Multiplication rules for polynomials

Abstract:

It is proved that the polynomial solutions of the functional equation \[ F(z)F( {z + 1/\sqrt a } ) = F( {\sqrt a {z^2} + (b/\sqrt a + 1)z + c/\sqrt a } )\] are precisely ${(a{z^2} + bz + c)^n}$ if ${b^2} - 4ac \ne 0$ and ${(\sqrt a z + b/2\sqrt a )^n}$ if ${b^2} - 4ac = 0$.