  We construct new families of (q-) difference and (contour) integral operators
having nice actions on Koornwinder's multivariate orthogonal polynomials. We
further show that the Koornwinder polynomials can be constructed by suitable
sequences of these operators applied to the constant polynomial 1, giving the
difference-integral representation of the title. Macdonald's conjectures (as
proved by van Diejen and Sahi) for the principal specialization and norm follow
immediately, as does a Cauchy-type identity of Mimachi.
