Monic Non-commutative Orthogonal Polynomials

For a measure μ on R, the situation is more subtle. One can always orthogonalize the subspaces of polynomials of different total degree (so that one gets a family of pseudo-orthogonal polynomials). The most common approach is to work directly with these subspaces, without producing individual orthogonal polynomials; see, for example [DX01]. One can also further orthogonalize the polynomials of the same total degree, for example to make them orthonormal [BCJ05, BC04]; however this requires a choice of an order on the monomials of the same degree, and there is no canonical choice of such order. The third approach, and (it is easy to see) the only one that will produce monic orthogonal polynomials, is to require that the pseudo-orthogonal polynomials obtained in the first step already be orthogonal. The price one pays is that this can be done only for some measures μ.