查理耶多项式

Charlie polynomial

Charlie polynomial

查理耶多项式（Charlier polynomials)是一个以瑞典天文学家Carl Charlier命名的正交多项式，由下列拉盖尔多项式定义

$\sum _{x=0}^{\mathrm{\infty }}\frac{{\mu }^x}{x!}{C}_n(x;\mu )C_m(x;\mu )={\mu }^{-n}{e}^{\mu }n!{\delta }_{nm},\mu >0.$

前几个查理耶多项式为

$\begin{array}{cc}C0& =1\\ C1& =x-1/mu\\ C2& =-x+x^2+2/mu-2\ast x/mu+2/((2-2\ast x)\ast m{u}^2)-2\ast x/((2-2\ast x)\ast m{u}^2)\\ C3& =2\ast x-3\ast x^2+x^3-6/mu+9\ast x/mu-3\ast x^2/mu-12/((2-2\ast x)\ast m{u}^2)+18\ast x/((2-2\ast x)\ast m{u}^2)-6\ast x^2/((2-2\ast x)\ast m{u}^2)-12/((2-2\ast x)\ast (6-3\ast x)\ast m{u}^3)+18\ast x/((2-2\ast x)\ast (6-3\ast x)\ast m{u}^3)-6\ast x^2/((2-2\ast x)\ast (6-3\ast x)\ast m{u}^3)\\ \end{array}$