  We work out a theory of integrability on the braided covector Hopf algebra
and braided vector Hopf algebra of type A_n as introduced by Kempf and Majid.
Starting by their definition of braided Fourier transform we prove
n-dimensional analogues to results by Koornwinder expressing the correspondence
between products of q^2-Gaussians times monomials and products of q^2-Gaussians
times q^2-Hermite polynomials under the transform. We invert the correspondence
finding a suitable inverse to the transform, different from that by Kempf and
Majid (our integral is not bosonic) and we show that in this case the
(q-)Plancherel measure will depend on the parity of the generalized functions
that we are transforming.
