The norm of the Legendre polynomial, where $P_k(t) = \frac{(2k)!}{2^k(k!)^2}q_k(t), k = 0,1,2,....$, is based on the following $L^2$ inner product:
Note: $q_k(t) = t^k - \sum\limits_{j=0}^{k-1} \frac {\langle t^k,q_j \rangle}{||q_j||^2}q_j(t)$ for $k= 1,2,...$
First, prove that $||R_{k,k}||^2 = (-1)^k(2k)! \int^{1}_{-1} (t^2 -1)^kdt$ by a repeated integration by parts, where $R_{j,k}(t) = \frac {d^j}{dt^j}(t^2 -1)^k$ which is a polynomial of degree $2k -j$. In particular, the Rodrigues formula claims that $P_k(t)$ is a multiple of $R_{k,k}(t)$.
Second, prove that $\int^{1}_{-1} (t^2 -1)^kdt = (-1)^k \frac{2^{2k+1}(k!)^2}{(2k+1)!}$ by using the change of variables $t = cos\theta$ in the integral.
Finally, use Rodrigues' formula to complete the proof and also use Rodrigues' formula to prove $P_k(1) = 1$.
How will I be able to prove this?