divided differences of powers
In this entry, we will prove the claims about divided differences of polynomials. Because the divided difference is a linear operator, we can focus our attention on powers.
Theorem 1.
If and , then
If , then .
Proof.
We proceed by induction. The formula is trivially true when . Assume that the formula is true for a certain value of . Then we have
Using the identity for the sum of a geometric series,
this becomes
Note that when , we have , which is consistent with the formula given above because, in that case, there are no solutions to , so the sum is empty and, by convention, equals zero. Likewise, when , then the only solution to is , so the sum only consists of one term, so , hence taking further differences produces zero. ∎
Title | divided differences of powers |
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Canonical name | DividedDifferencesOfPowers |
Date of creation | 2013-03-22 16:47:59 |
Last modified on | 2013-03-22 16:47:59 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 11 |
Author | rspuzio (6075) |
Entry type | Theorem |
Classification | msc 39A70 |