  Let $\mathcal{D}=G/K$ be a complex bounded symmetric domain of tube type in a
complex Jordan algebra $V$ and let $\mathcal{D}_{\mathbb{R}}=H/L\subset
\mathcal{D}$ be its real form in a formally real Euclidean Jordan algebra
$J\subset V$. We consider representations of $H$ that are gotten by the
generalized Segal-Bargmann transform from a unitary $G$-space of holomorphic
functions on $\mathcal{D}$ to an $L^2$-space on $\mathcal{D_{\mathbf{R}}}$. We
prove that in the unbounded realization the inverse of the unitary part of the
restriction map is actually the Laplace transform. We find the extension to
$\mathcal{D}$ of the spherical functions on $\mathcal{D}_{\mathbb{R}}$ and find
the expansion in terms of the $L$-spherical polynomials on $\mathcal{D}$, which
are Jack symmetric polynomials. We prove that the coefficients are orthogonal
polynomials in an $L^2$-space, the measure being the Harish-Chandra Plancherel
measure multiplied by the symbol of the Berezin transform. We prove the
difference equation and recurrence relation for those polynomials by
considering the action of the Lie algebra and the Cayley transform on the
polynomials on $\mathcal D$.
