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---
title: "Laguerre polynomials"
firstLetter: "L"
publishDate: 2021-09-08
categories:
- Mathematics
date: 2021-09-08T17:00:48+02:00
draft: false
markup: pandoc
---
# Laguerre polynomials
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:
$$\begin{aligned}
L_n(x)
&= \frac{1}{n!} \exp\!(x) \dv[n]{x} \big(x^n \exp\!(-x)\big)
\\
&= \frac{1}{n!} \Big( \dv{x} - 1 \Big)^n x^n
\end{aligned}$$
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 $L_n^\alpha = L_n$:
$$\begin{aligned}
L_n^\alpha(x)
&= \frac{1}{n!} x^{-\alpha} \exp\!(x) \dv[n]{x} \big( x^{n + \alpha} \exp\!(-x) \big)
\\
&= \frac{x^{-\alpha}}{n!} \Big( \dv{x} - 1 \Big)^n x^{n + \alpha}
\end{aligned}$$
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:
$$\begin{aligned}
\boxed{
\dv[k]{x} L_n^\alpha(x)
= (-1)^k L_{n - k}^{\alpha + k}(x)
}
\end{aligned}$$
Noteworthy is that these polynomials (both normal and associated)
are all mutually orthogonal for $x \in [0, \infty[$,
with respect to the weight function $w(x) \equiv x^\alpha \exp\!(-x)$:
$$\begin{aligned}
\boxed{
\braket{L_m^\alpha}{w L_n^\alpha}
= \int_0^\infty L_m^\alpha(x) \: L_n^\alpha(x) \: w(x) \dd{x}
= \frac{\Gamma(n + \alpha + 1)}{n!} \delta_{nm}
}
\end{aligned}$$
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 $\braket{f}{w f}$ is finite.
Any such $f$ can thus be expanded as follows:
$$\begin{aligned}
\boxed{
f(x)
= \sum_{n = 0}^\infty a_n L_n^\alpha(x)
= \sum_{n = 0}^\infty \frac{\braket{L_n}{w f}}{\braket{L_n}{w L_n}} L_n^\alpha(x)
}
\end{aligned}$$
Finally, the $L_n^\alpha(x)$ are related to
the [Hermite polynomials](/know/concept/hermite-polynomials/) $H_n(x)$ like so:
$$\begin{aligned}
H_{2n(x)} &= (-1)^n 2^{2n} n! \: L_n^{-1/2}(x^2)
\\
H_{2n + 1(x)} &= (-1)^n 2^{2n + 1} n! \: L_n^{1/2}(x^2)
\end{aligned}$$
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