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GeezerVison is showing mathworld.wolfram.com — Forward Difference -- from Wolfram MathWorld

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Forward Difference

The forward difference is a finite difference defined by

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

For a positive integer k, 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)

Writing Delta_n=Deltaa_n, the second forward difference is

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)

For a nonnegative integer k,

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).

For a step h, the first and nth forward differences are

Delta_hf(x)=f(x+h)-f(x)
(9)
Delta_h^nf(x)=sum_(j=0)^(n)(-1)^j(n; j)f(x+(n-j)h),
(10)

where n is a nonnegative integer and Delta_h^0f(x)=f(x).

The forward finite difference is implemented in the Wolfram Language as DifferenceDelta[f, i]. A step h is specified by DifferenceDelta[f[x], {x, 1, h}], and the nth difference by DifferenceDelta[f[x], {x, n, h}].

For a nonnegative integer n, Newton's forward difference formula gives the finite identity

a_n=sum_(k=0)^n(n; k)Delta^ka_0.
(11)

More generally, if the forward differences a_m, Deltaa_m, Delta^2a_m, ..., are known, then a formula for a later term is

a_(n+m)=sum_(k=0)^n(n; k)Delta^ka_m
(12)

for nonnegative integers m and n (Sloane and Plouffe 1995, 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 Resource. https://mathworld.wolfram.com/ForwardDifference.html

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