symmetry of divided differences

Theorem 1.

If y0,,yn is a permutationMathworldPlanetmath of x0,,xn, then


We proceed by inductionMathworldPlanetmath. When n=1, we have, from the definition,


Since the only permutations of two elements are the identityPlanetmathPlanetmathPlanetmath and the transpositionMathworldPlanetmath, we see that the first divided differrence is symmetricPlanetmathPlanetmath.

Now suppose that we already know that the n-th divided differenceMathworldPlanetmath is symmetric under permutation of its arguments for some n1. We will prove that the n+1-st divided difference is also symmmetric under all permutations of its arguments.

The divided difference is symmetric under transposing x0 with x1:

Δn+1f[x0,x1,x2,,xn+1] =Δnf[x1,x2,xn+1]-Δnf[x0,x2,xn+1]x1-x0

The divided difference is symmetric under transposing x1 with x2:

Δn+1 f[x0,x1,x2,,xn+1]

The divided difference is symmetric under transposing xk with xk+1 when k>1:

Δn+1 f[x0,x1,x2,,xk,xk+1,,xn+1]

Since any permutation of x0,x1,xn+1 can be genreated from the transpositions of xk with xk+1 for k between 0 and n, it follows that Δn+1f[x0,x1,,xk,xk+1,,xn+1] is symmetric under all permutaions of x0,x1,xn+1. ∎

Title symmetry of divided differences
Canonical name SymmetryOfDividedDifferences
Date of creation 2013-03-22 16:48:29
Last modified on 2013-03-22 16:48:29
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 22
Author rspuzio (6075)
Entry type Theorem
Classification msc 39A70