泽尔尼克多项式

**澤爾尼克多項式**是一個以 1953 年獲諾貝爾物理學獎荷蘭物理學家弗里茨・澤爾尼克命名的正交多項式，分為奇、偶兩類

奇多項式：

　
: $Z_{n}^{m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\cos(m\,\varphi )\!$

偶多項式

　
: $Z_{n}^{-m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\sin(m\,\varphi ),\!$

 
其中 $n\geq m$ 為非負整數，

 $\phi $ 為方位角

```
$0\leq \rho \leq 1$ 为径向距离
```

如果 _n_-_m_ 為偶數則

 

　
: $R_{n}^{m}(\rho )=\sum _{k=0}^{\tfrac {n-m}{2}}{\frac {(-1)^{k}\,(n-k)!}{k!\left({\tfrac {n+m}{2}}-k\right)!\left({\tfrac {n-m}{2}}-k\right)!}}\;\rho ^{n-2\,k}$

如果 _n_-_m_ 為奇數，則

　
: $R_{n}^{m}(\rho )=0$

## 澤爾尼克多項式的超幾何函數表示

澤爾尼克多項式也可以表示為超幾何函數

　
: ${\begin{aligned}R_{n}^{m}(\rho )&={\binom {n}{\tfrac {n+m}{2}}}\rho ^{n}\ {}_{2}F_{1}\left(-{\tfrac {n+m}{2}},-{\tfrac {n-m}{2}};-n;\rho ^{-2}\right)\\&=(-1)^{\tfrac {n+m}{2}}{\binom {\tfrac {n+m}{2}}{\tfrac {n-m}{2}}}\rho ^{m}\ {}_{2}F_{1}\left(1+n,1-{\tfrac {n-m}{2}};1+{\tfrac {n+m}{2}};\rho ^{2}\right)\end{aligned}}$

 

## Noll 序列

Noll 用一個 J 數字表示 [n,m]: 如下表

## 澤爾尼克多項式

由於

　
: $I_{j}=\int _{0}^{2\pi }\int _{0}^{1}Z_{j}^{2}\,\rho \,d\rho \,d\theta =k_{j}*\pi .$

其中 $k_{j}$ 因 j 而異，

　
: $k_{1}=1$

　
: $k_{2}={\frac {1}{4}}$

　
: $k_{3}={\frac {1}{4}}$

　
: $k_{4}={\frac {1}{3}}$

　
: $k_{5}={\frac {1}{6}}$

必須先歸一化

令 $Z_{j}=Z_{j}/{\sqrt {(}}k_{j})$

使得

　
: $I_{j}=\int _{0}^{2\pi }\int _{0}^{1}Z_{j}^{2}\,\rho \,d\rho \,d\theta =\pi .$

 
歸一化澤爾尼克多項式以 Noll 序列排列如下：

## 正交性

**徑向正交性**
: $\int _{0}^{1}\rho {\sqrt {2n+2}}R_{n}^{m}(\rho )\,{\sqrt {2n'+2}}R_{n'}^{m}(\rho )\,d\rho =\delta _{n,n'}.$

**角度正交性**
: $\int _{0}^{2\pi }\cos(m\varphi )\cos(m'\varphi )\,d\varphi =\epsilon _{m}\pi \delta _{|m|,|m'|},$

　
: $\int _{0}^{2\pi }\sin(m\varphi )\sin(m'\varphi )\,d\varphi =(-1)^{m+m'}\pi \delta _{|m|,|m'|};\quad m\neq 0,$

　
: $\int _{0}^{2\pi }\cos(m\varphi )\sin(m'\varphi )\,d\varphi =0,$

其中 $\epsilon _{m}$ 稱為 Neumann 因子，其數值為 _2_ 如果滿足 $m=0$ ，數值為 _1_，如果 $m\neq 0$ . 

**徑向與角度正交性**
: $\int Z_{n}^{m}(\rho ,\varphi )Z_{n'}^{m'}(\rho ,\varphi )\,d^{2}r={\frac {\epsilon _{m}\pi }{2n+2}}\delta _{n,n'}\delta _{m,m'},$

其中 $d^{2}r=\rho \,d\rho \,d\varphi $ 為 雅可比矩陣

 $n-m$ 與 $n'-m'$ 都是偶數.

 

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