Well, I was just doing some random reading, and came across this:
Let $P(n)$ be a polynomial of degree $d$, further, define, $p_k = \Delta^k(P(n))|_{n=0}$. (Here, $\Delta$ indicates the forward difference operator.)
Then, $P(n)= \displaystyle\sum^d_{i=0} p_i \binom{n}{i}$, where, $\binom {n}{i}$ is the generalised binomial co-efficient.
This seems like a pretty marvellous result, and I notice the similarity to Maclaurin or Taylor series expansions, but, well, does anybody know where I can find a proof? And where could I read more about results like this?
Apply the operators $\Delta^k|_{n=0}$ on both sides for all $k=0,1,...$ and check you get the same every time.
The operators applied on the binomial coefficients act nicely. Use the basic property of the binomial coefficients.
Of course, you also need to prove that if a polynomial gives you zero after applying all these operators then you get zero. But this clearly gives you that the polynomial is constant at too many points.