---
title: "Laguerre polynomials"
sort_title: "Laguerre polynomials"
date: 2021-09-08
categories:
- Mathematics
layout: "concept"
---

The **Laguerre polynomials** are a set of useful functions that arise in physics.
They are the non-singular eigenfunctions $$u(x)$$ of **Laguerre's equation**,
with the corresponding eigenvalues $$n$$ being non-negative integers:

$$\begin{aligned}
    \boxed{
        x u'' + (1 - x) u' + n u = 0
    }
\end{aligned}$$

The $$n$$th-order Laguerre polynomial $$L_n(x)$$
is given in the form of a *Rodrigues' formula* by:

$$\begin{aligned}
    L_n(x)
    &= \frac{1}{n!} \exp(x) \dvn{n}{}{x}\big(x^n \exp(-x)\big)
    \\
    &= \frac{1}{n!} \Big( \dv{}{x}- 1 \Big)^n x^n
\end{aligned}$$

The first couple of Laguerre polynomials $$L_n(x)$$ are therefore as follows:

$$\begin{gathered}
    L_0(x) = 1
    \qquad \quad
    L_1(x) = 1 - x
    \qquad \quad
    L_2(x) = \frac{1}{2} (x^2 - 4 x + 2)
\end{gathered}$$

Based on Laguerre's equation,
**Laguerre's generalized equation** is as follows,
with an arbitrary real (but usually integer) parameter $$\alpha$$,
and $$n$$ still a non-negative integer:

$$\begin{aligned}
    \boxed{
        x u'' + (\alpha + 1 - x) u' + n u = 0
    }
\end{aligned}$$

Its solutions, denoted by $$L_n^\alpha(x)$$,
are the **generalized** or **associated Laguerre polynomials**,
which also have a Rodrigues' formula.
Note that if $$\alpha = 0$$ then $$L_n^\alpha = L_n$$:

$$\begin{aligned}
    L_n^\alpha(x)
    &= \frac{1}{n!} x^{-\alpha} \exp(x) \dvn{n}{}{x}\big( x^{n + \alpha} \exp(-x) \big)
    \\
    &= \frac{x^{-\alpha}}{n!} \Big( \dv{}{x}- 1 \Big)^n x^{n + \alpha}
\end{aligned}$$

The first couple of associated Laguerre polynomials $$L_n^\alpha(x)$$ are therefore as follows:

$$\begin{aligned}
    L_0^\alpha(x) = 1
    \qquad
    L_1^\alpha(x) = \alpha + 1 - x
    \qquad
    L_2^\alpha(x) = \frac{1}{2} (x^2 - 2 \alpha x - 4 x + \alpha^2 + 3 \alpha + 2)
\end{aligned}$$

And then more $$L_n^\alpha$$ can be computed quickly
using the following recurrence relation:

$$\begin{aligned}
    \boxed{
        L_{n + 1}^\alpha(x)
        = \frac{(\alpha + 2 n + 1 - x) L_n^\alpha(x) - (\alpha + n) L_{n - 1}^\alpha(x)}{n + 1}
    }
\end{aligned}$$

The derivatives are also straightforward to calculate
using the following relation:

$$\begin{aligned}
    \boxed{
        \dvn{k}{}{x}L_n^\alpha(x)
        = (-1)^k L_{n - k}^{\alpha + k}(x)
    }
\end{aligned}$$

Noteworthy is that these polynomials (both normal and associated)
are all mutually orthogonal for $$x \in [0, \infty[$$,
with respect to the weight function $$w(x) \equiv x^\alpha \exp(-x)$$:

$$\begin{aligned}
    \boxed{
        \Inprod{L_m^\alpha}{w L_n^\alpha}
        = \int_0^\infty L_m^\alpha(x) \: L_n^\alpha(x) \: w(x) \dd{x}
        = \frac{\Gamma(n + \alpha + 1)}{n!} \delta_{nm}
    }
\end{aligned}$$

Where $$\delta_{nm}$$ is the Kronecker delta.
Moreover, they form a basis in
the [Hilbert space](/know/concept/hilbert-space/)
of all functions $$f(x)$$ for which $$\Inprod{f}{w f}$$ is finite.
Any such $$f$$ can thus be expanded as follows:

$$\begin{aligned}
    \boxed{
        f(x)
        = \sum_{n = 0}^\infty a_n L_n^\alpha(x)
        = \sum_{n = 0}^\infty \frac{\Inprod{L_n}{w f}}{\Inprod{L_n}{w L_n}} L_n^\alpha(x)
    }
\end{aligned}$$

Finally, the $$L_n^\alpha(x)$$ are related to
the [Hermite polynomials](/know/concept/hermite-polynomials/) $$H_n(x)$$ like so:

$$\begin{aligned}
    H_{2n(x)} &= (-1)^n 2^{2n} n! \: L_n^{-1/2}(x^2)
    \\
    H_{2n + 1(x)} &= (-1)^n 2^{2n + 1} n! \: L_n^{1/2}(x^2)
\end{aligned}$$