泽尔尼克多项式的超几何函数表示

泽尔尼克多项式也可以表示为超几何函数

Noll 序列

Noll 用一个J数字表示 [n,m]:如下表

泽尔尼克多项式

由于

其中kj{\displaystyle k_{j}}因j而异，

必须先归一化

令Zj=Zj/(kj){\displaystyle Z_{j}=Z_{j}/{\sqrt {(}}k_{j})}

使得

归一化泽尔尼克多项式以Noll序列排列如下：

正交性

其中 ϵ ϵ -->m{\displaystyle \epsilon _{m}} 称为Neumann因子，其数值为 2 如果满足 m=0{\displaystyle m=0} ，数值为 1，如果 m≠ ≠ -->0{\displaystyle m\neq 0}.

其中 d2r=ρ ρ -->dρ ρ -->dφ φ -->{\displaystyle d^{2}r=\rho \,d\rho \,d\varphi } 为 雅可比矩阵

n− − -->m{\displaystyle n-m} 与 n′− − -->m′{\displaystyle n"-m"} 都是偶数.

参考文献

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