From e85acc31dbf0c244d34a806f5c700990d374f14c Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 9 Sep 2021 17:25:09 +0200
Subject: Expand knowledge base

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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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