 Solution

The weight is strictly positive and all its polynomial moments are finite. Thus the given integral is a inner product: for a nonzero polynomial, its square is positive on a nonempty interval, so its weighted integral is strictly positive.

In the finite-dimensional vector space $V_n$, subtract the orthogonal projection onto a finite-dimensional subspace of $x^n$ onto $V_{n-1}$:This is monic, since the subtracted polynomial has lower degree, and is orthogonal to every lower-degree polynomial. If ${\varphi }_n,{\psi }_n$ were two such monic polynomials, their difference would lie in $V_{n-1}$ and be orthogonal to that space. Taking its inner product with itself gives zero, hence the difference vanishes. This proves existence and uniqueness for every $n$, with ${\varphi }_0=1$.

Odd moments vanish because the weight is even; the even moments areConsequently ${\varphi }_1=x$, while a monic quadratic orthogonal to $x$ has no linear term, and orthogonality to one gives its constant term $-4/2=-2$. ThusFinally $(-1{)}^n{\varphi }_n(-x)$ is also monic and orthogonal to every lower-degree polynomial, by the change of variable $x\mapsto -x$. Uniqueness implies ${\varphi }_n(-x)=(-1{)}^n{\varphi }_n(x)$. In particular ${\varphi }_{2007}$ is odd andThis is parity of monic orthogonal polynomials for an even weight.