Let F(x) be a polynomial with real coefficients. Find all functions F(x) satisfying:

F(x)*F(2x^2-1) = F(x^2)*F(2x-1)

F(x) = a*(x-1)^n  for all possible reals a and all whole numbers n

i.e.

a
ax-a
ax^2-2ax+a
ax^3-3ax^2+3ax-a
...

Proving they work (now that I have them) is not that hard.
F(x)*F(2x^2-1) =
a*(x-1)^n*a*(2x^2-2)^n =
a^2*[(x-1)*2*(x-1)^2]^n =
a*(x^2-1)^n*a*(2x-2)^n =
F(x^2)*F(2x-1)

Proving that these are the only ones seems much harder.

Posted by Joel on 2007-01-27 16:50:24