查理耶多項式

查理耶多項式（Charlier polynomials)是一個以瑞典天文學家Carl Charlier命名的正交多項式，由下列拉蓋爾多項式定義^[1]

Charlie polynomial

Charlie polynomial

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

前幾個查理耶多項式為

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

參考文獻

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