## Divided Difference

The divided difference on points , , ..., of a function is defined by and

 (1)

for . The first few differences are
 (2) (3) (4)

Defining
 (5)

and taking the Derivative
 (6)

gives the identity
 (7)

Consider the following question: does the property

 (8)

for and a given function guarantee that is a Polynomial of degree ? Aczél (1985) showed that the answer is yes'' for , and Bailey (1992) showed it to be true for with differentiable . Schwaiger (1994) and Andersen (1996) subsequently showed the answer to be yes'' for all with restrictions on or .

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877-878, 1972.

Aczél, J. A Mean Value Property of the Derivative of Quadratic Polynomials--Without Mean Values and Derivatives.'' Math. Mag. 58, 42-45, 1985.

Andersen, K. M. A Characterization of Polynomials.'' Math. Mag. 69, 137-142, 1996.

Bailey, D. F. A Mean-Value Property of Cubic Polynomials--Without Mean Values.'' Math. Mag. 65, 123-124, 1992.

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 439-440, 1987.

Schwaiger, J. On a Characterization of Polynomials by Divided Differences.'' Aequationes Math. 48, 317-323, 1994.