The reciprocal of the beta function and \(GL(n,R)\) Whittaker functions (Q1323151)

We derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler's beta function. This expression is similar in form to several well-known integrals for the beta function itself. We then apply our new formula to the study of \(\text{GL} (n,\mathbb{R})\) Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral representations of ``fundamental'' Whittaker functions for \(\text{GL}(3,\mathbb{R})\) and \(\text{GL} (4,\mathbb{R})\). The integrals obtained are seen to resemble very closely known integrals for the ``class-one'' Whittaker functions on these groups. It is expected that this correspondence between expressions for different types of Whittaker functions will carry over to groups of arbitrary rank.