\[H_{n}\left(x\right)=2^{n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^{2}\right)=2^{n}%
xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},x^{2}\right)=2^{\frac{1}{2}n}%
{\mathrm{e}}^{\frac{1}{2}x^{2}}U\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right),\]