n=sum_(i=0)^k(-1)^i(k; i)a_(n+k-i),
(8)

where (k; m) 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 a_n as the sum of the nth forward differences

a_n=a_0+nDelta_0+1/(2!)n(n+1)Delta_0^2+1/(3!)n(n+1)(n+2)Delta_0^3+...
(9)

where Delta_0^n is the first nth difference computed from the difference table. Furthermore, if the differences a_m, Deltaa_m, Delta^2a_m, ..., are known for some fixed value of m, then a formula for the nth term is given by

a_(n+m)=sum_(k=0)^n(n; k)Delta^ka_m
(10)

(Sloane and Plouffe 1985, p. 10).

See also

Backward Difference, Central Difference, Difference Equation, Divided Difference, Newton's Forward Difference Formula, Reciprocal Difference

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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.

Referenced on Wolfram|Alpha

Forward Difference

Cite this as:

Weisstein, Eric W. "Forward Difference." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ForwardDifference.html

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