From ac6a9c683d9a8f94d626838745d9b2bbd459b748 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Thu, 28 Apr 2022 13:31:38 +0200
Subject: Add definition with hypergeometric function 2F1

---
 notes/FourierOnS2.tex | 33 +++++++++++++++++++++++++++++----
 1 file changed, 29 insertions(+), 4 deletions(-)

(limited to 'notes/FourierOnS2.tex')

diff --git a/notes/FourierOnS2.tex b/notes/FourierOnS2.tex
index 24b972f..edd4813 100644
--- a/notes/FourierOnS2.tex
+++ b/notes/FourierOnS2.tex
@@ -227,7 +227,7 @@ Finding the solutions to this equation is so involved, that it deserves its own
 
 \subsection{The associated Legendre polynomials}
 
-In this section we would like to find the solutions to the \emph{associated} Legendre equation, which is actually a generalization of Legendre equation:
+In this section we would like to find the solutions to the \emph{associated} Legendre equation, which is actually a generalization of Legendre's equation:
 \begin{equation} \label{eqn:legendre}
   \left( 1 - x^2 \right) \frac{d^2 y}{dx^2}
     - 2x \frac{dy}{dx} + n(n + 1) y(x) = 0.
@@ -240,15 +240,40 @@ Thus we first need examine the solutions to this equation before constructing th
     P_n(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} 
     \frac{(-1)^k (2n-2k)!}{2^n k! (n-k)!(n-2k)!} x^{n-2k},
   \end{equation}
-  known as Legendre's polynomials are solutions to Legendre's equation \eqref{eqn:legendre}.
+  are solutions to Legendre's equation \eqref{eqn:legendre} when \(n > 0\).
 \end{proposition}
 
-\begin{lemma} The expression
+The proof for this proposition is quite algebraically involved and is thus left in the appendix. Since this is a power series \eqref{eqn:legendre-poly} can also be rewritten using Gauss' Hypergeometric function.
+
+\begin{proposition}
+  The polynomial \eqref{eqn:legendre-poly} can we rewritten using Gauss' Hypergeometric function
+  \[
+    {}_2F_1 \left( \begin{matrix}
+      a_1, & a_2 \\ \multicolumn{2}{c}{b}
+    \end{matrix} ; \frac{1 - x}{2} \right)
+    =
+    \sum_{k = 0}^\infty \frac{(a_1)_k (a_2)_k}{(b)_k} \frac{x^k}{k!},
+  \]
+  where the notation \((a)_k\) is for the Pochhammer Symbol
+  \[
+    (a)_k = a (a + 1) \ldots (a + k - 1).
+  \]
+  Hence for \(x \in (-1, 1)\) and \(n \in \mathbb{R}\):
+  \[
+    P_n (x) = {}_2F_1 \left( \begin{matrix}
+      n + 1, & -n \\ \multicolumn{2}{c}{1}
+    \end{matrix} ; \frac{1 - x}{2} \right).
+  \]
+\end{proposition}
+
+In some applications, such as in quantum mechanics, it is more common to see it written yet in another form using Rodrigues' Formula.
+
+\begin{proposition} The expression
   \begin{equation} \label{eqn:legendre-rodrigues}
     P_n(x) = \frac{1}{n!2^n}\frac{d^n}{dx^n}(x^2-1)^n.
   \end{equation}
   is equivalent to \eqref{eqn:legendre-poly}.
-\end{lemma}
+\end{proposition}
 
 \begin{proof}
   We start expanding the term \((x^2-1)^n\); According to the binomial theorem
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