Orthogonal monic polynomials $p_{0}, p_{1}, \ldots, p_{n}, \ldots$ are defined with respect to the inner product $\langle p, q\rangle=\int_{-1}^{1} w(x) p(x) q(x) d x$, where $p_{n}$ is of degree $n$. Show that such polynomials obey a three-term recurrence relation

$$p_{n+1}(x)=\left(x-\alpha_{n}\right) p_{n}(x)-\beta_{n} p_{n-1}(x)$$

for appropriate choices of $\alpha_{n}$ and $\beta_{n}$.

Now suppose that $w(x)$ is an even function of $x$. Show that the $p_{n}$ are even or odd functions of $x$ according to whether $n$ is even or odd.