I am looking at Trefethen and Bau Exercise 37.1: I have two normalizations of the Legendre polynomials with corresponding recurrence relations: $$P_n(1)=1$$ which follows $$P_n(x) = \frac{2n-1}{n} x P_{n-1}(x) -\frac{n-1}{n}P_{n-2}(x)$$ and $$\|q_n\|=1$$ which follows $$xq_n(x)=\beta_{n-1}q_{n-1}(x)+\alpha_n q_n(x) + \beta_n q_{n+1}(x)$$ where $$\beta_n = \tfrac12 (1-(2n)^{-2})^{-1/2}$$
Apparently, $q_{n+1}$ is proportional to $P_n$. I know that the entries of the Jacobi matrix $t_{ij} = (q_i,xq_j)$ in the $L^2([-1,1])$ inner-product space, and I managed to get the right answer for the $q_j$ with $0$ along the main diagonal, $\beta_{j-1}$ on the upper diagonal and $\beta_j$ along the lower diagonal. However, it doesn't seem that I am getting the right answer for the 1st normalization.
Also, I am asked to find the relationship between the two Jacobi matrices--my intuition is that they are equal but I can't quite justify it--I think it has to do with the fact that the characteristic polynomials are the same monic polynomial.
Furthermore, I am asked to find a formula for $q_{n+1}(1)=\|P_n\|$ using the Jacobi matrices. I've tried to solve this by equating the entries for the Jacobi matrices (where one includes a term of $||P_n||^2$) and solving. But I can't get the answer $\|P_n\|=\sqrt{\frac{2}{2n+1}}$.