(a) What are real orthogonal polynomials defined with respect to an inner product $\langle\cdot, \cdot\rangle ?$ What does it mean for such polynomials to be monic?

(b) Real monic orthogonal polynomials, $p_{n}(x)$, of degree $n=0,1,2, \ldots$, are defined with respect to the inner product,

$$\langle p, q\rangle=\int_{-1}^{1} w(x) p(x) q(x) d x$$

where $w(x)$ is a positive weight function. Show that such polynomials obey the three-term recurrence relation,

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

for appropriate $\alpha_{n}$ and $\beta_{n}$ which should be given in terms of inner products.