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Preprint

30 December 2018

Using a recursive formula for the Mellin transform \(T_{n,a}(s)\) of a spherical, principal series \(GL(n,\mathbb{R})\) Whittaker function, we develop an explicit recurrence relation for this Mellin transform. This relation, for any \(n\ge2\), expresses \(T_{n,a}(s)\) in terms of a number of "shifted" transforms \(T_{n,a}(s+\Sigma)\), 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 \(T_{4,a}(s)\) in terms of \(T_{4,a}(s+\Sigma)\) and \(T_{4,a}(s+\Omega)\), where for some \(1\le k\le 3\), the \(k\)th coordinates of both \(\Sigma\) and \(\Omega\) are strictly positive. Finally, we deduce a recurrence relation for \(T_{4,a}(s)\) involving strictly positive shifts in all three \(s_k\)'s at once. (That is, the condition "for some \(1\le k\le 3\)" above becomes "for all \(1\le k\le 3\).") These additional relations on \(GL(4,\mathbb{R})\) may be applied to the explicit understanding of certain poles and residues of \(T_{4,a}(s)\). This residue information is, as we describe below, in turn relevant to work of Goldfeld and Woodbury, concerning orthogonality of Fourier coefficients of \(SL(4,\mathbb{R})\) Maass forms, and the \(GL(4)\) Kuznetsov formula.

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Bertram Kostant (1978)

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Goro Shimura (1976)

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J. A. Shalika (1974)