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

TL;DR

This paper develops recursive relations for the Mellin transforms of $GL(n,R)$ Whittaker functions, especially for $n=4$, to analyze poles, residues, and applications in automorphic forms and number theory.

Contribution

It introduces explicit recurrence relations for Mellin transforms of $GL(n,R)$ Whittaker functions, with new relations involving positive shifts for $n=4$, aiding in understanding poles and residues.

Findings

Derived recursive formulas for $T_{n,a}(s)$ for all $n",

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Abstract

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.

Equations

$$y=diag(y_1{y}_2\cdots y_{n-1},y_2{y}_3\cdots y_{n-1},\dots ,y_{n-1},1),$$

$${\mathcal{H}}^n=GL(n,\mathbb{R})/(O(n,\mathbb{R})\cdot {\mathbb{R}}^{\ast })$$

$$\{z=xy|x\in X_n,y\in Y_n\}.$$

$$\prod _{j=1}^{n-1}{y}_j^{j(n-j)/2}\prod _{k=1}^j{y_j}^{2a_k},$$

$$d{H}_a={\lambda }_a(d)H_a(d\in D),$$

$$\Theta (x)=exp(2\pi i(x_{1,2}+x_{2,3}+\cdots +x_{n-1,n}))$$

$$T_{n,a}(s)=2^{n-1}\int_{0}^{\infty}\ldots\int_{0}^{\infty}W_{n,a}(y_{1},y_{2},\ldots,y_{n-1})\biggl{\{}\frac{1}{y^{\rho_{n}}}\prod_{j=1}^{n-1}(\pi y_{j})^{2s_{j}}\,\frac{dy_{j}}{y_{j}}\biggr{\}}$$

$$y^{{\rho }_n}=\prod _{j=1}^{n-1}{y}_j^{j(n-j)/2},$$

$$W_{n,a}(y)$$

$$\Gamma (s)={\int }_0^{\infty }{e}^{-t}{t}^s\frac{dt}{t}$$

$$\mathrm{Res}(\Gamma (s),s=-n)=\frac{(-1{)}^n}{n!}(n=0,1,2,\dots ),$$

$$\Gamma (s+1)=s\Gamma (s),\Gamma (s)\Gamma (1-s)=\frac{\pi }{\sin \pi s}.$$

$$\displaystyle K_{\nu}(z)=\frac{1}{4}\cdot\frac{1}{2\pi i}\int_{s}\Gamma\biggl{(}\frac{s+\nu}{2}\biggr{)}\Gamma\biggl{(}\frac{s-\nu}{2}\biggr{)}\biggl{(}\frac{z}{2}\biggr{)}^{-s}\,ds,$$

$$K_{\nu }(z)=\frac{1}{2}{\int }_0^{\infty }{e}^{-y(t+t^{-1})/2}{t}^{\nu }\frac{dt}{t}(\mathrm{Re}y>0).$$

$$W_{2,a}(y)=2\sqrt{y}{K}_{2a}(2\pi y).$$

$$T_{n,a}(s)=\frac{\Gamma (s_1+a_1)\Gamma (s_{n-1}-a_1)}{(2\pi i{)}^{n-2}}$$

$$\displaystyle\times\int_{z}\Bigg{(}\prod_{j=1}^{n-2}\Gamma\Bigl{(}z_{j}+s_{j}-\frac{ja_{1}}{n-1}\Bigr{)}\Gamma\Bigl{(}z_{j}+s_{j+1}+\frac{(n-1-j)a_{1}}{n-1}\Bigr{)}\Bigg{)}\,T_{n-1,b}(-z)\,dz$$

$$=\frac{\Gamma (s_1+a_1)\Gamma (s_{n-1}-a_1)}{(2\pi i{)}^{n-2}}$$

$$\displaystyle\times\int_{z}\Bigg{(}\prod_{j=1}^{n-2}\Gamma(z_{j}+s_{j})\Gamma(z_{j}+s_{j+1}+a_{1})\Bigg{)}\,T_{n-1,b}\biggl{(}\Big{(}-z_{j}-\frac{ja_{1}}{n-1}\Big{)}_{1\leq j\leq n-2}\biggr{)}\,dz,$$

$$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr 0.0pt{\hfil$\displaystyle z=(z_{1},z_{2},\ldots,z_{n-2}),\quad dz=dz_{1}\,dz_{2}\cdots dz_{n-2},\cr 0.0pt{\hfil$\displaystyle b=(b_{1},b_{2},\ldots,b_{n-1})=\biggl{(}a_{2}+\frac{a_{1}}{n-1},a_{3}+\frac{a_{1}}{n-1},\ldots,a_{n}+\frac{a_{1}}{n-1}\biggr{)}.\crcr}}}$$

$$T_{2,a}(s)=\Gamma (s_1+a_1)\Gamma (s_1-a_1),$$

$$U_{n-1}=\biggl{\{}\mu=(\mu_{1},\cdots,\mu_{n-1})\in\{0,1\}^{n-1}\hbox{ \rm and }\sum_{i=1}^{n-2}\mu_{i}\mu_{i+1}=0\biggr{\}}$$

$$I_{\mu }=\{i\in \{1,2,\cdots ,n-1\}:{\mu }_i=1\}(\mu \in U_{n-1}).$$

$${\alpha }_k=\frac{(z_{k-1}+s_k+a_1)(z_k+s_k)}{(s_{k-1}-s_k-a_1)(s_k-s_{k+1}-a_1)}.$$

$$\sum _{\mu \in U_{n-1}}\prod _{k\in I_{\mu }}{\alpha }_k=0.$$

$$\sum _{\mu \in U_1}\prod _{k\in I_{\mu }}{\alpha }_k=\prod _{k\in I_{(0)}}{\alpha }_k+\prod _{k\in I_{(1)}}{\alpha }_k=\prod _{k\in \varnothing }{\alpha }_k+{\alpha }_1=1+\frac{(s_1+a_1)(-a_1+s_1)}{(-s_1-a_1)(s_1-a_1)}=0,$$

$${\beta }_k={\alpha }_k(1+{\delta }_{k,n-1}{\beta }_n)for\; 1\le k\le n-1,$$

$$\sum _{\mu \in U_n}\prod _{k\in I_{\mu }}{\beta }_k=0.$$

$$\sum _{\mu \in U_n}\prod _{k\in I_{\mu }}{\beta }_k=\sum _{\genfrac{}{}{0ex}{}{\mu \in U_n}{{\mu }_{n-1}=0}}\prod _{k\in I_{\mu }}{\beta }_k+\sum _{\genfrac{}{}{0ex}{}{\mu \in U_n}{({\mu }_{n-1},{\mu }_n)=(1,0)}}\prod _{k\in I_{\mu }}{\beta }_k.$$

$$\sum _{\genfrac{}{}{0ex}{}{\mu \in U_n}{{\mu }_{n-1}=0}}\prod _{k\in I_{\mu }}{\beta }_k$$

$$=\sum _{\genfrac{}{}{0ex}{}{\mu \in U_{n-1}}{{\mu }_{n-1}=0}}\prod _{k\in I_{\mu }}{\alpha }_k+\sum _{\genfrac{}{}{0ex}{}{\mu \in U_{n-1}}{{\mu }_{n-1}=0}}{\beta }_n\prod _{k\in I_{\mu }}{\alpha }_k=(1+{\beta }_n)\sum _{\genfrac{}{}{0ex}{}{\mu \in U_{n-1}}{{\mu }_{n-1}=0}}\prod _{k\in I_{\mu }}{\alpha }_k$$

$$=-(1+{\beta }_n)\sum _{\genfrac{}{}{0ex}{}{\mu \in U_{n-1}}{{\mu }_{n-1}=1}}\prod _{k\in I_{\mu }}{\alpha }_k,$$

$$\sum _{\genfrac{}{}{0ex}{}{\mu \in U_n}{({\mu }_{n-1},{\mu }_n)=(1,0)}}\prod _{k\in I_{\mu }}{\beta }_k=\sum _{\genfrac{}{}{0ex}{}{\mu \in U_{n-1}}{{\mu }_{n-1}=1}}\prod _{k\in I_{\mu }}{\beta }_k=(1+{\beta }_n)\sum _{\genfrac{}{}{0ex}{}{\mu \in U_{n-1}}{{\mu }_{n-1}=1}}\prod _{k\in I_{\mu }}{\alpha }_k.$$

Full text

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 $1\leq k\leq 3$, the $k$th coordinates of both $\Sigma$ and $\Omega$ 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 $1\leq k\leq 3$” above becomes “for all $1\leq k\leq 3$.”)

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 recent results concerning orthogonality of Fourier coefficients of $SL(4,\mathbb{Z})$ Maass forms, and the $GL(4)$ Kuznetsov formula.

1 Introduction; definitions and notation

Friedberg and Goldfeld, in [3], demonstrated the existence of recurrence relations for the Mellin transform $T_{n,a}(s)$ of a spherical, principal series $GL(n,\mathbb{R})$ Whittaker function. Additionally, they described a method for producing certain relations of this kind explicitly, given explicit expressions for the $GL(n,\mathbb{R})$-invariant differential operators on the generalized upper half-plane $\mathscr{H}_{n}=GL(n,\mathbb{R})/(O(n,\mathbb{R})\times\mathbb{R}^{*})$.

In the present work, we develop such relations by a different method, which does not require explicit knowledge of the differential operators on $\mathscr{H}_{n}$, and which provides relations more readily, and in more manageable forms. Our approach uses a recursive expression, developed in [7], for $T_{n,a}(s)$ in terms of $T_{n-1,a^{\prime}}(s^{\prime})$ (for certain $a^{\prime}$ and $s^{\prime}$ depending on $a$ and $s$).

We then focus on the case $n=4$. We combine our above recurrence relation, in this case, with translates of this relation under a certain action of the Weyl group for $GL(4,\mathbb{R})$ on the variable $a$. We thereby obtain new relations that are particularly useful for studying meromorphic continuation, poles, and residues of $T_{4,a}(s)$.

Such information is, in turn, relevant to recent work [6] concerning orthogonality of Fourier coefficients of Maass forms for $GL(4)$. This work extends results previously developed in the context of $GL(3)$, cf. [5] (particularly Theorem 1.3 there). In the $GL(4)$ context, the analyses entail a test function $p_{T,R}(y_{1},y_{2},y_{2})$ depending on parameters $T$ and $R$. In [6], $p_{T,R}$ is defined in terms of its explicitly specified Lebedev-Whittaker transform $p_{T,R}^{\sharp}$ (cf. [4], [23]). Because the former equals the integral of the latter against a spherical, principal series $GL(4,\mathbb{R})$ Whittaker function, integrals – specifically, Mellin transforms – of these Whittaker functions arise. To understand $p_{T,R}$ (especially its growth properties), then, one must have information concerning various properties of $T_{4,a}(s)$. We will investigate these properties in this paper.

To establish the framework for our results, and to provide detail concerning the ideas discussed above, let us first discuss harmonic analysis on $GL(n,\mathbb{R})$. To this end, we let $X_{n}\subset GL(n,\mathbb{R})$ be the group of upper triangular matrices with diagonal elements equal to one, and let $Y_{n}\subset GL(n,\mathbb{R})$ be the group of diagonal matrices $y$ of the form

[TABLE]

where $y_{j}\in\mathbb{R}^{+}$ for $1\leq j\leq n-1.$ The Iwasawa decomposition for $GL(n,\mathbb{R})$ identifies the “generalized upper half-plane”

[TABLE]

with the set

[TABLE]

We wish to consider certain (non-zero) eigenfunctions of the algebra $D$ of $GL(n,\mathbb{R})$-invariant differential operators on $\mathscr{H}^{n}$. To do so, it is convenient to start with the simplest such eigenfunction, namely, the power function

[TABLE]

where $a_{k}\in\mathbb{C}$ for $1\leq j\leq n-1$. Let us put $a_{n}=-a_{1}-a_{2}-\cdots-a_{n-1}$ and $a=(a_{1},a_{2},\ldots,a_{n})$, and denote the corresponding power function (1) by $H_{a}(z)\equiv H_{a}(y)$. It is then shown [15] that $H_{a}$ is indeed an eigenfunction of $D$, and that its eigenvalues $\lambda_{a}(d)$, defined by

[TABLE]

are invariant under any permutation of the $a_{j}$’s.

$GL(n,\mathbb{R})$ Whittaker functions may now be defined in terms of the above power function $H_{a}$, and the character

[TABLE]

of $X_{n}$. We have:

Definition 1

A $GL(n,\mathbb{R})$ Whittaker function of type $a$ is a function $f_{a}$, smooth on $\mathcal{H}^{n}$ and meromorphic in $a_{1},a_{2},\ldots,a_{n-1}$, such that:

(a) $df_{a}=\lambda_{a}(d)f_{a}$ for all $d\in D$;

(b) $f_{a}(x^{\prime}z)=\Theta(x^{\prime})f_{a}(z)$ for all $x^{\prime}\in X_{n},z\in\mathcal{H}^{n}$.

The systematic study of Whittaker functions was initiated, in the more general context of Chevalley groups over local fields, by Jacquet [9]. We will restrict our attention here to the context of $GL(n,\mathbb{R})$.

The space $V_{n,a}$ of $GL(n,\mathbb{R})$ Whittaker functions of type $a$ has dimension $n!$: this follows from unpublished work of Casselman and Zuckerman, and independent work of Kostant [12]. In this paper, we will be concerned with the spherical principal series Whittaker function $W_{n,a}(z)\in V_{n,a}$, which is (up to scalars) the unique element of $V_{n,a}$ that decays rapidly as $y_{j}\rightarrow\infty$ for $1\leq j\leq n-1$. This uniqueness property of $W_{n,a}(z)$ follows from multiplicity-one theorems of Shalika [16] and Wallach [22].

The Whittaker function $W_{n,a}(z)$ is central to the harmonic analysis of automorphic forms on $GL(n,\mathbb{R})$. Specifically, suppose $\varphi\colon\mathcal{H}^{n}\rightarrow\mathbb{C}$ is a cusp form for $GL(n,\mathbb{Z})$ – in other words, $\varphi(\gamma z)=\varphi(z)$ for all $\gamma\in GL(n,\mathbb{Z})$ and $z\in\mathcal{H}^{n}$; $\varphi$ decays rapidly in each $y_{j}$, as $y_{j}\rightarrow\infty$; and $\varphi$ is an eigenfunction of $D$ with $d\varphi=\lambda_{a}(d)\varphi$ for all $d$. Then $\varphi$ has a “Fourier-Whittaker expansion” (cf. [14], [16]), all of whose terms are expressible in terms of $W_{n,a}(z)$. This follows from the multiplicity-one theorems mentioned above. If $\varphi$ is instead a Maass form, which is similar to a cusp form but may satisfy somewhat less stringent growth conditions, it is still the case that “most” of the Fourier coefficients of $\varphi$ are expressible in terms of $W_{n,a}(z)$.

The (normalized) Mellin transform $T_{n,a}(s)$ of $W_{n,a}(z)$, given by

[TABLE]

for $s=(s_{1},s_{2},\ldots,s_{n-1})\in{\mathbb{C}}^{n-1}$ and

[TABLE]

plays a significant role in the theory of automorphic forms, and especially in the study of automorphic $L$-functions. (See, for example, [2], [10], [11], [13], [17], [20], [21].) We remark that $W_{n,a}(z)$, properly normalized, is invariant under permutations of the $a_{j}$’s, and consequently so is $T_{n,a}(s)$. The standard normalization of $W_{n,a}(z)$, which is the one originally given by Jacquet [9], and to which we adhere throughout this paper, ensures this invariance.

By Mellin inversion, we have

[TABLE]

with the path of integration in each $s_{j}$ being a vertical line in the complex plane, of sufficiently large real part to keep the poles of $T_{n,{a}}(s)$ on its left.

We conclude this section with some basic facts – to be of importance in what follows – concerning gamma and Bessel functions. Proofs of all of these facts may be found, for example, in [24].

For the gamma function, defined by

[TABLE]

for $\mathop{\rm Re}\nolimits{s}>0$, and having analytic continuation to $\mathbb{C}\backslash\{0,-1,-2,\ldots\}$, we have the residue formula

[TABLE]

and the translation and reflection formulas

[TABLE]

Next, the modified Bessel function $K_{\nu}(z)$ of the second kind is defined as follows:

[TABLE]

the path of integration being a vertical line to the right of any poles of the integrand; or, equivalently

[TABLE]

As is well known (see, for example, [1]), the spherical principal series Whittaker function on $GL(2,\mathbb{R})$ is closely related to the Bessel function $K_{\nu}(z)$ – specifically, we have

[TABLE]

(Here, by a slight abuse of notation, we identify $(a,-a)\in\mathbb{C}^{2}$ with the complex number $a$, and identify the matrix $y=\hbox{diag}(y,1)\in Y_{n}$ with the positive number $y$.)

2 A recurrence formula for general $n$

We begin with the following recursive formula, derived in [7], for the Mellin transform $T_{n,a}(s)$:

[TABLE]

where $a=(a_{1},a_{2},\ldots,a_{n})$ and $s=(s_{1},s_{2},\ldots,s_{n-1})$, as in the previous section; also,

[TABLE]

(The second equality in (9) follows from the change of variable $z_{j}\to z_{j}+ja_{1}/(n-1)$ applied to the first.) It is shown in [18] that the above formula defines $T_{n,a}(s)$ as an analytic function of $s$, in a product of half-planes where the real parts of the $s_{k}$’s are sufficiently large with respect to the real parts of the $a_{j}$’s.

Remark 2

The right-hand side of (9) makes sense for $n\geq 2$, provided we agree that the “zero-fold” integral that appears there, in this case, simply equals one. With these conventions in force, (9) implies that

[TABLE]

a result that also follows from (6), (8), and the transform pair (2), (4).

In this section, we will use (9) to deduce a different sort of recurrence relation for $T_{n,a}(s)$ – one that expresses this Mellin transform in terms of “translates” $T_{n,a}(s+\Sigma)$, for certain integer $(n-1)$-tuples $\Sigma$. The main idea behind our derivation of this relation is as follows. Because of the functional equation $\Gamma(s+1)=s\Gamma(s)$, replacing $s$ by $s+\Sigma$ in (9) will result in some “extra” factors in the integrand on the right-hand side. If some linear combination of these factors, with coefficients independent of the $z_{j}$’s, yields a nonzero expression that is also independent of the $z_{j}$’s, then we have a representation of $T_{n,a}(s)$ in terms of the corresponding translates $T_{n,a}(s+\Sigma)$.

To this end, we have the following combinatorial lemma.

Lemma 3

Let $n$ be an integer with $n\geq 2$. Let

[TABLE]

be the set of binary sequences of length $n-1$ that do not contain two adjacent 1’s, and define

[TABLE]

Let $z_{0}=s_{0}=s_{n}=0$ and $z_{n-1}=-a_{1}$, and define

[TABLE]

Then (defining the empty product to equal $1$) we have

[TABLE]

{@proof}

[Proof.] We proceed by induction on $n$. Since $U_{1}=\{(0),(1)\}$, we see that

[TABLE]

so (12) is true in the base case $n=2$. Now assume that (12) is true for a given integer $n\geq 2$. For clarity of notation, we write $\beta_{k}$ for the quantity defined on the right-hand side of (11), but with $n+1$ in place of $n$. It follows readily that

[TABLE]

where $\delta_{k,n-1}$ denotes the usual Kronecker delta function. To complete our induction proof, we need to show that

[TABLE]

No $\mu\in U_{n}$ can end in a pair of $1$’s, so we may write

[TABLE]

We now investigate each of the two sums on the right-hand side of (15). On the one hand, since $\beta_{k}=\alpha_{k}$ for $1\leq k\leq n-2$, the first of these two sums may be rewritten as follows:

[TABLE]

the last step by the induction hypothesis (12). On the other hand, the second sum in (15) may be rewritten using (13), together with the fact that, by appending a zero to each element of $U_{n-1}$ that ends in a $1$, we obtain exactly those elements of $U_{n}$ that end in $1,0$. Thus

[TABLE]

Putting (16) and (17) into (15) gives

[TABLE]

and we are done. $\Box$

Remark 4

If we append $0,1$ to each element of $U_{n-2}$, append [math] to each element of $U_{n-1}$, and take the union of the results, we get exactly the set $U_{n}$. Thus $|U_{n}|=|U_{n-1}|+|U_{n-2}|$. From this, and the facts that $|U_{1}|=|\{(0),(1)\}|=2$ and $|U_{2}|=|\{(0,0),(0,1),(1,0)\}|=3$, we see that the number of summands in (12) (and hence in (18) below) equals the $(n+1)$st Fibonacci number $F_{n+1}$.

Now let

[TABLE]

where again $s_{0}=s_{n}=0$. The main result of this section is the following.

Theorem 5

[TABLE]

{@proof}

[Proof.] By (9) and the functional equation $\Gamma(s+1)=s\Gamma(s)$, the left-hand side of (18) equals

[TABLE]

which is zero by Lemma 12. $\Box$

Remark 6

In the case $n=2$, (18) yields

[TABLE]

a result that also follows from (10) and the equation $\Gamma(s+1)=s\Gamma(s)$. We may consider (19) to be something of a prototype for the various recurrence relations that follow. Compare, for example, with Propositions 7, 9, 10, and 11 below.

3 The case $n=4$

We now restrict our attention to the case $n=4$, and investigate some analytic properties of $T_{4,a}(s)$. Note that Theorem 5 reads, in this case,

[TABLE]

From this relation, we wish to derive new identities that relate our Mellin transform $T_{4,a}(s)$ to shifts of that transform that are “strictly positive” in one or more of the variables $s_{1}$, $s_{2}$, and $s_{3}$. Such identities will be useful for deriving meromorphic continuation and residue properties of our transform, cf. Section 4 below.

We begin with a pair of relations entailing strictly positive shifts in $s_{1}$. Each of these relations also involves a shift in one of the other $s_{k}$’s – in the first case, $k=3$, and in the second, $k=2$.

Proposition 7

Suppose $s_{1}\neq-a_{k}$ for $1\leq k\leq 4$.

(a)* We have*

[TABLE]

where

[TABLE]

(b)* If also $s_{2}\neq 0$, then*

[TABLE]

where

[TABLE]

Remark 8

The equivalence of the above three expressions for $C_{a}(p,q)$ is readily checked using straightforward algebra. The first two of these expressions highlight the invariance of $C_{a}(p,q)$ under permutations of the $a_{k}$’s; the third expression will be of use in simplifying some calculations in Sections 3 and 4 below.

{@proof}

[Proof.] We first prove (21). We multiply equation (20) through by $(s_{1}-s_{2}-a_{1})(s_{2}-s_{3}-a_{1})$. From the resulting new equation, we subtract the same equation but with $a_{1}$ and $a_{2}$ interchanged. Using the fact that $T_{4,a}(s)$ is invariant under permutations of the $a_{j}$’s, we thereby eliminate $T_{4,a}(s_{1},s_{2}+1,s_{3})$. We find, after dividing everything through by $a_{1}-a_{2}$, that

[TABLE]

We now multiply (25) through by $s_{3}-a_{2}$ and then subtract, from the resulting new equation, the same equation but with $a_{2}$ and $a_{3}$ interchanged. This eliminates the term $T_{4,a}(s_{1},s_{2},s_{3}+1)$ and gives, upon division through by $(s_{1}+a_{4})(a_{2}-a_{3})$, the equation

[TABLE]

which is the desired result.

We next prove (b). We eliminate $T_{4,a}(s_{1}+1,s_{2},s_{3})$ and $T_{4,a}(s_{1}+1,s_{2},s_{3}+1)$ from (20), much in the same way as we eliminated $T_{4,a}(s_{1},s_{2}+1,s_{3})$ and $T_{4,a}(s_{1},s_{2},s_{3}+1)$ above. We obtain the following relation:

[TABLE]

Into (27), we substitute $s_{1}\to s_{1}+1$, yielding a relation among $T_{4,a}(s_{1}+1,s_{2},s_{3})$, $T_{4,a}(s_{1}+1,s_{2}+1,s_{3})$, and $T_{4,a}(s_{1}+1,s_{2},s_{3}+1)$. We may combine this latter relation with (21) to eliminate $T_{4,a}(s_{1}+1,s_{2},s_{3}+1)$; the result is

[TABLE]

as desired. $\Box$ We now derive a relation entailing strictly positive shifts in $s_{2}$.

Proposition 9

If $s_{2}\neq-a_{j}-a_{k}$ for $1\leq j<k\leq 4$, then

[TABLE]

{@proof}

[Proof.] We proceed as in the proofs of parts (a) and (b) of Proposition 7, but this time we eliminate, from (20), the transforms $T_{4,a}(s_{1},s_{2},s_{3}+1)$ and $T_{4,a}(s_{1}+1,s_{2},s_{3}+1)$. The result is the relation

[TABLE]

We form a linear combination of (30) and (23) to eliminate $T_{4,a}(s_{1}+1,s_{2},s_{3})$; the result is (29). $\Box$ We now deduce some relations entailing strictly positive shifts in $s_{3}$.

Proposition 10

(a)* If $s_{3}\neq a_{k}$ for $1\leq k\leq 4$, then*

[TABLE]

(b) If $s_{3}\neq a_{k}$ for $1\leq k\leq 4$ and $s_{2}\neq 0$, then

[TABLE]

(c) If $s_{2}\neq-a_{j}-a_{k}$ for $1\leq j<k\leq 4$, then

[TABLE]

{@proof}

[Proof.] It is well-known – and may, in fact, be deduced from (9), and from induction – that the Mellin transform $T_{n,a}(s)$ is invariant under the transformation

[TABLE]

Parts (a), (b), and (c) of Proposition 10 then follow from Proposition 7(a), Proposition 7(b), and Proposition 9 respectively. $\Box$ For some applications, it is useful to have a recurrence relation expressing $T_{4,a}(s)$ in terms of “strictly positive shifts in all $s_{k}$’s” – that is, in terms of transforms $T_{4,a}(s+\mu)$ where $\mu\in(\mathbb{Z}^{+})^{3}$. We conclude this section with such a relation.

Proposition 11

Let $B(p,q,r)$ be as in (22), and $C_{a}(p,q)$ as in (24). If $s_{1}\neq-a_{k}$ and $s_{3}\neq a_{k}$ for $1\leq k\leq 4$, and $s_{2}\neq-a_{j}-a_{k}$ for $1\leq j<k\leq 4$, then

[TABLE]

{@proof}

[Proof.] We substitute $s_{2}\to s_{2}+1$ into (21), to get

[TABLE]

We then apply this result to (29) to eliminate $T_{4,a}(s_{1},s_{2}+1,s_{3})$; we get

[TABLE]

Next, we recall that $T_{4,a}(s)$ is invariant under the transformation (34), so that (36) yields

[TABLE]

Substituting $s_{1}\to s_{1}+1$ into this then gives

[TABLE]

We now put (39) into (37) to eliminate $T_{4,a}(s_{1}+1,s_{2}+1,s_{3})$; the result is

[TABLE]

Some rearrangement of these terms, using, for example, the fact that

[TABLE]

then yields the stated result. $\Box$

4 Poles and residues of ${T_{n,a}(s)}$

As described in Section 1 above, recent work [6] on orthogonality of Maass forms for $GL(4)$ involves a test function $p_{T,R}$ that is defined as an integral of a known function $p_{T,R}^{\sharp}$ (the Lebedev-Whittaker transform of $p_{T,R}$) against a Whittaker function.

To obtain estimates of the desired strength for $p_{T,R}$, it is necessary to move certain lines of integration, in this integral defining $p_{T,R}$, a finite distance to the left, and to analyze the poles and residues of $T_{4,a}(s)$ that are thereby encountered. We develop the requisite results concerning these poles and residues here.

From the analyticity of $T_{4,a}(s)$ for $s_{k}$’s of sufficiently large real parts, and from Proposition 11, we may readily deduce the following.

Proposition 12

The Mellin transform $T_{4,a}(s)$ extends to a meromorphic function of the variable $(s_{1},s_{2},s_{3})\in\mathbb{C}^{3}$, with poles at

[TABLE]

and no other poles or polar divisors in $\mathbb{C}^{3}$.

The analyses of [6] require explicit information concerning the residues of $T_{4,a}(s)$ at the above poles. Using our above recurrence relations, we will develop some formulas, below, for these residues. These formulas will entail a certain polynomial $p_{\delta}$, the salient properties of which we now describe.

Lemma 13

For $a\in\mathbb{C}$, define

[TABLE]

Then, for $\delta\in\mathbb{Z}_{\geq 0}$ and $b,c,d,e,f,g\in\mathbb{Z}$, define the polynomial

[TABLE]

of degree $\leq 3\delta$, by

[TABLE]

(a)* We have*

[TABLE]

(b)* Suppose $e+f+g+\delta-b-c-d=1$. Then*

[TABLE]

(c)* If any of the variables $b,c$, or $d$ equals a nonpositive integer $-\gamma$, where $0\leq\gamma\leq\delta$, then $p_{\delta}(b,c,d;e,f,g)$ is divisible by*

[TABLE]

{@proof}

[Proof.] We begin with part (a). By definition of $p_{\delta}$, we see that

[TABLE]

the last step because

[TABLE]

and

[TABLE]

We substitute $\kappa\to\kappa-1$ into the second sum on the right-hand side of (46), to get

[TABLE]

the last step by separating the $\kappa=0$ term off from the first sum, separating the $\kappa=\delta+1$ term off from the second sum, and then combining the remaining two sums on $\kappa$ (from $\kappa=1$ to $\kappa=\delta$). But

[TABLE]

and

[TABLE]

So (47) gives

[TABLE]

and part (a) of our lemma is proved.

The proof of part (b) is similar (though somewhat messier). We omit the details.

To prove part (c), we assume that $b=-\gamma$ where $\gamma$ is an integer and $0\leq\gamma\leq\delta$. Then, by (41), we see that $(b)_{\kappa}=0$ for $\kappa>\gamma$, so (43) gives

[TABLE]

the last step because, by (41),

[TABLE]

for $\kappa\leq\gamma\leq\delta$. Certainly (48) implies that $(e+\gamma)_{\delta-\gamma}(f+\gamma)_{\delta-\gamma}(g+\gamma)_{\delta-\gamma}$ divides $p_{\delta}(b,c,d;e,f,g)$, so part (b) of our lemma is proved. $\Box$ Using the above lemma, we may now deduce some explicit results concerning the residues of $T_{4,a}(s)$ at the poles described in Proposition 12.

Proposition 14

(a)* For $m\in\{1,2,3,4\},$ $\{n,p,q\}=\{1,2,3,4\}\backslash\{m\}$, and $\delta_{1}\in\mathbb{Z}_{\geq 0}$, we have*

[TABLE]

(b)* For $\{m,n\}\subset\{1,2,3,4\},$ $\{p,q\}=\{1,2,3,4\}\backslash\{m,n\}$, and $\delta_{2}\in\mathbb{Z}_{\geq 0}$, we have*

[TABLE]

(c)* For $m\in\{1,2,3,4\},$ $\{n,p,q\}=\{1,2,3,4\}\backslash\{m\}$, and $\delta_{3}\in\mathbb{Z}_{\geq 0}$, we have*

[TABLE]

{@proof}

[Proof.] We begin with part (a) of our proposition. We will prove the desired result by induction on $\delta_{1}$.

The case $\delta_{1}=0$ is given given by [18, Theorem 3.2], in the case $n=4$. So let us now assume that (49) is true for a nonnegative integer $\delta_{1}=\delta$. From the recurrence relation (21), we have

[TABLE]

so that, by the induction hypothesis,

[TABLE]

Using the translation equation $\Gamma(s+1)=s\Gamma(s)$, and the definition (22) of $B(s_{1},s_{2},s_{3})$, we may rewrite the above as follows:

[TABLE]

But by Lemma 13(a), the quantity in large parentheses, in (51), equals

[TABLE]

So (51) yields

[TABLE]

which tells us that (49) is true for $\delta_{1}=\delta+1$ as well. So by induction, part (a) of our lemma is proved.

We now prove part (b) of our proposition, by induction on $\delta_{2}$. We first note that the anchor step – the case $\delta_{2}=0$ – is given by [18, Theorem 3.2], in the case $n=4$. Now assume that (50) is true for a nonnegative integer $\delta_{2}=\delta$. From the recurrence relation (29), we have

[TABLE]

so that, by the induction hypothesis,

[TABLE]

Using the relation $\Gamma(s+1)=s\,\Gamma(s)$, we rewrite the above as follows:

[TABLE]

If we now define

[TABLE]

(so that $e+f+g+\delta-b-c-d=1$), then we check that the quantity in large braces, in (52), is equal to

[TABLE]

But by Lemma 13(b), this quantity equals

[TABLE]

So (52) reads

[TABLE]

which is to say that (50) holds for $\delta_{2}=\delta+1$. So, by induction, part (b) of our proposition is proved.

Part (c) of our proposition follows from part (a), and the fact that $T_{4,a}(s)$ is invariant under the transformation (34). $\Box$

Note that, because $T_{4,a}(s)$ is invariant under permutations of the $a_{k}$’s, so are the results of the above proposition.

We now wish to compute residues of the above residues in either of the remaining variables. For example, we wish to compute

[TABLE]

To reduce the total number of computations that we’ll need, we make several observations, which are clear from general principles and from the above proposition.

• •

“Order doesn’t matter;” for example,

[TABLE]

• •

The residue given in Proposition 14(a), as a function of $s_{3}$, is analytic at $s_{3}=a_{m}-\delta_{3}$, for $\delta_{3}\in\mathbb{Z}_{\geq 0}$. (Note that the factor $\Gamma(s_{3}-a_{m})$ is “missing” from our above formula for this residue.)

• •

The residue given in Proposition 14(b), as a function of $s_{3}$, is analytic at $s_{3}=a_{m}-\delta_{3}$ and $s_{3}=a_{n}-\delta_{3}$, for $\delta_{3}\in\mathbb{Z}_{\geq 0}$. (Note that the factors $\Gamma(s_{3}-a_{m})$ and $\Gamma(s_{3}-a_{n})$ are “missing” from our above formula for this residue.) Similarly, as a function of $s_{1}$, this residue is analytic at $s_{1}=-a_{p}-\delta_{1}$ and $s_{1}=-a_{q}-\delta_{1}$, for $\delta_{1}\in\mathbb{Z}_{\geq 0}$.

In light of the above observations, it will suffice to compute the following “two-variable-at-a-time” residues.

Proposition 15

(a)* For $\delta_{1},\delta_{2}\in\mathbb{Z}_{\geq 0}$, we have*

[TABLE]

where $f_{\delta_{1},\delta_{2}}$ is a polynomial of degree at most $2\delta_{1}+\delta_{2}$.

(b)* For $\delta_{1},\delta_{3}\in\mathbb{Z}_{\geq 0}$, we have*

[TABLE]

where $g_{\delta_{1},\delta_{3}}$ is a polynomial of degree at most $2\delta_{1}+\delta_{3}$.

(c)* For $\delta_{2},\delta_{3}\in\mathbb{Z}_{\geq 0}$, we have*

[TABLE]

where $h_{\delta_{2},\delta_{3}}$ is a polynomial of degree at most $\delta_{2}+2\delta_{3}$.

{@proof}

[Proof.] We prove part (a) only; proofs of the other parts are similar. The first equality in (54) follows immediately from Proposition 14(b). We rewrite this equality as follows:

[TABLE]

Denote the quantity in large braces, in (57), by $f_{\delta_{1},\delta_{2}}(s_{3},a)$: then we wish to show that $f_{\delta_{1},\delta_{2}}(s_{3},a)$ is a polynomial of degree at most $2\delta_{1}+\delta_{2}$. We consider two cases: (i) $\delta_{1}>\delta_{2}$ and (ii) $\delta_{1}\leq\delta_{2}$. In the first case, we note that

[TABLE]

is a polynomial of degree $2(\delta_{1}-\delta_{2})$. Since $p_{\delta_{2}}$ has degree at most $3\delta_{2}$, it follows that $f_{\delta_{1},\delta_{2}}(s_{3},a)$ has degree at most $2(\delta_{1}-\delta_{2})+3\delta_{2}=2\delta_{1}+\delta_{2}$, as claimed. On the other hand, if $\delta_{1}\leq\delta_{2}$, then by part (c) of Lemma 13, there is a polynomial $q_{\delta_{1},\delta_{2}}$ such that

[TABLE]

The last step because $\Gamma(s)\Gamma(1-s)=\pi/\sin(\pi s)$. The quantity in large braces, in (57), then equals

[TABLE]

The degree of $q_{\delta_{1},\delta_{2}}$ is, by (58), less than or equal to $3\delta_{2}-2(\delta_{2}-\delta_{1})=2\delta_{1}+\delta_{2}$. So part (a) of our proposition is proved. $\Box$

Remark 16

The above information on the maximum degrees of the polynomials $f_{\delta_{1},\delta_{2}}$, $g_{\delta_{1},\delta_{3}}$, and $h_{\delta_{2},\delta_{3}}$ will be critical to the calculations in [6].

Finally, we compute our three-variable residue. This may be accomplished by computing the appropriate residue of any of the three parts of Proposition 15. We choose to begin with part (a) of this proposition. (Other approaches will yield results that look different, a priori, but can be transformed into each other by various functional equations for $p_{\delta}$.)

Proposition 17

Let $\delta_{1},\delta_{2},\delta_{3}\in\mathbb{Z}_{\geq 0}$. Then

[TABLE]

{@proof}

[Proof.] This follows immediately from Proposition 15(a). $\Box$

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