Three-term recurrence for orthogonal polynomials

Remarks

Infinite support makes $\ip{\cdot}{\cdot}$ an inner product on all polynomials, so the Gram–Schmidt process applied to $1, x, x^2, \dots$ gives the $p_n$. The polynomial $x p_n - p_{n+1}$ has degree at most $n$, so it is a combination of $p_0, \dots, p_n$. Its coefficient on $p_k$ is $\frac{\ip{x p_n}{p_k}}{\ip{p_k}{p_k}} = \frac{\ip{p_n}{x p_k}}{\ip{p_k}{p_k}}$, which is $0$ for $k \le n - 2$ because $x p_k$ has degree less than $n$. This key step uses that multiplication by $x$ is symmetric for $\ip{\cdot}{\cdot}$. The same symmetry gives $\ip{x p_n}{p_{n-1}} = \ip{p_n}{x p_{n-1}} = \ip{p_n}{p_n}$, which yields the formula for $b_n$. Example: for the Hermite polynomials made monic, $a_n = 0$ and $b_n = \frac{n}{2}$, so $p_2 = x^2 - \frac{1}{2}$.

Going deeper

For a student

The recurrence coefficients encode the measure: they form a symmetric tridiagonal Jacobi matrix whose spectral measure is $\mu$, normalised to mass $1$, when $\mu$ is determined by its moments. The converse, that any such recurrence with $b_n > 0$ comes from a measure, is Favard's theorem. The zeros of $p_n$ are the eigenvalues of the $n \times n$ truncated Jacobi matrix, which is the basis of the Golub–Welsch method for Gaussian quadrature. The Christoffel–Darboux formula is a direct consequence of the recurrence.