We know that $\frac{d^k}{dx^k} e^{-\frac{x^2}{2}}$ can be written in terms of Hermite polynomials as \begin{align} \frac{d^k}{x^k} e^{-\frac{x^2}{2}}= (-1)^k e^{-\frac{x^2}{2}} H_{e_k}(x) \end{align}
It appears that in this case, we have to use complex values that is \begin{align} \frac{d^k}{dx^k} e^{\frac{x^2}{2}}= (-1)^k e^{\frac{x^2}{2}} H_{e_k}(ix) \end{align}
My questions: Is this a correct way to go about this? Also, are $H_{e_k}(ix)$ well defined?
$e^{x^2/2} = e^{-y^2/2}$ where $y = ix$. You should get $$ \dfrac{d^k}{dx^k} e^{x^2/2} = (-i)^k e^{x^2/2} H_k(ix)$$ (for the "probabilists" Hermite polynomials: physicists would not have the "$/2$").