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Let $x_1,\dots ,x_N$ be the distinct real roots of ${\mathrm{He}}_N$. The Gauss-Hermite quadrature rule iswhere $w_j$ is the weighted integral of the $j$th Lagrange interpolation polynomial. Equivalently,The orthogonality of ${\mathrm{He}}_N$ makes the rule exact for every polynomial of degree at most