TITLE: Does anyone know of a proof ot the recursion for Legendre polynomials which follows directly from Gram Schmidt?
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: 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.