# 伴随勒让德多项式

${\displaystyle (1-x^{2})\,{\frac {d^{2}\,y}{dx^{2}}}-2x{\frac {dy}{dx}}+\left(\ell [\ell +1]-{\frac {m^{2}}{1-x^{2}}}\right)\,y=0}$

${\displaystyle m\,=0}$${\displaystyle \ell }$为整数时，方程的解即为一般的勒让德多项式

## 正交性

${\displaystyle \int _{-1}^{1}P_{l}^{m}(x)P_{k}^{m}(x)\mathrm {d} x={\frac {(l+m)!}{(l-m)!}}{\frac {2}{2l+1}}\delta _{kl}}$

${\displaystyle \left\{{\frac {m^{2}}{1-x^{2}}}-{\frac {\mathrm {d} }{\mathrm {d} x}}\left[(1-x^{2}){\frac {\mathrm {d} }{\mathrm {d} x}}\right]\right\}P_{l}^{m}(x)=\lambda P_{l}^{m}(x),\quad \lambda =l(l+1),l\in \mathbb {Z} _{0}^{+}}$

${\displaystyle \int _{0}^{\pi }P_{l}^{m}(\cos \theta )P_{k}^{m}(\cos \theta )\sin \theta \mathrm {d} \theta ={\frac {(l+m)!}{(l-m)!}}{\frac {2}{2l+1}}\delta _{kl}}$

## 与勒让德多项式的关系

${\displaystyle P_{l}^{m}(x)=(1-x^{2})^{m/2}P_{l}^{(m)}(x)}$

## 与超几何函数的关系

${\displaystyle P_{\nu }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left({\frac {z-1}{z+1}}\right)^{\mu /2}\,_{2}F_{1}(-\nu ,\nu +1,1-\mu ,{\frac {1-z}{2}})}$

## 负数阶连带勒让德多项式

${\displaystyle P_{l}^{-m}(x)=(-1)^{m}{\frac {(l-m)!}{(l+m)!}}P_{l}^{m}(x),\quad m=1,\ldots ,l;l\in \mathbb {Z} ^{+}}$

${\displaystyle P_{l}^{-m}(x)=P_{l}^{m}(x)}$

## 与球谐函数的关系

${\displaystyle Y_{l}^{m}(\theta ,\phi )={\sqrt {{\frac {(l-m)!}{(l+m)!}}{\frac {2l+1}{4\pi }}}}P_{l}^{m}(\cos \theta )e^{im\phi }}$

${\displaystyle \int Y_{l}^{m}(\theta ,\phi )Y_{k}^{n*}(\theta ,\phi )\mathrm {d} \Omega =\delta _{kl}\delta _{mn}}$

## 参考文献

1. ^ 吴崇试. 16. 数学物理方法（第二版）. 北京大学出版社. [2003]. ISBN 9787301068199.