Formula_21168

\[\begin{gathered}H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x), \\H_{n}^{\prime}(x)=2 n H_{n-1}(x), \\H_{n}(x)=\sum_{k=0}^{\left[\frac{n}{2}\right]} \frac{(-1)^{k} n !}{k !(n-2 k) !}(2 x)^{n-2 k}, \\\exp \left(2 x w-w^{2}\right)=\sum_{n=0}^{\infty} \frac{H_{n}(x)}{n !} w^{n}\end{gathered}\]