Monic orthogonal polynomials satisfy a three-term recurrence

For $j<m$, $p_j^2$ and $x{p}_j^2$ have degrees at most $2j$ and $2j+1$, both at most $2m-1$. Thus the moments in $h_j$ and ${\alpha }_j$ are finite by [F1]. Since $p_j$ is nonzero, [F2] gives $h_j>0$, so ${\alpha }_j$ is defined; for $j\ge 1$, the same positivity for $p_{j-1}$ gives ${\beta }_j>0$.

The polynomials $p_0,\dots ,p_j$ form a basis for the polynomials of degree at most $j$: their distinct degrees and unit leading coefficients permit successive subtraction of the top term. Because $x{p}_j-p_{j+1}$ has degree at most $j$ by cancellation of the leading term, write it in this basis.

For $k\le j-2$, symmetry of the real pairing gives $⟨x{p}_j,p_k{⟩}_{\mu }=⟨p_j,x{p}_k{⟩}_{\mu }=0$, since $x{p}_k$ has degree at most $j-1$. As the basis in step 1.2 is orthogonal and each squared norm is positive by [F2], the expansion has no $p_k$ term for $k\le j-2$. Its $p_j$ coefficient is $⟨x{p}_j,p_j{⟩}_{\mu }\mathrm{/}{h}_j={\alpha }_j$.

For $j\ge 1$, the $p_{j-1}$ coefficient is $\frac{⟨x{p}_j,p_{j-1}{⟩}_{\mu }}{h_{j-1}}=\frac{⟨p_j,x{p}_{j-1}{⟩}_{\mu }}{h_{j-1}}=\frac{h_j}{h_{j-1}}={\beta }_j\mathrm{.}$ Indeed, $x{p}_{j-1}-p_j$ has degree at most $j-1$, so it is orthogonal to $p_j$ by [F3].

The expansion in step 1.2 therefore has only the displayed $p_j$ and, when $j\ge 1$, $p_{j-1}$ terms. Rearranging it yields the stated recurrence for every $0\le j<m$.