I have long been familiar with the method of common differences for finding the equation of a sequence: you subtract consecutive terms going down level by level, until you get a constant difference at level $n$. Then you infer that the sequence is generated by a polynomial of degree $n$ and find the coefficients.
I can also easily convince myself that a sequence generated by a polynomial of degree $n$ will have common differences at level $n$ (e.g. the proof given here).
However, it recently occurred to me that I have never seen a proof in the other direction: given that the differences at level $n$ are a constant (say, $C$), a polynomial of degree $n$ is the unique solution for the sequence equation.
I have thought of the following proof:
Prove by any means other than common differences that the sum of the $n^\text{th}$ power of the first $k$ integers is a polynomial of degree $(n+1)$ (e.g. use Gauss' trick for $n=1$, and some other trick for higher $n$, like $\sum_{i=1}^{k} i^{n+1} = \sum_{i=1}^{k} (i-1)^{n+1} + k^{n+1}$).
Then it's fairly easy to conclude that starting from a common difference at level $n$ and summing up at each level, you end up with a polynomial of degree $n$ at the first level.
My questions are:
- Is my concern valid to begin with? Or should it be obvious because I'm missing something?
- Is my proof valid?
- Are there cleaner proofs?
- Is this related to the issue of uniqueness of solutions to differential equations? (I guess so)
- It says on Wikipedia that difference equations can be solved analogously to differential equations, but I can't see how. Any pointers would be appreciated!
