Classical orthogonal polynomials

Remarks

The integral $\ip{f}{g} = \int_I f g\, w\, dx$ is an inner product on real polynomials, and an orthogonal sequence for $w$ is determined by $w$ up to a nonzero constant factor in each $p_n$. Example: the Chebyshev polynomials $T_n$ are Jacobi with $\alpha = \beta = -\frac{1}{2}$; for instance $\int_{-1}^1 T_0 T_1 (1 - x^2)^{-1/2}\, dx = \int_{-1}^1 \frac{x}{\sqrt{1 - x^2}}\, dx = 0$, since the integrand is odd. Example: the Hermite polynomials are the Hermite case, and the weight $e^{-x^2/2}$ of the probabilists' version is the Hermite weight after the change $x \mapsto x / \sqrt{2}$. A non-example: the orthogonal polynomials for $w(x) = e^{-x^4}$ on $\R$ exist, but they are not classical, since no affine change of variable turns $e^{-x^4}$ into one of the three weights. The classical families are exactly the orthogonal families that are eigenfunctions of an operator $\sigma(x) y'' + \tau(x) y'$ with $\deg \sigma \le 2$ and $\deg \tau = 1$. They are also exactly those whose derivatives $(p_{n+1}')_{n \ge 0}$ are again orthogonal, and those given by a Rodrigues formula.