  We give a characterization of a generalized Whittaker model of a degenerate
principal series representation of $GL(n,\R)$ as the kernel of some
differential operators. By this characterization, we investigate some examples
on $GL(4,\R)$. We obtain the dimensions of the generalized Whittaker models and
give their basis in terms of hypergeometric functions of one and two variables.
We show the multiplicity one of the generalized Whittaker models by using the
theory of hypergeometric functions.
