Parity of monic orthogonal polynomials for an even weight (${\varphi }_n(-x)=(-1{)}^n{\varphi }_n(x)$)

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, $(-1{)}^n{\varphi }_n(-x)$, so uniqueness gives the parity identity. Every coefficient with the opposite parity to its degree therefore vanishes.