In this paper we consider spaces of GL ( 4 , R ) {\text {GL}}(4,\mathbb {R}) -Whittaker functions, which are special functions that arise in the study of GL ( 4 , R ) {\text {GL}}(4,\mathbb {R}) automorphic forms. Our main result is to determine explicitly the series expansion for a GL ( 4 , R ) {\text {GL}}(4,\mathbb {R}) -Whittaker function that is "fundamental," in that it may be used to generate a basis for the space of all GL ( 4 , R ) {\text {GL}}(4,\mathbb {R}) -Whittaker functions of fixed eigenvalues. The series that we find in the case of GL ( 4 , R ) {\text {GL}}(4,\mathbb {R}) is particularly interesting in that its coefficients are not merely ratios of Gamma functions, as they are in the lower-rank cases. Rather, these coefficients are themselves certain series— namely, they are finite hypergeometric series of unit argument. We suspect that this is a fair indication of what will happen in the general case of GL ( n , R ) {\text {GL}}(n,\mathbb {R}) .