Chapter 12.22, Problem 7P

Prove that the functions $H_n(x)$ are orthogonal on $(-\infty ,\infty )$ with respect to the weight function $e^{-x^2}$.

Hint: Write the differential equation (22.14) as $e^{x^2}\frac{d}{dx}\left(e^{-x^2}{y}^{\prime }\right)+2ny=0$, and see Sections 7 and 19.