  We study degenerate Whittaker vectors in scalar type holomorphic discrete
series representations of tube type Hermitian Lie groups and their analytic
continuation. In four different realizations, the bounded domain picture, the
tube domain picture, the $L^2$-model and the Fock model, we find their explicit
$K$-type expansions. The coefficients are expressed in terms of the generalized
Laguerre functions on the corresponding symmetric cone, and we relate the
$K$-type expansions to the formula for the generating function of the Laguerre
polynomials and to their recurrence relations.
