= Hermite polynomial derivative recurrence
{c}
{title2=$H_n'=2nH_{n-1}$}
For the physicists' <Hermite polynomials> $H_n=(-1)^ne^{x^2}D^ne^{-x^2}$, the displayed recurrence holds for $n\geq1$, while $H_0'=0$ separately. Put $u=e^{-x^2}$. Differentiating $u'+2xu=0$ exactly $n$ times using the <Leibniz rule> yields
$$
(D+2x)D^nu=-2nD^{n-1}u.
$$
Differentiating the defining product for $H_n$ then gives
$$
H_n'=(-1)^ne^{x^2}(D+2x)D^nu
=2n(-1)^{n-1}e^{x^2}D^{n-1}u=2nH_{n-1}.
$$
The nonnegative integer convention is essential for the derivative-based definition; no negative-order derivative or $H_{-1}$ is needed.
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