\[\int_{-\infty}^{\infty}{\mathrm{e}}^{-xz}H_{n}\left(x\right){\mathrm{e}}^{-x^{%
2}}\,\mathrm{d}x={\pi}^{\frac{1}{2}}(-z)^{n}{\mathrm{e}}^{\frac{1}{4}z^{2}},\]