# 勒让德多项式

${\displaystyle (1-x^{2}){\frac {d^{2}P(x)}{dx^{2}}}-2x{\frac {dP(x)}{dx}}+n(n+1)P(x)=0.}$

${\displaystyle {d \over dx}\left[(1-x^{2}){d \over dx}P(x)\right]+n(n+1)P(x)=0.}$

## 正交性

${\displaystyle \int _{-1}^{1}P_{m}(x)P_{n}(x)\,dx={2 \over {2n+1}}\delta _{mn}}$

${\displaystyle {d \over dx}\left[(1-x^{2}){d \over dx}P(x)\right]=-\lambda P(x),}$

## 部分实例

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

## 在物理学中的应用

${\displaystyle {\frac {1}{\left|\mathbf {x} -\mathbf {x} ^{\prime }\right|}}={\frac {1}{\sqrt {r^{2}+r^{\prime 2}-2rr'\cos \gamma }}}=\sum _{\ell =0}^{\infty }{\frac {r^{\prime \ell }}{r^{\ell +1}}}P_{\ell }(\cos \gamma )}$

${\displaystyle \Phi (r,\theta )=\sum _{\ell =0}^{\infty }\left[A_{\ell }r^{\ell }+B_{\ell }r^{-(\ell +1)}\right]P_{\ell }(\cos \theta ).}$

## 其他性质

${\displaystyle P_{k}(-x)=(-1)^{k}P_{k}(x).\,}$

??? 递推关系 ??? 相邻的三个勒让德多项式具有三项递推关

${\displaystyle (n+1)P_{n+1}=(2n+1)xP_{n}-nP_{n-1}\,}$

${\displaystyle {x^{2}-1 \over n}{d \over dx}P_{n}=xP_{n}-P_{n-1}.}$
${\displaystyle (2n+1)P_{n}={d \over dx}\left[P_{n+1}-P_{n-1}\right].}$

#include <iostream>
using namespace std;

int main()
{
float n,x;
float polyaendl;

return 0;
}

float polya(float n, float x)
{
if (n == 0) return 1.0;
eurn x;
else return ((2.0 * n - 1.0) * x * polya(n - 1.0, x) - (n - 1.0) * polya(n - 2.0, x)) / n;
}


## 移位勒让德多项式

${\displaystyle \int _{0}^{1}{\tilde {P_{m}}}(x){\tilde {P_{n}}}(x)\,dx={1 \over {2n+1}}\delta _{mn}.}$

${\displaystyle {\tilde {P_{n}}}(x)=(-1)^{n}\sum _{k=0}^{n}{n \choose k}{n+k \choose k}(-x)^{k}.}$

${\displaystyle {\tilde {P_{n}}}(x)=(n!)^{-1}{d^{n} \over dx^{n}}\left[(x^{2}-x)^{n}\right].\,}$

 n ${\displaystyle {\tilde {P_{n}}}(x)}$ 0 1 1 ${\displaystyle 2x-1}$ 2 ${\displaystyle 6x^{2}-6x+1}$ 3 ${\displaystyle 20x^{3}-30x^{2}+12x-1}$

## 极限关系

${\displaystyle \lim _{q\to 1}P_{n}(x|q)=P_{n}(x)}$

${\displaystyle \lim _{q\to 1}p_{n}(x|q)=P_{n}(1-2x)}$

## 参考文献

1. ^ 严镇军编，《数学物理方程》，第二版，中国科学技术大学出版社，合肥，2002，ISBN 7-312-00799-6，第140页
• 2. Milton Abramowitz and Irene A. Stegun, eds. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4（参见 第8章第22章