  Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet
series for Phi is a Dirichlet series in r complex variables s_1,...,s_r,
initially converging for Re(s_i) sufficiently large, which has meromorphic
continuation to C^r and satisfies functional equations under the
transformations of C^r corresponding to the Weyl group of Phi. Two
constructions of such series are available, one based on summing products of
n-th order Gauss sums, the second based on averaging a certain group action
over the Weyl group. In this paper we study these constructions and the
relationship between them, and give evidence that when n=2 and Phi=A_r they
yield the same multiple Dirichlet series.
