\[\hat{H}_{n+3}\left(x\right)=\frac{(4x^{2}+2)H_{n+1}\left(x\right)+8xH_{n}\left%
(x\right)}{{\pi}^{1/4}\sqrt{2^{n+1}(n+3)n!}}=\frac{\mathscr{W}\left\{H_{1}%
\left(x\right),H_{2}\left(x\right),H_{n+3}\left(x\right)\right\}}{{\pi}^{1/4}%
\sqrt{2^{n+7}(n+1)(n+2)(n+3)!}},\]