Mellin Transforms of$\mathbf{\mathit{GL}}(\mathbf{\mathit{n}},{\Bbb R})$) Whittaker Functions

Using a known recursive formula for the class one principal series$GL(n,{\Bbb R})$Whittaker function, we deduce a recursive formula for the multiple Mellin transform of this function. From the latter formula, we verify a conjecture of Goldfeld regarding the location of poles of our Mellin transform. We further express the residues at these poles in terms of Mellin transforms of lower-rank Whittaker functions. Our next result concerns the simplification of our Mellin transform under a certain restriction on the transform parameter. We show, by applying a change of variable to our above result on poles of the Mellin transform, that the$GL(n,{\Bbb R})$transform reduces essentially to a$GL(n-1,{\Bbb R})$transform under this restriction. We then demonstrate that, under further restriction of the Mellin transform parameter, this Mellin transform in fact reduces to a ratio of products of gamma functions. Our result proves a conjecture of Bump and Friedberg that is motivated by the theory of exterior square automorphic L-functions. Finally, we show that a certain Mellin transform of a product of two Whittaker functions (one on$GL(n-1,{\Bbb R})$, and the other on$GL(n,{\Bbb R})$) reduces to a product of gamma functions. This last result verifies a conjecture of Bump regarding archimedean Euler factors of automorphic L-functions on$GL(n-1,{\Bbb R})\times GL(n,{\Bbb R})$.