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