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