Generating-function proof of Hermite orthogonality (${\int }_{\mathbb{R}}{e}^{-t^2}{H}_m(t)H_n(t)dt=\sqrt{\pi }{2}^{n}n!{\delta }_{mn}$)

Multiply two copies of the generating function of the physicists' Hermite polynomials and integrate against $e^{-t^2}$. Completing the square givesComparing coefficients gives the displayed orthogonality and normalization. Equivalently differentiation in $y,z$ at zero justifies coefficient extraction by the Gaussian integrable bound on compact parameter ranges. Hence the unnormalized Hermite functions $e^{-t^2/2}{H}_n(t)$ have norm ${\pi }^{1/4}\sqrt{2^{n}n!}$. This proves their mutual orthogonality; completeness of that family is an additional assertion.