Newton's Forward Difference Formula

Newton's forward difference formula is a finite difference identity giving an interpolated value between tabulated points  in terms of the first value  and the powers of the forward difference . For , the formula states

(1)

When written in the form

(2)

with  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

(3)

where the binomial coefficient  represents a polynomial of degree  in .

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

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.