A generalized hypergeometric function III. Associated Hilbert space transform

For generic parameters (a(+),a(-),c)is an element of(0,infinity)(2) x R-4, we associate a Hilbert space transform to the ''relativistic'' hypergeometric function (a(+),a(-),c; v, (v) over cap) studied in previous papers. Restricting the couplings c to a certain polytope, we show that the (renormalized) R-function kernel gives rise to an isometry from the even subspace of L-2(R, (w) over cap (v)d (v) over cap) to the even subspace of L-2(R,w(v)dv), where (w) over cap((v) over cap) and w(v) are positive and even weight functions. We prove that the orthogonal complement of the range of this isometry is spanned by Nis an element ofN pairwise orthogonal functions. The latter are in essence Askey-Wilson polynomials, arising from the R-function by choosing (v) over cap =ikappa(n), with kappa(0),...,kappa(N-1) distinct negative numbers. The two commuting analytic difference operators acting on the variable v for which R is a joint eigenfunction, give rise to two commuting self-adjoint Hamiltonians on the even subspace of L-2(R,w(v)dv). We explicitly determine the relation of the time-dependent scattering theory for these dynamics to their joint spectral transform.