---
tag: 13CH1ZF88V
kind: definition
name: Classical orthogonal polynomials
msc: [33C45, 42C05]
defines: 47WJ34NC9K
uses: [0P4JRMSWNH, 0NB0AWGJBW]
sources:
  - url: https://en.wikipedia.org/wiki/Classical_orthogonal_polynomials
---

## Definition

Let $I = (a, b)$ be an open interval with $-\infty \le a < b \le \infty$, and let $w \colon I \to (0, \infty)$ be continuous with $\int_I \abs{x}^k w(x)\, dx < \infty$ for every $k \ge 0$.
A sequence $(p_n)_{n \ge 0}$ of real polynomials with [[degree|0P4JRMSWNH]] $\deg p_n = n$ is *orthogonal* for $w$ if $\int_I p_m(x)\, p_n(x)\, w(x)\, dx = 0$ for all $m \ne n$.
Such a sequence is a family of *classical orthogonal polynomials* if an affine change of variable $x \mapsto c x + d$ with $c \ne 0$ does the following.
It turns the pair $(I, w)$, up to a constant factor in $w$, into one of these:

1. (Jacobi) $I = (-1, 1)$ and $w(x) = (1 - x)^\alpha (1 + x)^\beta$ with $\alpha, \beta > -1$.
2. (Laguerre) $I = (0, \infty)$ and $w(x) = x^\alpha e^{-x}$ with $\alpha > -1$.
3. (Hermite) $I = \R$ and $w(x) = e^{-x^2}$.

## Remarks

The integral $\ip{f}{g} = \int_I f g\, w\, dx$ is an [[inner product|0NB0AWGJBW]] 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|M6EAR1JHPP]] $T_n$ 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|MGACY98E9H]] 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.