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Recurrence relations for Mellin transforms of $GL(n,\mathbb{R})$ Whittaker functions

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 $T_{n,a}(s+Σ)$, 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 $T_{4,a}(s)$ in terms of $T_{4,a}(s+Σ)$ and $T_{4,a}(s+Ω)$, where for some $1\le k\le 3$, the $k$th coordinates of both $Σ$ and $Ω$ 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.