Table 2 Explicit expressions of the basis functions

Power

W _n (x)

0

1

1

x ⁿ

Legendre

P _n (x)

[n/2]

2⁻ⁿ

\( {\left(-1\right)}^m.\left(\underset{m}{\overset{n}{}}\right).\left(\underset{n}{\overset{2.n-2.m}{}}\right) \)

x ^n−2.m

Laguerre

L _n (x)

n

1

\( \frac{{\left(-1\right)}^m}{m!}.\left(\underset{n-m}{\overset{n}{}}\right) \)

x ^m

Hermite A

H _n (x)

\( \left[\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right] \)

n!

\( {\left(-1\right)}^m.\frac{1}{m!\left(n-2.m\right)!} \)

(2.x)^n−2.m