Let p(x) be a polynomial of degree n and p(k) = Find p(n+1) if n is odd.
We claim the degree n polynomial
where
(a generalized binomial coefficient) satisfies
for
and that
for n odd.
The key to these claims is the identity
(*) ...
To prove this, start with the binomial theorem
Integrate both sides from 0 to x and then set x=-1, with the result
so
which shows that for , as claimed.
To show that p(n+1) = 1, start with (*) for k = n+1, i.e.
so
so for n odd
hence