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

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

for $n\geq 2$. The first few differences are

$${[}x_0, x_1] = {f_0-f_1\over x_0-x_1}$$ (2)

$${[}x_0, x_1, x_2] = {[x_0, x_1]-[x_1, x_2]\over x_0-x_2}$$ (3)

$${[}x_0, x_1, \ldots, x_n] = {[x_0, \ldots, x_{n-1}]-[x_1, \ldots, x_n]\over x_0-x_n}.$$ (4)

Defining

$$\pi_n(x)\equiv (x-x_0)(x-x_1)\cdots(x-x_n)$$ (5)

and taking the Derivative

$$\pi_n'(x_k)=(x_k-x_0)\cdots(x_k-x_{k-1})(x_k-x_{k+1})\cdots(x_k-x_n)$$ (6)

gives the identity

$$[x_0, x_1, \ldots, x_n]=\sum_{k=0}^n {f_k\over \pi_n'(x_k)}.$$ (7)

Consider the following question: does the property

$$f[x_1, x_2, \ldots, x_n]=h(x_1+x_2+\ldots+x_n)$$ (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

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.