symmetry of divided differences
Theorem 1.
If is a permutation of , then
Proof.
We proceed by induction. When , we have, from the definition,
Since the only permutations of two elements are the identity and the
transposition
, we see that the first divided differrence is symmetric
.
Now suppose that we already know that the -th divided difference
is symmetric under permutation of its arguments for some .
We will prove that the -st divided difference is also symmmetric
under all permutations of its arguments.
The divided difference is symmetric under transposing with :
The divided difference is symmetric under transposing with :
The divided difference is symmetric under transposing with when :
Since any permutation of can be genreated from the transpositions of with for between and , it follows that is symmetric under all permutaions of . ∎
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 |