Bochner's classification of orthogonal polynomial eigenfunctions

Remarks

Every operator $\mathcal{D}$ of this form maps polynomials of degree at most $n$ to themselves. In fact $\mathcal{D} x^n = \left( n(n-1)\sigma_2 + n \tau_1 \right) x^n + \text{lower terms}$, where $\sigma_2$ and $\tau_1$ are the coefficients of $x^2$ in $\sigma$ and of $x$ in $\tau$. The proof sorts $\sigma$ by its degree and its roots, up to affine changes. Orthogonality for a positive weight forces $w$ to solve the Pearson equation $(\sigma w)' = \tau w$, and only the Jacobi, Laguerre and Hermite cases give a positive weight with finite moments. Example: $\sigma = 1$ and $\tau = -2x$ give $\mathcal{D} = \frac{d^2}{dx^2} - 2x \frac{d}{dx}$, with eigenfunctions the Hermite polynomials and $\lambda_n = -2n$; the check for $H_2 = 4x^2 - 2$ is $8 - 2x \cdot 8x = -16x^2 + 8 = -4 H_2$. The orthogonality hypothesis is necessary: $\sigma = x^2$ and $\tau = 2x + 2$ have the Bessel polynomials as eigenfunctions, which are not orthogonal for any positive weight on a real interval. Without the orthogonality hypothesis, Bochner's list also contains the Bessel polynomials and the powers $x^n$, which are eigenfunctions of $x^2 \frac{d^2}{dx^2}$.

Going deeper

For a student

This is the reason the classical families are singled out among all orthogonal polynomials. The operator $\mathcal{D}$ becomes symmetric for the inner product $\int f g\, w\, dx$ when $(\sigma w)' = \tau w$, the Pearson equation for the weight, and $\sigma w$ vanishes fast enough at the ends of $I$. Then $\mathcal{D}$ is a Sturm–Liouville operator, and the $p_n$ are its eigenfunctions, orthogonal because they belong to different eigenvalues. Discrete analogues with difference operators lead to the Hahn, Meixner, Krawtchouk and Charlier polynomials, and $q$-analogues to the Askey scheme.