Let

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

be an inner product. The Hermite polynomials $H_{n}(x), n=0,1,2, \ldots$ are polynomials in $x$ of degree $n$ with leading term $2^{n} x^{n}$ which are orthogonal with respect to the inner product, with

$$\left\langle H_{m}, H_{n}\right\rangle= \begin{cases}\gamma_{m}>0 & \text { if } m=n \\ 0 & \text { otherwise }\end{cases}$$

and $H_{0}(x)=1$. Find a three-term recurrence relation which is satisfied by $H_{n}(x)$ and $\gamma_{n}$ for $n=1,2,3$. [You may assume without proof that

$$\left.\langle 1,1\rangle=\sqrt{\pi}, \quad\langle x, x\rangle=\frac{1}{2} \sqrt{\pi}, \quad\left\langle x^{2}, x^{2}\right\rangle=\frac{3}{4} \sqrt{\pi}, \quad\left\langle x^{3}, x^{3}\right\rangle=\frac{15}{8} \sqrt{\pi} .\right]$$

Next let $x_{0}, x_{1}, \ldots, x_{k}$ be the $k+1$ distinct zeros of $H_{k+1}(x)$ and for $i, j=0,1, \ldots, k$ define the Lagrangian polynomials

$$L_{i}(x)=\prod_{j \neq i} \frac{x-x_{j}}{x_{i}-x_{j}}$$

associated with these points. Prove that $\left\langle L_{i}, L_{j}\right\rangle=0$ if $i \neq j$.