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