Does anyone know of a proof ot the recursion for Legendre polynomials which follows directly from Gram Schmidt? If we start with the functions $\{1,x,x^2,...\}$ on $[-1,1]$ and apply the Gram-Schmidt process, we obtain the nonnormalized Legendre Polynomials. Does anyone know a proof of the recursive relation
$$
(n+1)P_{n+1}(x) = (2n+1)xP_n(x)-nP_{n-1}(x)
$$
which uses nothing more than this, and which does not depend on the differential equation, the generating function, Rodrigues' formula, etc?
 A: This an answer to the version of the question given in Mike's comment. Note first that for any polynomials $p$ and $q$ we have $(xp,q)=(p,xq)$. Second, note that $P_n$ is orthogonal to all polynomials of degree less than $n$. 
So if $m \le n-2$, then
$$
  (xP_n,P_m) = (P_n,xP_m) = 0.
$$
But we know that $xP_n$ must be a linear combination of $P_0,\ldots,P_n,P_{n+1}$
(because this holds for any polynomial of degree at most $n+1$), and the coefficient of $P_m$ in this expansion if $(xP_,P_m)/(P_m,P_m)$. Therefore $xP_n$ is a linear combination of $P_{n-1}$, $P_n$ and $P_{n+1}$.
Finally note that this works for any family of orthogonal polynomials, not just Legendre polynomials.
A: Let $(f,g)=\int_{-1}^1f(x)g(x)\,dx$ denote the $L^2$ inner product.
We seed the process with $P_0(x)=1,P_1(x)=x$. We'll prove this by induction and try to first prove $n=2$. We know $P_2(x)$ satisfies $(P_2,P_0)=\int_{-1}^1 P_2(x)\cdot 1=0$ and $(P_2,P_1)=\int_{-1}^1 P_2(x)\cdot x=0$. We don't actually have a normal condition on this basis, though Gram-Schmidt has built into it a normalising process after every step.
So instead we're just going to prove that this recursive relation starts building up an orthogonal basis.  We have $P_2(x)=x^2-\frac{(x^2,1)}{(1,1)}1-\frac{(x^2,x)}{(x,x)}x$. This gives us
$$
\frac{(x^2,1)}{(1,1)}=\frac{\int_{-1}^1 x^2\,dx}{\int_{-1}^1 1\,dx}=\frac{2/3}{2}=\frac{1}{3},\quad \frac{(x^2,x)}{(x,x)}=\frac{\int_{-1}^1 x^3\,dx}{\int_{-1}^1 x^2\,dx}=0 \\
P_2(x)=x^2-\frac{1}{3}\cdot1-0\cdot x=x^2-\frac{1}{3}.
$$
But this doesn't agree with the second Legendre polynomial. Maybe I messed up in here, but at least I'll post this here for you to ponder.
