= Generating function of the physicists' Hermite polynomials
{title2=$\sum_{n\geq0}H_n(t)z^n/n!=e^{2tz-z^2}$}
For the <Hermite polynomials> in the Rodrigues normalization $H_n=(-1)^ne^{t^2}D^ne^{-t^2}$, differentiating gives $H_{n+1}=2tH_n-H_n'$ with $H_0=1$. Induction gives
$$
H_n(t)=n!\sum_{j=0}^{\lfloor n/2\rfloor}\frac{(-1)^j2^{n-2j}t^{n-2j}}{j!(n-2j)!}.
$$
Insert this into the exponential generating <series> and reindex $n=m+2j$. The result factors into $\sum_m(2tz)^m/m!$ times $\sum_j(-z^2)^j/j!$, proving the displayed identity. Equivalently Taylor expansion of $e^{-(t-z)^2}$ in $z$ gives the same formula. This <generating function> keeps the normalization distinct from the <Probabilists' Hermite polynomial> convention.
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