How do we prove the generalized formula for orthogonal polynomials connecting Legendre polynomials (where $P_n(x)$ is given by Rodrigue's formula) obtained through Gram-Schmidt, using induction?
I tried by using Recursion relation but couldn't proceed further.
$$\phi_n(x) = \sqrt{\frac{2n+1}2}\,P_n(x)$$
Outline. Let $w_n$ be the $n$-th vector in the Gram-Schmidt process applied to the sequence $f_n(x) = x^n$. We show by induction that $w_n = \phi_n$. The inductive hypothesis is that $w_m = \phi_m$ for all $m < n$. Now consider $w_0,\ldots,w_{n-1},w_{n},\phi_{n}$. The space of polynomials of degree $\leq n$ is a vector space of dimension $n+1$, so there is a non-trivial linear relation $$c_0 w_0 + \ldots + c_{n-1} w_{n-1} + c_n w_{n} + {c_{n+1}} \phi_{n} = 0.$$ Taking the inner product with each $w_m$ for $m<n$ and using orthogonality reduces this to $$c_n w_{n} + {c_{n+1}} \phi_{n} = 0$$ with at least one of the coefficients non-zero. Since both $w_{n}$ and $\phi_{n}$ have norm 1, one can show that $|c_n| = |c_{n+1}|$. Furthermore, since both $w_{n}$ and $\phi_{n}$ are polynomials of degree $n$ whose leading coefficients are real and positive, one can show that $c_n = -c_{n+1}$.