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Newton's Forward Difference Formula

Newton's forward difference formula is a finite difference identity giving an interpolated value between tabulated points {f_p} in terms of the first value f_0 and the powers of the forward difference Delta. For a in [0,1], the formula states

f_a=f_0+aDelta+1/(2!)a(a-1)Delta^2+1/(3!)a(a-1)(a-2)Delta^3+....
(1)

When written in the form

f(x+a)=sum_(n=0)^infty((a)_nDelta^nf(x))/(n!)
(2)

with (a)_n the falling factorial, the formula looks suspiciously like a finite analog of a Taylor series expansion. This correspondence was one of the motivating forces for the development of umbral calculus.

An alternate form of this equation using binomial coefficients is

f(x+a)=sum_(n=0)^infty(a; n)Delta^nf(x),
(3)

where the binomial coefficient (a; n) represents a polynomial of degree n in a.

The derivative of Newton's forward difference formula gives Markoff's formulas.

See also

Finite Difference, Markoff's Formulas, Newton's Backward Difference Formula, Newton's Divided Difference Interpolation Formula

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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. 880, 1972.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 432, 1987.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Jordan, C. Calculus of Finite Differences, 3rd ed. New York: Chelsea, 1965.Nörlund, N. E. Vorlesungen über Differenzenrechnung. New York: Chelsea, 1954.Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1980.Whittaker, E. T. and Robinson, G. "The Gregory-Newton Formula of Interpolation" and "An Alternative Form of the Gregory-Newton Formula." §8-9 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 10-15, 1967.

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Newton's Forward Difference Formula

Cite this as:

Weisstein, Eric W. "Newton's Forward Difference Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NewtonsForwardDifferenceFormula.html

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