= Parity of monic orthogonal polynomials for an even weight
{title2=$\phi_n(-x)=(-1)^n\phi_n(x)$}
Assume a positive even weight on a symmetric domain has finite <polynomial> moments. Subtracting the <orthogonal projection onto a finite-dimensional subspace> of $x^n$ onto the lower-degree <polynomials> gives the unique monic <orthogonal polynomial>. Reflection produces another such <monic polynomial>, $(-1)^n\phi_n(-x)$, so uniqueness gives the parity identity. Every coefficient with the opposite parity to its degree therefore vanishes.
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