Past exam of the analysis course of the University of Oxford / 2007 / B / 6 / c

Complete the square to obtain, initially for real $y,z$,Compare coefficients of $y^m{z}^n$ on the two sides. The left coefficient is $(m!n!{)}^{-1}\int e^{-t^2}{H}_m(t)H_n(t)dt$, and the right is zero unless $m=n$, in which case it is $\sqrt{\pi }{2}^n/n!$. The polynomials are real on the real line, so these are the complex $L^2$ inner products of the ${\varphi }_n$. ThusPolynomial times Gaussian decay also directly shows each ${\varphi }_n$ lies in $L^2$.