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Divided Difference

The divided difference $f[x_1, x_2, \ldots, x_n]$ on $n$ points $x_1$, $x_2$, ..., $x_n$ of a function $f(x)$ is defined by $f[x_1]\equiv f(x_1)$ and

f[x_1, x_2, \ldots, x_n]={f[x_1, \ldots, x_n]-f[x_2, \ldots, x_n]\over x_1-x_n}
\end{displaymath} (1)

for $n\geq 2$. The first few differences are
$\displaystyle {[}x_0, x_1]$ $\textstyle =$ $\displaystyle {f_0-f_1\over x_0-x_1}$ (2)
$\displaystyle {[}x_0, x_1, x_2]$ $\textstyle =$ $\displaystyle {[x_0, x_1]-[x_1, x_2]\over x_0-x_2}$ (3)
$\displaystyle {[}x_0, x_1, \ldots, x_n]$ $\textstyle =$ $\displaystyle {[x_0, \ldots, x_{n-1}]-[x_1, \ldots, x_n]\over x_0-x_n}.$ (4)

\pi_n(x)\equiv (x-x_0)(x-x_1)\cdots(x-x_n)
\end{displaymath} (5)

and taking the Derivative
\end{displaymath} (6)

gives the identity
\begin{displaymath}[x_0, x_1, \ldots, x_n]=\sum_{k=0}^n {f_k\over \pi_n'(x_k)}.
\end{displaymath} (7)

Consider the following question: does the property

f[x_1, x_2, \ldots, x_n]=h(x_1+x_2+\ldots+x_n)
\end{displaymath} (8)

for $n\geq 2$ and $h(x)$ a given function guarantee that $f(x)$ is a Polynomial of degree $\leq n$? Aczél (1985) showed that the answer is ``yes'' for $n=2$, and Bailey (1992) showed it to be true for $n=3$ with differentiable $f(x)$. Schwaiger (1994) and Andersen (1996) subsequently showed the answer to be ``yes'' for all $n\geq 3$ with restrictions on $f(x)$ or $h(x)$.

See also Newton's Divided Difference Interpolation Formula, Reciprocal Difference


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.

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© 1996-9 Eric W. Weisstein