In a similar way, for an integral which goes like $$ \begin{equation*} I=\int_{-\infty}^{\infty}f(x)dx =\int_{-\infty}^{\infty}e^{-x^2}g(x)dx. \end{equation*} $$ we could use the Hermite polynomials in order to extract weights and mesh points. The Hermite polynomials are the solutions of the following differential equation $$ \begin{equation} \frac{d^2H(x)}{dx^2}-2x\frac{dH(x)}{dx}+ (\lambda-1)H(x)=0. \tag{18} \end{equation} $$