The authors study in detail the Mellin transform of Jacquet's Whittaker function for \(\text{GL}(n,\mathbb{R})\) and give a new proof for its meromorphic continuation, see also [\textit{H. Jacquet} and \textit{J. Shalika}, Automorphic forms, Shimura varieties and \(L\)-functions, Proc. Conf. Ann Arbor 1988, Perspect. Math. 11, 143-226 (1990; Zbl 0695.10025)]. They further show, that it satisfies certain explicit difference equations, an effective algorithm for obtaining these is presented and applied to some small \(n\). Finally the authors generalize their results to the case of a connected reductive algebraic group defined and quasi-split over \(\mathbb{R}\).