 Generating function of the physicists' Hermite polynomials

For the Hermite polynomials in the Rodrigues normalization $H_n=(-1{)}^n{e}^{t^2}{D}^n{e}^{-t^2}$, differentiating gives $H_{n+1}=2t{H}_n-H_n^{\prime }$ with $H_0=1$. Induction givesInsert 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.