Recurrence relations for Mellin transforms of [formula omitted] Whittaker functions

Abstract

Using a recursive formula for the Mellin transform Tn,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 Tn,a(s) in terms of a number of “shifted” transforms Tn,a(s+Σ), with each coordinate of Σ 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 T4,a(s) in terms of T4,a(s+Σ) and T4,a(s+Ω), where for some 1≤k≤3, the kth coordinates of both Σ and Ω are strictly positive.Next, we deduce a recurrence relation for T4,a(s) involving strictly positive shifts in all three sk'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 T4,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.