Three-term recurrence for monic orthogonal polynomials
= Three-term recurrence for monic orthogonal polynomials
For monic orthogonal polynomials,
$$
p_{n+1}(x)=(x-\alpha_n)p_n(x)-\beta_np_{n-1}(x),
$$
where
$$
\alpha_n=\frac{\langle xp_n,p_n\rangle}{\langle p_n,p_n\rangle},
\qquad
\beta_n=\frac{\langle p_n,p_n\rangle}{\langle p_{n-1},p_{n-1}\rangle}.
$$