Chapter 10.3, Problem 31E

The Hermite polynomial $H_n\left(x\right)$ are orthogonal on the interval $\left(-\infty ,\infty \right)$ with respect to the weight $W\left(x\right)=e^{-x^2}$. Verify this fact for the first three Hermite polynomial:

$H_0\left(x\right)=1,H_1\left(x\right)=2x,H_2\left(x\right)=4x^2-2$.