 Hermite polynomial derivative recurrence

For the physicists' Hermite polynomials $H_n=(-1{)}^n{e}^{x^2}{D}^n{e}^{-x^2}$, the displayed recurrence holds for $n\ge 1$, while $H_0^{\prime }=0$ separately. Put $u=e^{-x^2}$. Differentiating $u^{\prime }+2xu=0$ exactly $n$ times using the Leibniz rule yieldsDifferentiating the defining product for $H_n$ then givesThe nonnegative integer convention is essential for the derivative-based definition; no negative-order derivative or $H_{-1}$ is needed.