M2: Legendre Polynomials

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Nov 3, 2020
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- M1: Hermite Polynomials
- M3: Laguerre Polynomials

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Each Legendre polynomial $P_n(x)\,$ is an $n$ =th-degree polynomial. It may be expressed using Rodrigues' formula:

$P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right]$

That these polynomials satisfy the Legendre differential equation follows by differentiating (n+1) times both sides of the identity

$(x^2-1)\frac{d}{dx}(x^2-1)^n = 2nx(x^2-1)^n$

The first few Legendre polynomials are:

$n$	$P_n(x)\,$
0	$1$
1	$x$
2	$\begin{matrix}\frac12\end{matrix} (3x^2-1) \,$
3	$\begin{matrix}\frac12\end{matrix} (5x^3-3x) \,$
4	$\begin{matrix}\frac18\end{matrix} (35x^4-30x^2+3)\,$
5	$\begin{matrix}\frac18\end{matrix} (63x^5-70x^3+15x)\,$
6	$\begin{matrix}\frac1{16}\end{matrix} (231x^6-315x^4+105x^2-5)\,$
7	$\begin{matrix}\frac1{16}\end{matrix} (429x^7-693x^5+315x^3-35x)\,$
8	$\begin{matrix}\frac1{128}\end{matrix} (6435x^8-12012x^6+6930x^4-1260x^2+35)\,$
9	$\begin{matrix}\frac1{128}\end{matrix} (12155x^9-25740x^7+18018x^5-4620x^3+315x)\,$
10	$\begin{matrix}\frac1{256}\end{matrix} (46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63)\,$

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