= Solution
The <Probabilists' Hermite polynomial>[probabilists' Hermite polynomials] satisfy
$$
\mathrm{He}'_n=n\mathrm{He}_{n-1},
\qquad
x\mathrm{He}_n=\mathrm{He}_{n+1}+n\mathrm{He}_{n-1},
$$
which follow by differentiating the Rodrigues formula. Thus
$$
\boxed{\mathrm{He}_{n+1}=x\mathrm{He}_n-n\mathrm{He}_{n-1}},
$$
so $\alpha_n=0$ and $\beta_n=n$.
For the squared norm, substitute the Rodrigues formula and integrate by parts $n$ times. All boundary terms vanish under the Gaussian weight, and $\mathrm{He}_n^{(n)}=n!$, giving
$$
\begin{aligned}
\langle\mathrm{He}_n,\mathrm{He}_n\rangle
&=(-1)^n\int_{-\infty}^{\infty}
\mathrm{He}_n(x)\frac{d^n}{dx^n}e^{-x^2/2}\,dx\\
&=n!\int_{-\infty}^{\infty}e^{-x^2/2}\,dx
=\boxed{n!\sqrt{2\pi}}.
\end{aligned}
$$
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