- $\{p_n\}$ is orthogonal polynomial which satisfies
$$p_{n+1} = (A_n x +B_n) p_n +D_n p_{n-1}$$
- whose monic form $\tilde{p}_n$ satisfies
$$x \tilde{p}_n(x) = \tilde{p}_{n+1}(x) + b_n \tilde{p}_n(x) + d_n \tilde{p}_{n-1}(x),$$
- whose orthonormal form $\hat{p}_n$ satisfies
$$x \hat{p}_n(x) = E_n \hat{p}_{n+1} + F_n \hat{p}_n(x) + E_{n-1} \hat{p}_{n-1}(x).$$
Now the problem is how to get the other two from one? What's the relation among the coefficients?