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 .

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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.Jeffreys, H. and Jeffreys, B. S. "Divided Differences."
§9.012 in Methods
of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University
Press, pp. 260-264, 1988.Schwaiger, J. "On a Characterization
of Polynomials by Divided Differences." Aequationes Math.48,
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§11-12 in The
Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New
York: Dover, pp. 20-24, 1967.