The functions $H_{0}, H_{1}, \ldots$ are generated by the Rodrigues formula:

$$H_{n}(x)=(-1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}} e^{-x^{2}}$$

(a) Show that $H_{n}$ is a polynomial of degree $n$, and that the $H_{n}$ are orthogonal with respect to the scalar product

$$(f, g)=\int_{-\infty}^{\infty} f(x) g(x) e^{-x^{2}} d x$$

(b) By induction or otherwise, prove that the $H_{n}$ satisfy the three-term recurrence relation

$$H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x) .$$

[Hint: you may need to prove the equality $H_{n}^{\prime}(x)=2 n H_{n-1}(x)$ as well.]