Abstract Using a recursive formula for the Mellin transform T n , a ( s ) of a spherical, principal series G L ( n , R ) Whittaker function, we develop an explicit recurrence relation for this Mellin transform. This relation, for any n ≥ 2 , 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 ≤ k ≤ 3 , the kth coordinates of both Σ and Ω are strictly positive. Next, 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 ≤ k ≤ 3 ” above becomes “for all 1 ≤ k ≤ 3 .”) These additional relations on G L ( 4 , 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 recent results concerning orthogonality of Fourier coefficients of S L ( 4 , Z ) Maass forms, and the G L ( 4 ) Kuznetsov formula.

中文翻译：

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GL(n,R) Whittaker 函数的 Mellin 变换的递归关系

摘要 使用梅林变换 T​​ n 的递归公式，球面主级数 GL ( n , R ) Whittaker 函数的 a ( s )，我们开发了此梅林变换的显式递推关系。对于任何 n ≥ 2 ，这种关系用“移位”变换 T n , a ( s + Σ ) 的数量表示 T n , a ( s ) ，其中 Σ 的每个坐标都是一个非负整数。然后我们关注 n = 4 的情况。在这种情况下，我们使用上面引用的关系来推导出进一步的关系，每个关系都涉及 s 坐标之一的“严格正位移”。更具体地说：我们的每个新关系都用 T 4 , a ( s + Σ ) 和 T 4 , a ( s + Ω ) 表示 T 4 , a ( s ) ，其中对于某些 1 ≤ k ≤ 3 ，第 k 个Σ 和 Ω 的坐标都严格为正。接下来，我们推导出 T 4 的递推关系，a ( s ) 涉及所有三个 sk 一次严格的正变化。（也就是说，上面的条件“对于一些 1 ≤ k ≤ 3 ”变成了“对于所有 1 ≤ k ≤ 3 。”）GL ( 4 , R ) 上的这些附加关系可以应用于对某些极点和残差的明确理解T 4 , a(s) 。正如我们在下面描述的，这个残差信息又与最近关于 SL ( 4 , Z ) Maass 形式的傅立叶系数的正交性和 GL (4) Kuznetsov 公式相关的结果相关。