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