orthopoly - Maple Programming Help

orthopoly

 H
 Hermite polynomial

 Calling Sequence H(n, x)

Parameters

 n - non-negative integer x - algebraic expression

Description

 • The H(n, x) function computes the nth Hermite polynomial evaluated at x.
 • The Hermite polynomials are orthogonal on the interval ($-\mathrm{\infty },\mathrm{\infty }$) with respect to the weight function $w\left(x\right)={ⅇ}^{-{x}^{2}}$. They satisfy:

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}w\left(t\right)H\left(n,t\right)H\left(m,t\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt=\left\{\begin{array}{cc}0& n\ne m\\ \sqrt{\mathrm{\pi }}{2}^{n}n!& n=m\end{array}\right\$

 • Hermite polynomials satisfy the following recurrence relation:

$H\left(0,x\right)=1,$

$H\left(1,x\right)=2x,$

$H\left(n,x\right)=2xH\left(n-1,x\right)-2\left(n-1\right)H\left(n-2,x\right),\mathrm{for n>1.}$

Examples

 > with(orthopoly):
 > H(2,x);
 ${4}{}{{x}}^{{2}}{-}{2}$ (1)
 > H(3,x);
 ${8}{}{{x}}^{{3}}{-}{12}{}{x}$ (2)
 > H(10,5);
 ${3275529760}$ (3)