Alright, so I ran into a little problem while applying the Gram-Schmidt orthogonalization process. To the functions $\{1,x,x^2,x^3...\}$ over $x\in(0,\infty)$ with weight function $\sigma (x)=e^{-x}$. As a reminder:
$$<x^n,x^m>=\int_0^{\infty}x^n x^m e^{-x}dx=(n+m)!$$
The first three polynomials I obtained were: \begin{align} \phi_0 &=1\\ \phi_1 &=x-1\\ \phi_2 &=x^2-4x-2 \end{align}
It's very strange to me that $\phi_2$ is orthogonal to $\phi_1$, but NOT orthogonal to $\phi_0$. Note that $\phi_1$ and $\phi_0$ are not only orthogonal to one another, but also each already normalized. Using these two for the orthogonalization process, you get $\phi_2$... Why is this happening?