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}); ; ; .