\[\int_{0}^{\infty}x^{2m}{\mathrm{e}}^{-\frac{1}{2}x^{2}}L^{(2m)}_{n}\left(x^{2}%
\right)\cos\left(xy\right)\,\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}\frac{1}%
{n!}{\mathrm{e}}^{-\frac{1}{2}y^{2}}\mathit{He}_{n}\left(y\right)\mathit{He}_{%
n+2m}\left(y\right),\]