---
tag: J9367QFD3G
kind: statement
role: theorem
name: Rodrigues' formula
msc: [33C45, 42C05]
uses: [47WJ34NC9K, 0P4JRMSWNH]
about: [47WJ34NC9K]
sources:
  - url: https://en.wikipedia.org/wiki/Rodrigues%27_formula
same_as:
  wikidata: Q4491933
  wikipedia: Rodrigues' formula
---

## Statement

Let $(I, w, \sigma)$ be one of the following triples:

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

For $n \ge 0$ let
$$q_n(x) = \frac{1}{w(x)}\, \frac{d^n}{dx^n} \left( w(x)\, \sigma(x)^n \right), \qquad x \in I.$$
Then $q_n$ is a polynomial of [[degree|0P4JRMSWNH]] exactly $n$, and $\int_I q_m(x)\, q_n(x)\, w(x)\, dx = 0$ for $m \ne n$.
So $(q_n)$ is, up to constant factors, the family of [[classical orthogonal polynomials|47WJ34NC9K]] for $w$.

## Remarks

That $q_n$ is a polynomial follows by induction: each derivative of $w \sigma^n$ is $w \sigma^{n-k}$ times a polynomial of degree at most $k$.
This holds because $\sigma w' / w$ is a polynomial of degree at most $1$.
For orthogonality, take $m < n$ and integrate $\int_I x^m\, \frac{d^n}{dx^n}(w \sigma^n)\, dx$ by parts $n$ times.
The boundary terms vanish because $w \sigma^n$ and its first $n - 1$ derivatives vanish at the ends of $I$, and the final integrand contains $\frac{d^n}{dx^n} x^m = 0$.
The usual normalisations are $P_n^{(\alpha,\beta)} = \frac{(-1)^n}{2^n n!}\, q_n$ (Jacobi), $L_n^{(\alpha)} = \frac{1}{n!}\, q_n$ (Laguerre), and $H_n = (-1)^n q_n$ (Hermite).
Example: in the Hermite case with $n = 2$, $\frac{d^2}{dx^2} e^{-x^2} = (4x^2 - 2) e^{-x^2}$, so $q_2 = 4x^2 - 2 = H_2$.
Example: in the Jacobi case with $\alpha = \beta = 0$, the formula gives the Legendre polynomials $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n$.

## Explanations

### L1 human

Each classical family of orthogonal polynomials comes from one recipe.
Multiply the weight by a power of a simple polynomial, differentiate $n$ times, and divide by the weight again.
For the Legendre polynomials this means differentiating $(x^2 - 1)^n$ exactly $n$ times.
The formula makes orthogonality almost automatic, through integration by parts.

### L2 human

Rodrigues' formula is the quickest way to compute norms, leading coefficients and generating functions of the classical families.
With Cauchy's integral formula it gives contour integral representations, from which generating functions such as $\sum_n P_n(x)\, t^n = (1 - 2xt + t^2)^{-1/2}$ follow.
The existence of such a formula characterises the classical families among orthogonal polynomials, together with Bochner's characterisation by differential equations.
The polynomial $\sigma$ is the same one that appears in the second-order differential equation $\sigma y'' + \tau y' + \lambda_n y = 0$ satisfied by $q_n$.