The equation w"(z) + (X -p(z))w(z) = 0 (z e C,A e R) with a fixed p(z) = a2mz2m + a2m_2z2m-2 + ---+ a2z2(a,>0Vi,a2m>0), possesses a set of solutions {i//"(z)}^L0 (with associated {a"}^°=0) which form a complete orthonormal set for L2(-co, oo) (as a real space). Here it is shown that any g(z) 1 when q = 0) cannot be expressed as a finite linear combination of {ipn(z))^L0, when deg(p) is a multiple of 4. It is well known that g(z)\¡fj¡j\z) can always be expressed as such when p(z) = z2 (the Hermite case).