HermiteH - Maple Help

HermiteH

Hermite function

 Calling Sequence HermiteH(n, x)

Parameters

 n - algebraic expression x - algebraic expression

Description

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

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}w\left(t\right)\mathrm{HermiteH}\left(n,t\right)\mathrm{HermiteH}\left(m,t\right)ⅆt=\left\{\begin{array}{cc}0& n\ne m\\ \sqrt{\mathrm{\pi }}{2}^{n}n!& n=m\end{array}$

 • Hermite polynomials satisfy the following recurrence relation:

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

 where HermiteH(0,x) = 1 and HermiteH(1,x) = 2*x.
 • For n different from a non-negative integer, the analytic extension of the Hermite polynomial is given by:

$\mathrm{HermiteH}\left(n,x\right)={2}^{n}\sqrt{\mathrm{\pi }}\left(\frac{\mathrm{KummerM}\left(-\frac{1}{2}n,\frac{1}{2},{x}^{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\frac{1}{2}n\right)}-\frac{2x\mathrm{KummerM}\left(\frac{1}{2}-\frac{1}{2}n,\frac{3}{2},{x}^{2}\right)}{\mathrm{\Gamma }\left(-\frac{1}{2}n\right)}\right)$

Examples

 > $\mathrm{HermiteH}\left(3,x\right)$
 ${\mathrm{HermiteH}}{}\left({3}{,}{x}\right)$ (1)
 > $\mathrm{simplify}\left(,'\mathrm{HermiteH}'\right)$
 ${8}{}{{x}}^{{3}}{-}{12}{}{x}$ (2)
 > $\mathrm{HermiteH}\left(3.2,2.1\right)$
 ${59.58210770}$ (3)