Hermite Polynomial

Hermite polynomials are used for solution of the QHO

$H_n(\xi )=(-1{)}^n{e}^{{\xi }^2}{\left(\frac{\mathrm{d}}{\mathrm{d}\xi }\right)}^n{e}^{-{\xi }^2}$

• This formula is the Rodrigues formula
• Expands to $n$th order polynomials due to cancellations of $e^{{\xi }^2}$ with $e^{-{\xi }^2}$
• Example:
• $H_1(\xi )=(-1{)}^1{e}^{{\xi }^2}\frac{\partial }{\partial \xi }{e}^{-{\xi }^2}=-1\cdot e^{{\xi }^2}(-2\xi e^{-{\xi }^2})=2\xi$

${\int }_{-\infty }^{\infty }{H}_n(x)H_m(x)e^{-x^2}\mathrm{d}x={\delta }_{nm}(x)2^{n}n!\sqrt{\pi }$

• Orthogonality condition of Hermite polynomials
• ${\delta }_{nm}(x)$ is the Kronecker delta

$H_n(\xi )=\sum _{j=0}^{\infty }{a}_j{X}^j,a_{j+2}=\frac{2j-k+1}{(j+1)(j+2)}{a}_j$

• Power series/recurrence relation representation of the above equation

$H_n(z)=\frac{n!}{2\pi i}\oint e^{-t^2+2tz}{t}^{-n-1}\mathrm{d}t$

• Contour integral definition
• Wolfram article

Example Values

TableForm[Table[Simplify[(-1)^n*E^\[Xi]^2*D[E^(-\[Xi]^2), {\[Xi], n}]], {n, 4}]] (* 1st Version*)
TableForm[Table[Simplify[HermiteH[n,\[Xi]]],{n,4}]] (* Mathematica version*)

Hermite Polynomial	Value
$H_0(\xi )$	$1$
$H_1(\xi )$	$2\xi$
$H_2(\xi )$	$4{\xi }^2-2$
$H_3(\xi )$	$4\xi (2{\xi }^2-3)$
$H_4(\xi )$	$4(4{\xi }^4-12{\xi }^2+3)$