查理耶多项式

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^[1]

${\displaystyle \sum _{x=0}^{\infty }{\frac {\mu ^{x}}{x!}}C_{n}(x;\mu )C_{m}(x;\mu )=\mu ^{-n}e^{\mu }n!\delta _{nm},\quad \mu >0.}$

前几个查理耶多项式为

${\displaystyle {\begin{aligned}C0&=1\\C1&=x-1/mu\\C2&=-x+x^{2}+2/mu-2*x/mu+2/((2-2*x)*mu^{2})-2*x/((2-2*x)*mu^{2})\\C3&=2*x-3*x^{2}+x^{3}-6/mu+9*x/mu-3*x^{2}/mu-12/((2-2*x)*mu^{2})+18*x/((2-2*x)*mu^{2})-6*x^{2}/((2-2*x)*mu^{2})-12/((2-2*x)*(6-3*x)*mu^{3})+18*x/((2-2*x)*(6-3*x)*mu^{3})-6*x^{2}/((2-2*x)*(6-3*x)*mu^{3})\\\end{aligned}}}$

1. ^ Szegő, Gabor (1939), Orthogonal Polynomials, Colloquium Publications – American Mathematical Society, ISBN 978-0-8218-1023-1, MR 0372517