= Solution
{c}
The displayed recurrence requires monic normalization, which is not explicitly stated in the preamble. Arbitrary independent rescaling of <orthogonal polynomials> preserves orthogonality but changes the recurrence coefficients. Choose their monic representatives for this and the following calculations.
Expand $x\phi_n$ in the orthogonal <polynomial> basis:
$$
x\phi_n=\phi_{n+1}+\sum_{j=0}^na_j\phi_j.
$$
The leading coefficient of $\phi_{n+1}$ is one. For $j\leq n-2$, $(x\phi_n,\phi_j)=(\phi_n,x\phi_j)=0$, because the second factor has degree below $n$. Thus only $j=n,n-1$ remain. Inner products give
$$
a_n=\frac{(\phi_n,x\phi_n)}{\|\phi_n\|^2},\qquad
a_{n-1}=\frac{(\phi_n,x\phi_{n-1})}{\|\phi_{n-1}\|^2}
=\frac{\|\phi_n\|^2}{\|\phi_{n-1}\|^2}.
$$
The last equality uses $x\phi_{n-1}-\phi_n\in P_{n-1}$. Rearranging yields exactly the <three-term recurrence for monic orthogonal polynomials>
$$
\boxed{\phi_{n+1}=(x-a_n)\phi_n-
\frac{\|\phi_n\|^2}{\|\phi_{n-1}\|^2}\phi_{n-1}.}
$$
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