divided differences of powers
If and , then
If , then .
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,
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. ∎