\[\mathit{He}_{n}\left(x\right)=2^{\frac{1}{2}n}U\left(-\tfrac{1}{2}n,\tfrac{1}{%
2},\tfrac{1}{2}x^{2}\right)=2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac{%
1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)={\mathrm{e}}^{\tfrac{1}{4}x^{2}}U%
\left(-n-\tfrac{1}{2},x\right).\]