Recurrence relations for Mellin transforms of GL(n, R) Whittaker functions

Using a recursive formula for the Mellin transform T-n,T-a(s) of a spherical, principal series GL(n, R) Whittaker function, we develop an explicit recurrence relation for this Mellin transform. This relation, for any n >= 2, expresses T-n,T-a(s) in terms of a number of shifted transforms T-n,T-a(s + Sigma), with each coordinate of Sigma being a non-negative integer. We then focus on the case n = 4. In this case, we use the relation referenced above to derive further relations, each of which involves strictly positive shifts in one of the coordinates of s. More specifically: each of our new relations expresses T-4,T-a(s) in terms of T-4,T-a(s+Sigma) and T-4,T- a (s+Omega), where for some 1 <= k <= 3, the kth coordinates of both Sigma and Omega are strictly positive. Next, we deduce a recurrence relation for T-4,T-a(s) involving strictly positive shifts in all three s(k) 's at once. (That is, the condition for some 1 <= k <= 3 above becomes for all 1 <= k <= 3.) These additional relations on GL(4, R) may be applied to the explicit understanding of certain poles and residues of T-4,T-a(s). This residue information is, as we describe below, in turn relevant to recent results concerning orthogonality of Fourier coefficients of SL(4, Z) Maass forms, and the GL(4) Kuznetsov formula. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

In this paper, an explicit recurrence relation is developed for the Mellin transform T-n,T-a(s) of a spherical, principal series GL(n, R) Whittaker function using a recursive formula, involving strictly positive shifts. The focus is placed on the case n = 4, where further relations are derived involving strictly positive shifts in the coordinates of s. A recurrence relation for T-4,T-a(s) involving shifts in all three s(k)'s simultaneously is then deduced, providing insight into certain poles and residues. This residue information is connected to recent results on orthogonality of Fourier coefficients of SL(4, Z) Maass forms and the GL(4) Kuznetsov formula.