Solution
= Solution
In the physicists' <Hermite polynomial> convention, begin by differentiating the <Gaussian function>:
$$
\frac d{dx}e^{-x^2}=-2xe^{-x^2},\qquad
\frac{d^2}{dx^2}e^{-x^2}=(4x^2-2)e^{-x^2}.
$$
The defining prefactor then gives
$$
\boxed{H_0=1,\qquad H_1=2x,\qquad H_2=4x^2-2}.
$$
The derivative-based definition uses nonnegative integers; negative orders are not part of this <polynomial> definition.