Let D ¼ G=K be a complex bounded symmetric domain of tube type in a Jordan algebra V C ; and let
The analytic continuation of the holomorphic discrete series on D forms a family of interesting representations of G: We consider the restriction on D and the branching rule under H of the scalar holomorphic representations. The unitary part of the restriction map gives then a generalization of the Segal-Bargmann transform. The group L is a spherical subgroup of K and we find a canonical basis of L-invariant polynomials in the components of the Schmid decomposition and we express them in terms of the Jack symmetric polynomials. We prove that the Segal-Bargmann transform of those L-invariant polynomials are, under the spherical transform on D; multi-variable Wilson-type polynomials and we give a simple alternative proof of their orthogonality relation. We find the expansion of the spherical functions on D; when extended to a holomorphic function in a neighborhood of 0 2 D; in terms of the L-spherical holomorphic polynomials on D; the coefficients being the Wilson polynomials.