# Construction of eigenfunctions for the elliptic
Ruijsenaars difference operators

Edwin Langmann111Department of Physics, KTH Royal Institute of Technology,
SE-106 91 Stockholm, Sweden, Masatoshi Noumi222Department of Mathematics, KTH
Royal Institute of Technology, SE-100 44 Stockholm, Sweden (on leave from:
Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan) and
Junichi Shiraishi333Graduate School of Mathematical Sciences, The University
of Tokyo, Komaba, Tokyo 153-8914, Japan

###### Abstract

We present a perturbative construction of two kinds of eigenfunctions of the
commuting family of difference operators defining the elliptic Ruijsenaars
system. The first kind corresponds to elliptic deformations of the Macdonald
polynomials, and the second kind generalizes asymptotically free
eigenfunctions previously constructed in the trigonometric case. We obtain
these eigenfunctions as infinite series which, as we show, converge in
suitable domains of the variables and parameters. Our results imply that, for
the domain where the elliptic Ruijsenaars operators define a relativistic
quantum mechanical system, the elliptic deformations of the Macdonald
polynomials provide a family of orthogonal functions with respect to the
pertinent scalar product.

Keywords: Ruijsenaars difference operators, perturbative eigenfunctions,
elliptic deformation of Macdonald polynomials

2010 Mathematics Subject Classification: 81Q80; 33E30, 33D67

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Notations
  2. 2 Summary of results
    1. 2.1 Elliptic Ruijsenaars difference operators
    2. 2.2 Elliptic deformation of Macdonald polynomials
    3. 2.3 Asymptotically free eigenfunctions in the domain $|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$
    4. 2.4 Plan of this paper
  3. 3 Integrable transform to the physical domain: a prototype
    1. 3.1 Integral transform
    2. 3.2 From eigenfunctions in the asymptotic domain to Macdonald polynomials
  4. 4 Existence of formal joint eigenfunctions
    1. 4.1 Separation of eigenvalues
    2. 4.2 Elliptic Ruijsenaars operators as power series of $p$
    3. 4.3 Elliptic deformation of Macdonald polynomials
    4. 4.4 Asymptotically free eigenfunctions in the domain $|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$
  5. 5 Convergence of perturbative eigenfunctions
    1. 5.1 Ruijsenaars functions in the asymptotic domain $|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$
    2. 5.2 Elliptic deformation of Macdonald polynomials
    3. 5.3 Orthogonality relation for $\mathcal{P}_{\lambda}(x;p)$
  6. 6 Symmetries of the Ruijsenaars function
    1. 6.1 Symmetry of $f(x;s;p|q,t)$ with respect the rotation of variables
    2. 6.2 Transformation of $f(x;s;p|q,t)$ under the reflection $t\leftrightarrow q/t$
  7. 7 Concluding remarks

##  1 Introduction

We consider the relativistic generalization of $A$-type Calogero-Moser-
Sutherland systems discovered by Ruijsenaars [21]. It is well-known that, in
the trigonometric case, the eigenfunctions of the commuting family of
difference operators defining the Ruijsenaars system are given by the
Macdonald polynomials [16]. While the Macdonald polynomials provide
eigenfunctions that describe identical particles of a quantum mechanical
system, there exists another kind called asymptotically free eigenfunctions
[25, 1, 18, 20] that are interesting from a mathematical point of view. The
aim of this paper is to generalize these two kinds of eigenfunctions to the
general elliptic Ruijsenaars system [21]; see [4, 5, 6, 23, 24] preceding
works in this direction. For that purpose, we generalize an approach based on
kernel functions that was used to construct eigenfunctions of the elliptic
Calogero-Moser-Sutherland system [13, 14].

###  1.1 Motivation

We recall that the Macdonald polynomials,
$P_{\lambda}(x_{1},\ldots,x_{n}|q,t)$, can be interpreted as quantum
mechanical eigenstates of a Hamiltonian, and the quantum numbers
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$ with integers $\lambda_{i}$ such
that $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$ correspond to
quasi-momenta; this Hamiltonian describes a system of an arbitrary number,
$n$, of massive particles moving on the circle of circumference $L>0$, and it
depends on two parameters $mc>0$ and $g>0$ [21].4)4)4)We mention in passing
that $m$, $c$ and $g$ have the physical interpretation of particle mass,
vacuum velocity of light, and coupling constant, respectively [21]; note that
$u_{i}$ here corresponds to $q_{i}$ in [21]. Moreover, the particle positions
$u_{i}\in[-L/2,L/2]$ and parameters $mc$, $g$ in this quantum mechanical
system are related to the variables $x_{i}$ and Macdonald parameters $q$, $t$
as follows [3, 7],5)5)5) This relation between the trigonometric Ruijsenaars
system and the Macdonald polynomials was already noted in by Koornwinder in
1987 [3].

$x_{i}=\exp(2\pi\sqrt{-1}u_{i}/L)\quad(i=1,\ldots,n),\quad q=\exp(-1/mc),\quad
t=\exp(-g/mc).$ (1.1)

Thus, from a physical point of view, one is mainly interested in the
eigenfunctions on the physical domain of variables and parameters given by

$|x_{i}|=1\quad(i=1,\ldots,n),\quad 0<q<1,\quad 0<t<1.$ (1.2)

However, mathematically, it is convenient to extend the variables $x_{i}$ and
parameter $q$, $t$ to the complex domain [22].

The asymptotically free eigenfunctions,
$f(x_{1},\ldots,x_{n};s_{1},\ldots,s_{n}|q,t)$, generalize the Macdonald
polynomials in that they are defined for complex $s_{i}$ and essentially6)6)6)
The precise relation is
$P_{\lambda}(x|q,t)=x^{\lambda}f(x;t^{\rho}q^{\lambda}|q,t)$ with
$\rho_{i}=n-i$, using a common shorthand notation introduced in the main text.
reduce to the Macdonald polynomials $P_{\lambda}(x_{1},\ldots,x_{n}|q,t)$ in
the special case $s_{i}=t^{n-i}q^{\lambda_{i}}$ $(i=1,\ldots,n)$; however, the
series defining the asymptotically free eigenfunctions are only convergent in
the asymptotically free domain $|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|$ [20]. We
mention in passing that one remarkable property of the functions
$f(x_{1},\ldots,x_{n};s_{1},\ldots,s_{n}|q,t)$ is that they can be represented
by simple explicit formulas [20, Eqs. (1.10)–(1.11)].7)7)7) Note that
$f(x;s|q,t)$ here is denoted as $p_{n}(x;s|q,t)$ in [20].

Our first main result is a construction of elliptic deformations of the
Macdonald polynomials, $\mathcal{P}_{\lambda}(x_{1},\ldots,x_{n};p|q,t)$,
depending on the elliptic deformation parameter $p$ with $p\to 0$
corresponding to the trigonometric limit, such that

$\mathcal{P}_{\lambda}(x_{1},\ldots,x_{n};p|q,t)=P_{\lambda}(x_{1},\ldots,x_{n}|q,t)+\sum_{k=1}^{\infty}p^{k}\,\mathcal{P}_{\lambda,k}(x_{1},\ldots,x_{n}|q,t)$
(1.3)

where $(x_{1}\cdots
x_{n})^{k-\lambda_{n}}\mathcal{P}_{\lambda,k}(x_{1},\ldots,x_{n}|q,t)$ are
symmetric polynomials for all $k=1,2,\ldots$. Moreover, we show that the
series defining these functions is absolutely convergent in a suitable domain
of the variables and parameters, including the physical domain in (1.1) and
real nonnegative $p$. We also show that these eigenfunctions, restricted to
the physical domain, provide a family of orthogonal functions with respect to
the pertinent scalar product. Our second main result is a construction of
asymptotically free eigenfunctions,
$f(x_{1},\ldots,x_{n};s_{1},\ldots,s_{n};p|q,t)$, of the elliptic Ruijsenaars
operators in the asymptotic domain
$|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$ such that

$\begin{split}&f(x_{1},\ldots,x_{n};s_{1},\ldots,s_{n};p|q,t)\\\
&=f(x_{1},\ldots,x_{n};s_{1},\ldots,s_{n}|q,t)+\sum_{k=1}^{\infty}p^{k}f_{k}(x_{1},\ldots,x_{n};s_{1},\ldots,s_{n}|q,t),\end{split}$
(1.4)

where $(x_{n}/x_{1})^{k}f_{k}(x_{1},\ldots,x_{n};s_{1},\ldots,s_{n}|q,t)$ are
convergent power series in the variables $x_{2}/x_{1}$, $\ldots$,
$x_{n}/x_{n-1}$, with coefficients depending rationally on $s_{i}$, for all
$k=1,2,\ldots$. In a similar way to the trigonometric case, they essentially
recover the elliptic deformations
$\mathcal{P}_{\lambda}(x_{1},\ldots,x_{n};p|q,t)$ of Macdonald polynomials as
the special case $s_{i}=t^{n-i}q^{\lambda_{i}}$ ($i=1,\ldots,n$).

In the sense of formal power series in the deformation parameter $p$, the
construction of the eigenfunctions
$\mathcal{P}_{\lambda}(x_{1},\ldots,x_{n};p|q,t)$ can be achieved by the
standard approach of perturbation of eigenvalue problems [10, 12]. It seems
technically demanding, however, to directly prove their convergence in
suitable domains of the variables and parameters. To overcome this difficulty,
we first establish the convergence of asymptotically free eigenfunctions
$f(x_{1},\ldots,x_{n};s_{1},\ldots,s_{n};p|q,t)$ in the asymptotic domain
$|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$ by the method of majorants for their
coefficients which are determined by nonlinear recurrences. Then, by applying
an integral transform to those eigenfunctions in the asymptotic domain
$|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$, we obtain symmetric eigenfunctions
around the physical domain $|x_{i}|=1$ ($i=1,\ldots,n$) which provide the
elliptic deformation of Macdonald polynomials; this is a new type of integral
transform based on a kernel function for the elliptic Ruijsenaars system.

In Section 2, we give a detailed presentation of our results, ending with the
plan (Subsection 2.4) for Sections 3 – 6 where we present the proofs of our
results. In our concluding remarks in Section 7, we summarize our results and
discuss open problems.

###  1.2 Notations

We collect below some notations which will be used in various places of this
paper.

(0) We use the standard notations $\mathbb{Z}$, $\mathbb{R}$, $\mathbb{C}$ for
the sets of integers, real numbers and complex numbers, respectively. We
denote by $\mathbb{N}=\mathbb{Z}_{\geq 0}$ and $\mathbb{Z}_{<0}$ the sets of
nonnegative and negative integers, and by
$\mathbb{C}^{\ast}=\mathbb{C}\backslash\\!\left\\{0\right\\}$ the set of
nonzero complex numbers. We also use the notation
$q^{S}=\left\\{{q^{k}}\big{|}{k\in S}\right\\}$ for a subset
$S\subseteq\mathbb{Z}$.

(1) For a set $x=(x_{1},\ldots,x_{n})$ of $n$ variables, we denote by
$\mathbb{C}[x]$, $\mathbb{C}[x^{\pm 1}]$, $\mathbb{C}\left\\{x\right\\}$ and
$\mathbb{C}[[x]]$ the rings of all polynomials, Laurent polynomials,
convergent power series and formal power series in $x$, respectively. The
symmetric group $\mathfrak{S}_{n}$ of degree $n$ acts on these rings by the
permutation of indices of $x_{i}$ ($i=1,\ldots,n$). If $\mathcal{R}$ is one of
them, we denote by $\mathcal{R}^{\mathfrak{S}_{n}}$ the subring of all
$\mathfrak{S}_{n}$-invariant (symmetric) elements in $\mathcal{R}$. When we
regard the elliptic nome $p$ as an indeterminate, we denote by
$\mathcal{R}[[p]]$ the ring of formal power series in $p$ with coefficients in
a ring $\mathcal{R}$.

(2) For each $r=0,1,\ldots,n$, we denote by

$e_{r}(x)=e_{r}(x_{1},\ldots,x_{n})=\sum_{1\leq i_{1}<\cdots<i_{r}\leq
n}x_{i_{1}}\cdots x_{i_{r}}$ (1.5)

the $r$th elementary symmetric function of $x=(x_{1},\ldots,x_{n})$. We remark
that the ring of symmetric polynomials $\mathbb{C}[x]^{\mathfrak{S}_{n}}$ is
generated by the elementary symmetric functions $e_{1}(x),\ldots,e_{n}(x)$,
and that the ring of symmetric Laurent polynomials $\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}$ is generated by $e_{1}(x),\ldots,e_{n}(x)=x_{1}\cdots
x_{n}$ and $(x_{1}\cdots x_{n})^{-1}$. We use the generating function

$\sum_{r=0}^{n}(-1)^{r}e_{r}(x)u^{r}=(1-x_{1}u)\cdots(1-x_{n}u)$ (1.6)

for the elementary symmetric functions throughout this paper.

(3) Let $\mu=(\mu_{1},\ldots,\mu_{n})\in\mathbb{Z}^{n}$ be an $n$-vector of
integers. For a nonzero complex number $c\in\mathbb{C}^{\ast}$, and an
$n$-vector $a=(a_{1},\ldots,a_{n})\in(\mathbb{C}^{\ast})^{n}$, we define

$c^{\mu}=(c^{\mu_{1}},\ldots,c^{\mu_{n}}),\quad
a^{\mu}=(a_{1}^{\mu_{1}},\ldots,a_{n}^{\mu_{n}})\in(\mathbb{C}^{\ast})^{n}$
(1.7)

respectively. Also, for an $n$-vector
$a=(a_{1},a_{2},\ldots,a_{n})\in\mathbb{C}^{n}$, we denote by
$a^{\vee}=(a_{n},\ldots,a_{2},a_{1})\in\mathbb{C}^{n}$ the reversal of $a$.

##  2 Summary of results

###  2.1 Elliptic Ruijsenaars difference operators

Fixing a base $p\in\mathbb{C}$ with $|p|<1$, throughout this paper we use the
notations

$\theta(z;p)=(z;p)_{\infty}(p/z;p)_{\infty},\quad(z;p)_{\infty}=\prod_{i=0}^{\infty}(1-p^{i}z),$
(2.1)

of an elliptic theta function with elliptic nome $p$ in the multiplicative
variable and the infinite $p$-shifted factorial, respectively. For
$n=1,2,\ldots$, we denote by $\mathcal{D}_{x}(p)=\mathcal{D}_{x}(p|q,t)$ the
elliptic Ruijsenaars $q$-difference operator of first order of type $A_{n-1}$,
in $n$ variables $x=(x_{1},\ldots,x_{n})$ with parameter
$t\in\mathbb{C}^{\ast}=\mathbb{C}\backslash\left\\{0\right\\}$;
$\mathcal{D}_{x}(p)$ is defined by

$\mathcal{D}_{x}(p)=\sum_{i=1}^{n}\prod_{\begin{subarray}{c}1\leq j\leq n\\\
j\neq
i\end{subarray}}\frac{\theta(tx_{i}/x_{j};p)}{\theta(x_{i}/x_{j};p)}\,T_{q,x_{i}},$
(2.2)

where $q\in\mathbb{C}^{\ast}$, $|q|<1$, and for $i=1,\ldots,n$, $T_{q,x_{i}}$
stands for the $q$-shift operator in $x_{i}$ such that
$T_{q,x_{i}}(x_{j})=q^{\delta_{i,j}}x_{j}$ ($j=1,\ldots,n$). We regard
$x=(x_{1},\ldots,x_{n})$ as the canonical coordinates of the direct product
$(\mathbb{C}^{\ast})^{n}$ of $n$ copies of $\mathbb{C}^{\ast}$, and denote by

$\mathbb{T}^{n}=\left\\{{x=(x_{1},\ldots,x_{n})\in(\mathbb{C}^{\ast})^{n}\,}\big{|}{\
|x_{i}|=1\ \ (i=1,\ldots,n)}\right\\}$ (2.3)

the $n$-dimensional torus in $(\mathbb{C}^{\ast})^{n}$.

For each $r=0,1,\ldots,n$, we define a $q$-difference operator
$\mathcal{D}^{(r)}_{x}(p)=\mathcal{D}^{(r)}_{x}(p|q,t)$ of $r$th order by

$\mathcal{D}^{(r)}_{x}(p)=t^{\binom{r}{2}}\sum_{\begin{subarray}{c}I\subseteq\left\\{1,\ldots,n\right\\}\\\
|I|=r\end{subarray}}\,\prod_{\begin{subarray}{c}i\in I\\\ j\notin
I\end{subarray}}\frac{\theta(tx_{i}/x_{j};p)}{\theta(x_{i}/x_{j};p)}\,\prod_{i\in
I}T_{q,x_{i}}$ (2.4)

so that $\mathcal{D}^{(0)}_{x}(p)=1$,
$\mathcal{D}^{(1)}_{x}(p)=\mathcal{D}_{x}(p)$ and
$\mathcal{D}^{(n)}_{x}(p)=t^{\binom{n}{2}}T_{q,x_{1}}\cdots T_{q,x_{n}}$. We
remark that the $q$-difference operators
$\mathcal{D}^{(-r)}_{x}(p)=\mathcal{D}^{(-r)}_{x}(p|q,t)$ of negative orders
defined by

$\mathcal{D}^{(-r)}_{x}(p)=\mathcal{D}^{(n-r)}_{x}(p)\mathcal{D}^{(n)}_{x}(p)^{-1}$
(2.5)

satisfy
$\mathcal{D}^{(-r)}_{x}(p|q,t)=\mathcal{D}^{(r)}_{x}(p|q^{-1},t^{-1})$. It is
known by the pioneering work of Ruijsenaars [21] that all these operators
$\mathcal{D}^{(r)}_{x}(p)$ ($r=\pm 1,\ldots,\pm n$) commute with each other.
It seems, however, that understanding of the eigenfunctions of this elliptic
Ruijsenaars system is rather limited. The purpose of this paper is to
investigate the joint eigenvalue problem

$\mathcal{D}^{(r)}_{x}(p)\psi(x;p)=\varepsilon^{(r)}(p)\psi(x;p)\qquad(r=\pm
1,\ldots,\pm n)$ (2.6)

for this family of commuting difference operators, in which the joint
eigenfunction $\psi(x;p)$ and the eigenvalues $\varepsilon^{(r)}(p)$ are
expressible as convergent power series in $p$:

$\psi(x;p)=\sum_{k=0}^{\infty}\,p^{k}\psi_{k}(x),\quad\varepsilon^{(r)}(p)=\sum_{k=0}^{\infty}\,p^{k}\varepsilon^{(r)}_{k}\quad(r=1,\ldots,n).$
(2.7)

We propose in particular two kinds of joint eigenfunctions8)8)8) Any joint
eigenfunction of the operators $\mathcal{D}^{(r)}_{x}(p)$ ($r=1,\ldots,n$) is
necessarily an eigenfunction of $\mathcal{D}^{(r)}_{x}(p)$ for
$r=-1,\ldots,-n$ as well. For this reason, we will mainly consider the
elliptic Ruijsenaars operators $\mathcal{D}^{(r)}_{x}(p)$ ($r=1,\ldots,n$) of
positive orders in the construction of joint eigenfunctions. :

(1) Symmetric joint eigenfunctions around the torus $\mathbb{T}^{n}$, and

(2) Asymptotically free joint eigenfunctions in the domain
$|x_{1}|\gg|x_{2}|\gg\cdots\gg|px_{1}/x_{n}|$.

The joint eigenfunctions of the first kind can be thought of as elliptic
deformations of the Macdonald polynomials ([16, 17]), and those of the second
kinds as elliptic deformations of the asymptotically free solutions of the
trigonometric case in the domain $|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|$ ([25],
[1], [18], [20]).

###  2.2 Elliptic deformation of Macdonald polynomials

In the limit as $p\to 0$, the operators $D^{(x)}_{x}=\mathcal{D}_{x}^{(r)}(0)$
appear as

$D_{x}^{(r)}=t^{\binom{r}{2}}\sum_{\begin{subarray}{c}I\subseteq\left\\{1,\ldots,n\right\\}\\\
|I|=r\end{subarray}}\prod_{\begin{subarray}{c}i\in I\\\ j\notin
I\end{subarray}}\frac{1-tx_{i}/x_{j}}{1-x_{i}/x_{j}}\prod_{i\in
I}T_{q,x_{i}}\qquad(r=0,1,\ldots,n),$ (2.8)

which we call the Macdonald-Ruijsenaars operators. They define a commuting
family of $\mathbb{C}$-linear operators $D_{x}^{(r)}:\
\mathbb{C}[x]^{\mathfrak{S}_{n}}\to\mathbb{C}[x]^{\mathfrak{S}_{n}}$ on the
ring of symmetric polynomials in $x=(x_{1},\ldots,x_{n})$. If the parameter
$t\in\mathbb{C}^{\ast}$ is generic, they are simultaneously diagonalized by
the Macdonald polynomials $P_{\lambda}(x)=P_{\lambda}(x|q,t)$ attached to
partitions $\lambda$ with number of positive parts $\ell(\lambda)\leq n$
([16]). In considering the elliptic deformations of Macdonald polynomials,
however, we need to consider the operators $D_{x}^{(r)}:\ \mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}\to\mathbb{C}[x^{\pm 1}]^{\mathfrak{S}_{n}}$ acting on
the ring of symmetric Laurent polynomials as well.

We say that an $n$-vector
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Z}^{n}$ of integers is
dominant if $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$. This
condition is equivalent to saying that it is expressed as
$\lambda=\mu+(k^{n})$, $(k^{n})=(k,\ldots,k)$, by a partition $\mu$ with
$\ell(\mu)\leq n$ and an integer $k\in\mathbb{Z}$. We extend the notation
$P_{\lambda}(x)$ of Macdonald polynomials to symmetric Laurent polynomials
attached to the dominant vectors $\lambda$ in $\mathbb{Z}^{n}$ by the rule

$P_{\lambda}(x)=P_{\mu+(k^{n})}(x)=(x_{1}\cdots
x_{n})^{k}P_{\mu}(x)\quad(k\in\mathbb{Z}).$ (2.9)

Then, the Macdonald-Ruijsenaars operators $D^{(r)}_{x}$ acting on
$\mathbb{C}[x^{\pm 1}]^{\mathfrak{S}_{n}}$ are simultaneously diagonalized by
them.

In the following, to each $n$-vector
$\mu=(\mu_{1},\ldots,\mu_{n})\in\mathbb{Z}^{n}$ of integers we attach the
monomial $x^{\mu}=x_{1}^{\mu_{1}}\cdots x^{\mu_{n}}$ of $x$ variables, and
denote by $|\mu|=\mu_{1}+\cdots+\mu_{n}$ the degree of $x^{\mu}$. For each
dominant vector $\lambda$ in $\mathbb{Z}^{n}$, we denote by

$m_{\lambda}(x)=\sum_{\mu\in\mathfrak{S}_{n}.\lambda}x^{\mu}$ (2.10)

the monomial symmetric function of type $\lambda$, i.e. the sum of all
monomials $x^{\mu}$ that are obtained from $x^{\lambda}$ by permuting the
variables. Then we have

$\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}=\bigoplus_{\begin{subarray}{c}\lambda\in\mathbb{Z}^{n}:\,\mbox{\scriptsize
dominant}\end{subarray}}\,\mathbb{C}\,m_{\lambda}(x).$ (2.11)

Note that $m_{\lambda+(k^{n})}(x)=(x_{1}\cdots x_{n})^{k}m_{\lambda}(x)$ for
any $k\in\mathbb{Z}$. We define the dominance order $\mu\leq\nu$ on
$\mathbb{Z}^{n}$ by the condition

$|\mu|=|\nu|\quad\mbox{and}\quad\mu_{1}+\cdots+\mu_{i}\leq\nu_{1}+\cdots+\nu_{i}\quad(i=1,\ldots,n-1).$
(2.12)

Also, for each dominant vector $\lambda$ in $\mathbb{Z}^{n}$, we denote by
$\mathbb{C}[x^{\pm 1}]^{\mathfrak{S}_{n}}_{\leq\lambda}$ the vector subspace
of all symmetric Laurent polynomials of the form

$\varphi(x)=\sum_{\mu\leq\lambda}\varphi_{\mu}m_{\mu}(x),$ (2.13)

where the sum is over all dominant vectors $\mu$ in $\mathbb{Z}^{n}$ such that
$\mu\leq\lambda$. Then the existence theorem of Macdonald (Laurent)
polynomials can be formulated as follows.

Suppose that the parameter $t\in\mathbb{C}^{\ast}$ satisfies the condition
$t^{k}\notin q^{\mathbb{Z}_{<0}}$ ($k=1,\ldots,n-1$). Then, for each dominant
vector $\lambda$ in $\mathbb{Z}^{n}$, there exists a unique symmetric Laurent
polynomial $P_{\lambda}(x)=P_{\lambda}(x|q,t)\in\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}$ such that $P_{\lambda}(x)$ is expressed in the form

$P_{\lambda}(x)=m_{\lambda}(x)+\sum_{\mu<\lambda}P_{\lambda,\mu}\,m_{\mu}(x)$
(2.14)

with leading term $m_{\lambda}(x)$ with respect the dominance order, and
satisfies the joint eigenfunction equations

$D_{x}^{(r)}P_{\lambda}(x)=e_{r}(t^{\rho}q^{\lambda})P_{\lambda}(x)\qquad(r=1,\ldots,n);$
(2.15)

the eigenvalues $e_{r}(t^{\rho}q^{\lambda})$ are elementary symmetric
functions of
$t^{\rho}q^{\lambda}=(t^{n-1}q^{\lambda_{1}},t^{n-2}q^{\lambda_{2}},\ldots,q^{\lambda_{n}})$.
We use hereafter the notation $\rho=(n-1,n-2,\ldots,0)$ for the staircase
partition, instead of $\delta$ in Macdonald’s monograph [16], reserving the
symbol $\delta$ for other uses.

Returning to the elliptic Ruijsenaars operators, we expand each $q$-difference
operator $\mathcal{D}^{(r)}_{x}(p)$ as a power series of $p$ in the form

$\mathcal{D}^{(r)}_{x}(p)=\sum_{k=0}^{\infty}p^{k}\mathcal{D}^{(r)}_{x,k}.$
(2.16)

###### Proposition 2.1

For each $r=1,\ldots,n$, the coefficients $\mathcal{D}^{(r)}_{x,k}$
$(k=0,1,2,\ldots)$ are $\mathfrak{S}_{n}$-invariant $q$-difference operators,
and all stabilize the ring $\mathbb{C}[x^{\pm 1}]^{\mathfrak{S}_{n}}$ of
symmetric Laurent polynomials. Furthermore, they have triangularity property

$\mathcal{D}^{(r)}_{x,k}\big{(}\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}_{\leq\mu}\big{)}\subseteq\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}_{\leq\mu+k\phi}$ (2.17)

for each dominant vector $\mu$ in $\mathbb{Z}^{n}$, where
$\phi=(1,0,\ldots,0,-1)$.

By Proposition 2.1, the elliptic Ruijsenaars operators define a commuting
family of linear operators

$\mathcal{D}^{(r)}_{x}(p):\ \mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}[[p]]\to\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}[[p]]\quad(r=1,\ldots,n)$ (2.18)

acting on the ring of formal power series in $p$ with coefficients in
$\mathbb{C}[x^{\pm 1}]^{\mathfrak{S}_{n}}$. We first investigate formal
solutions of the joint eigenfunction equations

$\mathcal{D}^{(r)}_{x}(p)\psi(x;p)=\varepsilon^{(r)}(p)\psi(x;p)\quad(r=1,\ldots,n)$
(2.19)

for $\psi(x;p)\in\mathbb{C}[x^{\pm 1}]^{\mathfrak{S}_{n}}[[p]]$ and
$\varepsilon^{(r)}(p)\in\mathbb{C}[[p]]$.

###### Theorem 2.2

Suppose that $t^{k}\notin q^{\mathbb{Z}_{<0}}$ $(k=1,\ldots,n-1)$. Then, for
each dominant vector $\lambda$ in $\mathbb{Z}^{n}$, the initial value problem
$\psi(x;0)=P_{\lambda}(x)$ of the joint eigenfunction equations (2.19) has a
formal solution $\mathcal{P}_{\lambda}(x;p)\in\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}[[p]]$ with uniquely determined formal eigenvalues
$\varepsilon^{(r)}_{\lambda}(p)\in\mathbb{C}[[p]]$ $(r=1,\ldots,n)$. The
formal solution $\mathcal{P}_{\lambda}(x;p)$ is determined uniquely up to
multiplication by a power series $\gamma(p)\in\mathbb{C}[[p]]$ such that
$\gamma(0)=1$.

Proposition 2.1 also implies that the formal joint eigenfunction
$\mathcal{P}_{\lambda}(x;p)$ has $p$-expansion of the form

$\mathcal{P}_{\lambda}(x;p)=\sum_{k=0}^{\infty}p^{k}\mathcal{P}_{\lambda,k}(x),\quad\mathcal{P}_{\lambda,k}(x)\in\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}_{\leq\lambda+k\phi},$ (2.20)

with $\mathcal{P}_{\lambda,0}(x)=P_{\lambda}(x)$. If $\lambda$ is a partition
with $\ell(\lambda)\leq n$, in particular, we have
$\mathcal{P}_{\lambda,k}(x)\in(x_{1}\cdots
x_{n})^{-k}\mathbb{C}[x]^{\mathfrak{S}_{n}}$ for $k=0,1,2,\ldots$.

We say that the formal solution $\mathcal{P}_{\lambda}(x;p)$ is normalized if
the coefficient of $m_{\lambda}(x)$ in $\mathcal{P}_{\lambda,k}(x)$ is 0 for
$k>0$, i.e.

$\mathcal{P}_{\lambda,k}(x)=\sum_{\mu\leq\lambda+k\phi}\mathcal{P}_{\lambda,k;\mu}\,m_{\mu}(x),\quad\mathcal{P}_{\lambda,k;\lambda}=\delta_{k,0}\quad(k=0,1,2,\ldots).$
(2.21)

This condition is equivalent to saying that
$x^{-\lambda}\mathcal{P}_{\lambda}(x;p)$ has constant term 1 with respect to
the $x$ variables. Hence, there exists a unique normalized formal solution
$\mathcal{P}_{\lambda}(x;p)$ to the initial value problem of Theorem 2.2, and
the other solutions are expressed as $\gamma(p)\mathcal{P}_{\lambda}(x;p)$
with some $\gamma(p)\in\mathbb{C}[[p]]$ with $\gamma(0)=1$. Note also that any
nonzero formal solution $\varphi(x;p)$ of the joint eigenfunction equations
(2.19) is expressed as $\varphi(x;p)=\gamma(p)\mathcal{P}_{\lambda}(x;p)$ for
some dominant vector $\lambda$ in $\mathbb{Z}^{n}$ and
$\gamma(p)\in\mathbb{C}[[p]]$.

We remark that, for each $k=0,1,2,\ldots$, the coefficient of the leading term
$m_{\lambda+k\phi}(x)$ in $\mathcal{P}_{\lambda,k}(x)$ of (2.21) is explicitly
computed as

$\mathcal{P}_{\lambda,k;\lambda+k\phi}=\frac{(t;q)_{k}(t^{n}q^{\lambda_{1}-\lambda_{n}};q)_{k}}{(q;q)_{k}(t^{n-1}q^{\lambda_{1}-\lambda_{n}+1};q)_{k}}(q/t)^{k}\quad(k=0,1,2,\ldots),$
(2.22)

where $(a;q)_{k}=(1-a)(1-qa)\cdots(1-q^{k-1}a)$. It should be noted that the
“groundstate” $\mathcal{P}_{0}(x;p)$ determined by the initial condition
$\mathcal{P}_{0}(x;0)=1$ is already a nontrivial power series in $x$.9)9)9)
Here, the subcript $0$ stands for the empty partition.  We also remark that
the normalized formal solutions $\mathcal{P}_{\lambda}(x;p)$ satisfy the
property

$\mathcal{P}_{\lambda+(k^{n})}(x;p)=(x_{1}\cdots
x_{n})^{k}\mathcal{P}_{\lambda}(x;p)\quad(k\in\mathbb{Z}),$ (2.23)

similarly to the case of Macdonald polynomials. In particular,
$\mathcal{P}_{\lambda}(x;p)$ for general dominant vectors $\lambda$ in
$\mathbb{Z}^{n}$ are determined from those attached to partitions $\lambda$
with $\ell(\lambda)\leq n$.

As to the convergence of the formal joint eigenfunctions, we propose the
following theorem.

###### Theorem 2.3

Suppose that the parameter $t\in\mathbb{C}^{\ast}$ satisfies the condition
$t^{k}\notin q^{\mathbb{Z}}$ $(k=1,\ldots,n-1)$ and let $\lambda$ be a
dominant vector in $\mathbb{Z}^{n}$. Then, there exists a constant
$\tau\in(0,1]$, not depending on $p$, such that the normalized formal joint
eigenfunction $\mathcal{P}_{\lambda}(x;p)$ is absolutely convergent in the
domain

$|p|/\tau<|x_{j}/x_{i}|<\tau/|p|\qquad(1\leq i<j\leq n),$ (2.24)

for $|p|<\tau$ ; the formal eigenvalues $\varepsilon^{(r)}_{\lambda}(p)$
$(r=1,\ldots,n)$ are also absolutely convergent for $|p|<\tau$.

Note that, if $t^{k}\neq 1$ ($k=1,\ldots,n-1$) and
$|q|<\min\left\\{|t|^{n-1},|t|^{-n+1}\right\\}$, then the condition
$t^{k}\notin q^{\mathbb{Z}}$ ($k=1,\ldots,n-1$) is automatically satisfied.

We remark that the symmetric joint eigenfunctions $\mathcal{P}_{\lambda}(x;p)$
of Theorem 2.3 are holomorphic around the torus $\mathbb{T}^{n}$; they can be
thought of as elliptic deformation of the Macdonald polynomials. In fact, they
satisfy the joint eigenfunction equations

$\mathcal{D}^{(r)}_{x}(p)\mathcal{P}_{\lambda}(x;p)=\varepsilon^{(r)}_{\lambda}(p)\mathcal{P}_{\lambda}(x;p)\quad(r=1,\ldots,n)$
(2.25)

for the elliptic Ruijsenaars operators, with initial condition
$\mathcal{P}_{\lambda}(x;0)=P_{\lambda}(x)$,
$\varepsilon^{(r)}_{\lambda}(0)=e_{r}(t^{\rho}q^{\lambda})$, in the domain
$|p|/|q|\tau<|x_{j}/x_{i}|<|q|\tau/|p|$ ($1\leq i<j\leq n$) for $|p|<|q|\tau$.
We expect that $\mathcal{P}_{\lambda}(x;p)$ could be continued to global
meromorphic functions on $(\mathbb{C}^{\ast})^{n}$, by means of the
$q$-difference equations (2.25), but we do not have a proof of this conjecture
yet.

With this existence theorem established, we can prove the orthogonality
relations for the normalized joint eigenfunctions
$\mathcal{P}_{\lambda}(x;p)$. We denote by $\Gamma(z;p,q)$ the Ruijsenaars
elliptic gamma function with bases $p,q$:

$\Gamma(z;p,q)=\frac{(pq/z;p,q)_{\infty}}{(z;p,q)_{\infty}},\quad(z;p,q)_{\infty}=\prod_{i,j=0}^{\infty}(1-p^{i}q^{j}z).$
(2.26)

We define a meromorphic function $w^{\mathrm{sym}}(x;p)$ on
$(\mathbb{C}^{\ast})^{n}$ by

$w^{\mathrm{sym}}(x;p)=\prod_{1\leq i<j\leq
n}\frac{\Gamma(tx_{i}/x_{j};p,q)\Gamma(tx_{j}/x_{i};p,q)}{\Gamma(x_{i}/x_{j};p,q)\Gamma(x_{j}/x_{i};p,q)}.$
(2.27)

We also denote by $d\omega_{n}(x)$ the normalized invariant measure on
$\mathbb{T}^{n}$:

$d\omega_{n}(x)=\frac{1}{(2\pi\sqrt{-1})^{n}}\frac{dx_{1}\cdots
dx_{n}}{x_{1}\cdots x_{n}},\qquad\int_{\mathbb{T}^{n}}d\omega_{n}(x)=1.$
(2.28)

###### Theorem 2.4

Suppose that $|t|<1$ and $t^{k}\notin q^{\mathbb{Z}_{>0}}$ $(k=1,\ldots,n-1)$.
Then, the normalized joint eigenfunctions $\mathcal{P}_{\lambda}(x;p)$ of the
elliptic Ruijsenaars operators, attached to the dominant vectors $\lambda$ in
$\mathbb{Z}^{n}$, are orthogonal with respect to the weight function
$w^{\mathrm{sym}}(x;p)$ of (2.27) in the sense that

$\int_{\mathbb{T}^{n}}\mathcal{P}_{\lambda}(x^{-1};p)\mathcal{P}_{\mu}(x;p)w^{\mathrm{sym}}(x;p)d\omega_{n}(x)=\delta_{\lambda,\mu}\,\mathcal{N}_{\lambda}(p)$
(2.29)

with some constants $\mathcal{N}_{\lambda}(p)$ depending on $p$.

(We expect that the assumption $t^{k}\in q^{\mathbb{Z}_{>0}}$
($k=1,\ldots,n-1$) could be removed.)

We remark that the integral (2.29) for $p=0$ is Macdonald’s scalar product
$\left\langle\ ,\ \right\rangle^{\prime}$ in [16] up to a factor $n!$. It
would be an intriguing problem to evaluate the norm $\mathcal{N}_{\lambda}(p)$
in (2.29), although it seems challenging at present. Notice here that the
weight function $w^{\mathrm{sym}}(x;p)=w^{\mathrm{sym}}(x;p,q,t)$ is invariant
by exchanging $p$ and $q$. This suggests that the orthogonal functions
$\mathcal{P}_{\lambda}(x;p)=\mathcal{P}_{\lambda}(x;p|q,t)$ should also be
symmetric with respect to $(p,q)$.

###  2.3 Asymptotically free eigenfunctions in the domain
$|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$

Denoting by

$\Delta(x)=\prod_{1\leq i<j\leq n}(x_{i}-x_{j})=x^{\rho}\prod_{1\leq i<j\leq
n}(1-x_{j}/x_{i})$ (2.30)

the difference product of $x$ variables, we write the Macdonald-Ruijsenaars
operators in the form

$D^{(r)}_{x}=\sum_{I\subseteq\left\\{1,\ldots,n\right\\};\
|I|=r}A_{I}(x)T_{q,x}^{I},\quad
A_{I}(x)=\frac{T_{t,x}^{I}(\Delta(x))}{\Delta(x)},$ (2.31)

where $T_{q,x}^{I}=\prod_{i\in I}T_{q,x_{i}}$. Then, from the expression

$A_{I}(x)=(t^{\rho})^{I}B_{I}(x),\quad
B_{I}(x)=\prod_{\begin{subarray}{c}1\leq i<j\leq n\\\ i\in I,\,j\notin
I\end{subarray}}\frac{1-x_{j}/tx_{i}}{1-x_{j}/x_{i}}\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\ i\notin I,\,j\in
I\end{subarray}}\frac{1-tx_{j}/x_{i}}{1-x_{j}/x_{i}}$ (2.32)

with $(t^{\rho})^{I}=\prod_{i\in I}t^{n-i}$, it follows that the coefficients
$A_{I}(x)$ are holomorphic in the domain
$|x_{2}/x_{1}|<1,\ldots,|x_{n}/x_{n-1}|<1$, and they are represented there by
convergent power series of the variables $x_{2}/x_{1},\ldots,x_{n}/x_{n-1}$.
Hence we see that, for each complex vector
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{C}^{n}$, the Macdonald-
Ruijsenaars operators define a commuting family of linear operators

$D^{(r)}_{x}:\
x^{\lambda}\mathbb{C}\left\\{x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}\to
x^{\lambda}\mathbb{C}\left\\{x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}\quad(r=1,\ldots,n)$
(2.33)

on the space of convergent power series of the form

$\psi(x)=x_{1}^{\lambda_{1}}\cdots
x_{n}^{\lambda_{n}}\sum_{k_{1},\ldots,k_{n-1}\in\mathbb{Z}_{\geq
0}}c_{k_{1},\ldots,k_{n-1}}(x_{2}/x_{1})^{k_{1}}\cdots(x_{n}/x_{n-1})^{k_{n-1}}.$
(2.34)

Regarding the joint eigenfunctions in this setting, it is known that, for each
complex vector $\lambda\in\mathbb{C}^{n}$ such that
$t^{i-j}q^{\lambda_{j}-\lambda_{i}}\notin q^{\mathbb{Z}}$ $(1\leq i<j\leq n)$,
the joint eigenvalue problem

$D^{(r)}_{x}\psi_{\lambda}(x)=e_{r}(t^{\rho}q^{\lambda})\psi_{\lambda}(x)\quad(r=1,\ldots,n)$
(2.35)

has a unique solution $\psi_{\lambda}(x)\in
x^{\lambda}\mathbb{C}\left\\{x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}$ with
leading term $x^{\lambda}$. Such a solution is called asymptotically free in
the domain $|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|$ (see Shiraishi [25],
Cherednik [1], van Meer-Stockman [18], Noumi-Shiraishi [20]).

It is also known that there exists a meromorphic function $f(x;s)$ in the
position variables $x=(x_{1},\ldots,x_{n})$ and the spectral variables
$s=(s_{1},\ldots,s_{n})$ such that
$\psi_{\lambda}(x)=x^{\lambda}f(x;t^{\rho}q^{\lambda})$ for generic
$\lambda\in\mathbb{C}^{n}$ under the identification
$s_{i}=t^{n-i}q^{\lambda_{i}}$ ($i=1,\ldots,n$). We remark that the
conjugation of the Macdonald-Ruijsenaars operators $D^{(r)}_{x}$ by
$x^{\lambda}$ are expressed as

$x^{-\lambda}D_{x}^{(r)}x^{\lambda}=\sum_{I\subseteq\left\\{1,\ldots,n\right\\};\
|I|=r}(t^{\rho}q^{\lambda})^{I}B_{I}(x)T_{q,x}^{I},$ (2.36)

where $(t^{\rho}q^{\lambda})^{I}=\prod_{i\in I}t^{n-i}q^{\lambda_{i}}$. In
view of this expression, replacing $(t^{\rho}q^{\lambda})^{I}$ by
$s^{I}=\prod_{i\in I}s_{i}$, we introduce the modified Macdonald-Ruijsenaars
operators $E^{(r)}_{x,s}$ with parameters $s=(s_{1},\ldots,s_{n})$ by

$E^{(r)}_{x,s}=\sum_{I\subseteq\left\\{1,\ldots,n\right\\};\
|I|=r}s^{I}B_{I}(x)T_{q,x}^{I}\qquad(r=0,1,\ldots,n),$ (2.37)

so that
$E^{(r)}_{x,\,t^{\rho}q^{\lambda}}=x^{-\lambda}D^{(r)}_{x}x^{\lambda}$. Then,
as a function of the $x$ variables, $f(x;s)$ mentioned above is characterized
as the unique solution in
$\mathbb{C}\left\\{x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}$ with leading
term 1 of the joint eigenfunction equations

$E^{(r)}_{x,s}f(x;s)=e_{r}(s)f(x;s)\qquad(r=1,\ldots,n).$ (2.38)

We remark that the unique existence of $f(x;s)$ is guaranteed under the
condition $s_{j}/s_{i}\notin q^{\mathbb{Z}}$ ($1\leq i<j\leq n$) [20]. We call
this $f(x;s)=f(x;s|q,t)$ the Macdonald function in the asymptotic domain
$|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|$. It is known that $f(x;s|q,t)$ can be
continued to a meromorphic function on
$(\mathbb{C}^{\ast})^{n}\times(\mathbb{C}^{\ast})^{n}$ with pole divisors
explicitly described, and it has various remarkable properties comparable to
those of Macdonald polynomials, such as combinatorial formula, bispectral
self-duality and transformation under the reflection $t\leftrightarrow q/t$ of
the parameter $t$ (see [20]; $f(x;s|q,t)$ is denoted there by
$p_{n}(x;s|q,t)$).

In the elliptic case, the elliptic Ruijsenaars operators are expressed as

$\mathcal{D}^{(r)}_{x}(p)=\sum_{I\subseteq\left\\{1,\ldots,n\right\\};\
|I|=r}\mathcal{A}_{I}(x;p)T_{q,x}^{I},\quad\mathcal{A}_{I}(x;p)=\frac{T_{t,x}^{I}(\Delta(x;p))}{\Delta(x;p)},$
(2.39)

in terms of the elliptic deformation $\Delta(x;p)$ of $\Delta(x)$ defined by

$\Delta(x;p)=\prod_{1\leq i<j\leq
n}x_{i}\theta(x_{j}/x_{i};p)=x^{\rho}\prod_{1\leq i<j\leq
n}\theta(x_{j}/x_{i};p).$ (2.40)

From the expression

$\mathcal{A}_{I}(x;p)=(t^{\rho})^{I}\mathcal{B}_{I}(x;p),\quad\mathcal{B}_{I}(x;p)=\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\ i\in I,\,j\notin
I\end{subarray}}\frac{\theta(x_{j}/tx_{i};p)}{\theta(x_{j}/x_{i};p)}\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\ i\notin I,\,j\in
I\end{subarray}}\frac{\theta(tx_{j}/x_{i};p)}{\theta(x_{j}/x_{i};p)}$ (2.41)

we have

###### Proposition 2.5

The coefficients $\mathcal{A}_{I}(x;p)$ of the elliptic Ruijsenaars operators
are holomorphic in the domain
$|px_{1}/x_{n}|<1,|x_{2}/x_{1}|<1,\ldots,|x_{n}/x_{n-1}|<1$ and, in
particular, they can be represented as convergent power series in the
variables $z_{0}=px_{1}/x_{n},z_{1}=x_{2}/x_{1},\ldots,z_{n-1}=x_{n}/x_{n-1}$.

Regarding the coefficients $\mathcal{A}_{I}(x;p)$ as convergent power series
in

$\mathbb{C}\left\\{px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\},$
(2.42)

we consider the commuting family of elliptic Ruijsenaars operators

$\mathcal{D}^{(r)}_{x}(p):\
x^{\lambda}\mathbb{C}\left\\{px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}\to
x^{\lambda}\mathbb{C}\left\\{px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}$
(2.43)

($r=1,\ldots,n$) for each complex vector $\lambda\in\mathbb{C}^{n}$. As we
will see below, if $\lambda\in\mathbb{C}^{n}$ is generic in the sense that
$t^{i-j}q^{\lambda_{j}-\lambda_{i}}\notin q^{\mathbb{Z}}$ ($1\leq i<j\leq n$),
these linear operators have a joint eigenfunction $\psi_{\lambda}(x;p)$ in
$x^{\lambda}\mathbb{C}\left\\{px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}$
with leading term $x^{\lambda}$, determined uniquely up to multiplication by a
power series $\gamma(p)\in\mathbb{C}\left\\{p\right\\}$ with $\gamma(0)=1$. We
call such an eigenfunction $\psi_{\lambda}(x;p)$ asymptotically free in the
domain $|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$.

Similarly to the trigonometric case explained above, we define the modified
elliptic Ruijsenaars operators $\mathcal{E}_{x,s}^{(r)}(p)$ ($r=0,1,\ldots,n$)
by

$\mathcal{E}_{x,s}^{(r)}(p)=\sum_{I\subseteq\left\\{1,\ldots,n\right\\};\
|I|=r}\,s^{I}\mathcal{B}_{I}(x;p)T_{q,x}^{I}$ (2.44)

so that $\mathcal{D}^{(r)}_{x}(p)=\mathcal{E}^{(r)}_{x,t^{\rho}}(p)$ and
$x^{-\lambda}\mathcal{D}^{(r)}_{x}(p)x^{\lambda}=\mathcal{E}^{(r)}_{x,t^{\rho}q^{\lambda}}(p)$.
Regarding $s=(s_{1},\ldots,s_{n})\in(\mathbb{C}^{\ast})^{n}$ as parameters, we
investigate the joint eigenvalue problem of the modified elliptic Ruijsenaars
operators

$\mathcal{E}^{(r)}_{x,s}(p)f(x;s;p)=\varepsilon^{(r)}(s;p)f(x;s;p)\quad(r=1,\ldots,n)$
(2.45)

for
$f(x;s;p)\in\mathbb{C}\left\\{px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}$
and $\varepsilon^{(r)}(s;p)\in\mathbb{C}\left\\{p\right\\}$. We first
establish an existence theorem for formal solutions such that
$f(x;s;p)\in\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]]$ and
$\varepsilon^{(r)}(s;p)\in\mathbb{C}[[p]]$, and then propose a convergence
theorem for them. We remark that the ring

$\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]]$ (2.46)

of formal power series in $(px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1})$
can be defined alternatively as the ring of all power series in $p$ of the
form

$f(x;p)=\sum_{k=0}^{\infty}p^{k}f_{k}(x),\quad
f_{k}(x)\in(x_{1}/x_{n})^{k}\mathbb{C}[[x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]].$
(2.47)

###### Theorem 2.6

Suppose that the parameters $s=(s_{1},\ldots,s_{n})\in(\mathbb{C}^{\ast})^{n}$
satisfy the condition $s_{j}/s_{i}\notin q^{\mathbb{Z}^{\ast}}$ $(1\leq
i<j\leq n)$, where
$\mathbb{Z}^{\ast}=\mathbb{Z}\backslash\\!\left\\{0\right\\}$. Then the joint
eigenvalue problem (2.45) for the modified elliptic Ruijsenaars operators
$\mathcal{E}^{(r)}_{x,s}(p)$ $(r=1,\ldots,n)$ has a formal solution
$f(x;s;p)\in\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]]$ with
uniquely determined formal eigenvalues
$\varepsilon^{(r)}(s;p)\in\mathbb{C}[[p]]$ $(r=1,\ldots,n)$. The solution
$f(x;s;p)$ is determined uniquely up to multiplication by a power series
$\gamma(s;p)\in\mathbb{C}[[p]]$ such that $\gamma(s;0)=1$.

We say that the solution $f(x;s;p)$ is normalized if it has constant term 1
with respect to $x=(x_{1},\ldots,x_{n})$. In terms of the expansion

$f(x;s;p)=\sum_{k_{0},k_{1},\ldots,k_{n-1}}f_{k_{0},k_{1},\ldots,k_{n-1}}(s)(px_{1}/x_{n})^{k_{0}}(x_{2}/x_{1})^{k_{1}}\cdots(x_{n}/x_{n-1})^{k_{n-1}}$
(2.48)

as a formal power series in $(px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1})$,
the constant term of $f(x;s;p)$ with respect to $x=(x_{1},\ldots,x_{n})$ is
given by

$\sum_{k=0}^{\infty}p^{k}f_{k,k,\ldots,k}(s)\in\mathbb{C}[[p]].$ (2.49)

Hence, the normalization condition for $f(x;s;p)$ mentioned above is
equivalent to

$f_{k,k,\ldots,k}(s)=\delta_{k,0}\qquad(k=0,1,2,\ldots).$ (2.50)

We remark that the joint eigenvalue problem (2.45) has a unique normalized
solution $f(x;s;p)$ in
$\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]]$, and the other
solutions are expressed as $\gamma(s;p)f(x;s;p)$ for some
$\gamma(s;p)\in\mathbb{C}[[p]]$.

If we specialize the $s$ variables by $s=t^{\rho}q^{\lambda}$ with a complex
vector $\lambda\in\mathbb{C}^{n}$, we have
$s_{j}/s_{i}=t^{i-j}q^{\lambda_{j}-\lambda_{i}}$. This means that, if
$t^{i-j}q^{\lambda_{j}-\lambda_{i}}\notin q^{\mathbb{Z}^{\ast}}$ ($1\leq
i<j\leq n$), $\psi_{\lambda}(x;p)=x^{\lambda}f(x;t^{\rho}q^{\lambda};p)$ gives
the formal asymptotically free solution of the joint eigenvalue problem of the
elliptic Ruijsenaars operators, with formal eigenvalues
$\varepsilon^{(r)}(t^{\rho}q^{\lambda};p)$, in the domain
$|x_{1}|\gg\dots\gg|x_{n}|\gg|px_{1}|$. Also, if $\lambda$ is a dominant
vector in $\mathbb{Z}^{n}$, and if $t^{k}\notin q^{\mathbb{Z}}$
($k=1,\ldots,n-1$), the normalized formal solution of Theorem 2.2 is recovered
as $\mathcal{P}_{\lambda}(x;p)=x^{\lambda}f(x;t^{\rho}q^{\lambda};p)$
consistently with the normalization condition.

As to the convergence of the formal solution $f(x;s;p)$ we have

###### Theorem 2.7

Suppose that $s_{j}/s_{i}\notin q^{\mathbb{Z}}$ $(1\leq i<j\leq n)$. Then,
there exist a constant $\sigma\in(0,1)$, not depending on $p$, such that the
normalized formal solution $f(x;s;p)$ of the joint eigenvalue problem (2.45)
is absolutely convergent in the domain

$|px_{1}/x_{n}|<\sigma,\ |x_{2}/x_{1}|<\sigma,\ \ldots,|x_{n}/x_{n-1}|<\sigma$
(2.51)

for $|p|<\sigma^{n}$.

Note that the joint eigenfunction equations (2.45) hold in the domain

$|px_{1}/x_{n}|<|q|\sigma,\ |x_{2}/x_{1}|<|q|\sigma,\
\ldots,|x_{n}/x_{n-1}|<|q|\sigma$ (2.52)

for $|p|<|q|^{n}\sigma$. We also remark that $f(x;s;p)=f(x;s;p|q,t)$ depend
holomorphically on $s=(s_{1},\ldots,s_{n})\in(\mathbb{C}^{\ast})^{n}$, and
$q\in\mathbb{C}^{\ast}$ with $|q|<1$ and $t\in\mathbb{C}^{\ast}$, provided
that the condition $1-q^{l}s_{j}/s_{i}\neq 0$ $(l\in\mathbb{Z})$ is satisfied
for any distinct $i,j\in\left\\{1,\ldots,n\right\\}$. More specifically,
$s_{i}\neq s_{j}$ for any distinct $i,j\in\left\\{1,\ldots,n\right\\}$ and
that $\tau_{0}\leq|s_{j}/s_{i}|\leq\tau_{0}^{-1}$ $(1\leq i<j\leq n)$ for some
$\tau_{0}\in(0,1]$, then $f(x;s;p|q,t)$ is a holomorphic function of $(q,t)$
in the domain $\left\\{{q\in\mathbb{C}}\big{|}{\
|q|<\tau_{0}}\right\\}\times\mathbb{C}^{\ast}$.

We refer to this normalized joint eigenfunction $f(x;s;p)=f(x;s;p|q,t)$ as the
Ruijsenaars function in the domain
$|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$. In the context of [26] and
[15], $f(x;s;p|q,t)$ is the stationary Ruijsenaars function, which is
conjectured to arise through a limiting procedure, as $\kappa\to 1$, from the
non-stationary Ruijsenaars function involving an extra parameter $\kappa$. We
expect that the stationary and non-stationary Ruijsenaars functions should
inherit various symmetry properties of the Macdonald function in the
trigonometric case.

We give below two results regarding symmetries of the Ruijsenaars function.

###### Theorem 2.8

The normalized Ruijsenaars function $f(x;s;p)=f(x;s;p|q,t)$ is invariant under
the simultaneous rotation of indices of the $x$ and $s$ variables in the
following sense $:$

$f(x_{2},\ldots,x_{n},px_{1};s_{2},\ldots,s_{n},s_{1};p)=f(x_{1},\ldots,x_{n};s_{1},\ldots,s_{n};p).$
(2.53)

The eigenvalues $\varepsilon^{(r)}(s;p)$ of the modified elliptic Ruijsenaars
operators are also invariant under the rotation of indices of the $s$
variables $:$

$\varepsilon^{(r)}(s_{2},\ldots,s_{n},s_{1};p)=\varepsilon^{(r)}(s_{1},\ldots,s_{n};p)\quad(r=1,\ldots,n).$
(2.54)

###### Theorem 2.9

The normalized Ruijsenaars function $f(x;s;p|q,t)$ satisfies the following
transformation formula under the reflection $t\leftrightarrow q/t$ of the
parameter $t$ $:$

$f(x;s;p|q,q/t)=\gamma(s;p|q,t)\prod_{1\leq i<j\leq
n}\frac{\Gamma(tx_{j}/x_{i};p,q)}{\Gamma(qx_{j}/tx_{i};p,q)}\cdot
f(x;s;p|q,t),$ (2.55)

for some constant $\gamma(s;p|q,t)$ depending on $p$ such that
$\gamma(s;p|q,t)\gamma(s;p|q,q/t)=1$. Also, the eigenvalues
$\varepsilon^{(r)}(s;p)=\varepsilon^{(r)}(s;p|q,t)$ are invariant under the
reflection $t\leftrightarrow q/t$ of the parameter $t$ $:$

$\varepsilon^{(r)}(s;p|q,q/t)=\varepsilon^{(r)}(x;p|q,t)\quad(r=1,\ldots,n).$
(2.56)

If we define

$\psi(x;s;p|q,t)=\prod_{1\leq i<j\leq
n}\frac{\Gamma(tx_{j}/x_{i};p,q)}{\Gamma(x_{j}/x_{i};p,q)}\cdot f(x;s;p|q,t),$
(2.57)

Theorem 2.9 is equivalently rewritten as

$\psi(x;s;p|q,q/t)=\gamma(s;p|q,t)\psi(x;s;p|q,t).$ (2.58)

It is known that $\gamma(s;0;q,t)=1$ by a result of [20]. We conjecture that
the undetermined constant $\gamma(s;p|q,t)$ is 1, namely the function
$\psi(x;s;p|q,t)$ is symmetric with respect to the reflection
$t\leftrightarrow q/t$ of the parameter $t$.

###  2.4 Plan of this paper

The rest of this paper is organized as follows.

Theorems 2.2 and 2.6 on existence of joint eigenfunctions in the formal sense
will be established in Section 4. Once we have set up appropriate classes of
formal power series on which the joint eigenvalue problems should be
investigated, their formal solutions are obtained along a more or less
standard procedure of perturbation theory as in Kato [10] and Komori-Takemura
[12] under the separation conditions of eigenvalues.

The convergence of formal joint eigenfunctions will be discussed in Section 5.
This problem for the elliptic Ruijsenaars operators requires careful analysis
of the recurrence relations for the coefficients if we follow the direct
approach. We first prove the convergence of the joint eigenfunction $f(x;s;p)$
(stationary Ruijsenaars function) of the modified elliptic Ruijsenaars
operators in the asymptotic domain $|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$.
The absolute convergence of $f(x;s;p)$ is established by the method of
majorants for the expansion coefficients defined through quadratic
recurrences. It seems technically demanding, however, to directly apply a
similar method to the case of symmetric joint eigenfunctions
$\mathcal{P}_{\lambda}(x;p)$ that deform the Macdonald polynomials. We
overcome this difficulty by employing an integral transform that generates
symmetric eigenfunctions in the physical domain (around the torus) from
eigenfunctions in the asymptotic domain.

Since this approach of integral transforms seems novel and characteristic in
this paper, we explain in advance in Section 3 how this method works in the
trigonometric case, presenting a prototype of our arguments of the elliptic
case in Section 5.

As a consequence of the convergence theorem, we include a proof of Theorem 2.4
on the orthogonality relations for $\mathcal{P}_{\lambda}(x;p|q,t)$ in Section
5. Theorems 2.8 and 2.9 on symmetries of $f(x;s;p|q,t)$ will be proved in
Section 6.

##  3 Integrable transform to the physical domain: a prototype

In this section, we deal with joint eigenfunctions of the Macdonald-
Ruijsenaars operators in the trigonometric case, and introduce an integral
transform that maps holomorphic eigenfunctions in the asymptotic domain
$|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|$ to symmetric eigenfunctions in the
physical domain around the torus. The method we explain below will be extended
to the elliptic case in Section 5 so as to establish the convergence of the
symmetric eigenfunctions that deform Macdonald polynomials.

###  3.1 Integral transform

We first recall some basic results from the theory of Macdonald polynomials
[16]. The Cauchy kernel

$K(x,y)=\prod_{i=1}^{n}\prod_{j=1}^{n}\frac{(tx_{i}/y_{j};q)_{\infty}}{(x_{i}/y_{j};q)_{\infty}}$
(3.1)

for the Macdonald polynomials in variables $x=(x_{1},\ldots,x_{n})$ and
$y=(y_{1},\ldots,y_{n})$ is a meromorphic function on
$(\mathbb{C}^{\ast})^{n}\times(\mathbb{C}^{\ast})^{n}$, and it has an
expansion

$K(x,y)=\sum_{\ell(\lambda)\leq
n}b_{\lambda}P_{\lambda}(x)P_{\lambda}(y^{-1}),\quad b_{\lambda}=\prod_{1\leq
i\leq j\leq
n}\frac{(t^{j-i+1}q^{\lambda_{i}-\lambda_{j}};q)_{\lambda_{j}-\lambda_{j+1}}}{(t^{j-i}q^{\lambda_{i}-\lambda_{j}+1};q)_{\lambda_{j}-\lambda_{j+1}}}$
(3.2)

in the domain $|x_{i}/y_{j}|<1$ ($i,j\in\left\\{1,\ldots,n\right\\}$), where
the sum is taken over all partitions $\lambda$ with $l(\lambda)\leq n$, and
$(a;q)_{k}=(1-a)(1-qa)\cdots(1-q^{k-1}a)$ ($k=0,1,2,\ldots$). We define the
weight function $w(y)$ by

$w(y)=\prod_{1\leq i<j\leq
n}(1-y_{j}/y_{i})\frac{(qy_{j}/ty_{i};q)_{\infty}}{(ty_{j}/y_{i};q)_{\infty}},$
(3.3)

which is holomorphic in the domain $|y_{j}/y_{i}|<|t|^{-1}$ ($1\leq i<j\leq
n$).

For each $\theta>0$, we denote by $V_{\theta}$ the domain in
$(\mathbb{C}^{\ast})^{n}$ specified by

$V_{\theta}=\left\\{{y=(y_{1},\ldots,y_{n})\in(\mathbb{C}^{\ast})^{n}}\big{|}{\
|y_{2}/y_{1}|<\theta,\ |y_{3}/y_{2}|<\theta,\ \ldots,\
|y_{n}/y_{n-1}|<\theta}\right\\}.$ (3.4)

Fixing positive constants $r,\sigma$ such that
$\sigma<\min\left\\{1,\theta,|t|^{-1}\right\\}$, we define an $n$-cycle
$C_{r,\sigma}$ in $(\mathbb{C}^{\ast})^{n}$ by

$C_{r,\sigma}=\left\\{{y=(y_{1},\ldots,y_{n})\in(\mathbb{C}^{\ast})^{n}}\big{|}{\
|y_{i}|=\sigma^{i-1}r\ \ (i=1,\ldots,n)}\right\\}$ (3.5)

with positive orientation. Note that $w(y)$ is holomorphic in an neighborhood
of $C_{r,\sigma}$ since $\sigma<\min\left\\{1,|t|^{-1}\right\\}$. For each
holomorphic function $\psi=\psi(y)$ in $V_{\theta}$, we define its integral
transform $\mathcal{K}(\psi)$ by

$\mathcal{K}(\psi)(x)=\int_{C_{r,\sigma}}K(x,y)w(y)\psi(y)d\omega_{n}(y).$
(3.6)

###### Lemma 3.1

The function $\varphi(x)=\mathcal{K}(\psi)(x)$ defined as above is continued
to an entire symmetric holomorphic function in $\mathbb{C}^{n}$.

Proof : The symmetry of $\varphi(x)$ follows from that of $K(x,y)$ in $x$.
Since $K(x,y)$ is holomorphic in the domain $|x_{i}/y_{j}|<1$
($i,j=1,\ldots,n$), it is is holomorphic in the domain $|x_{i}|<\sigma^{n-1}r$
$(i=1,\ldots,n)$ when $y\in C_{r,\sigma}$. This implies that $\varphi(x)$ is
holomorphic in the same domain. Since the integral does not depend on
$r\in\mathbb{R}_{>0}$, $\varphi(x)$ is continued to an entire function in
$\mathbb{C}^{n}$. $\square$

###  3.2 From eigenfunctions in the asymptotic domain to Macdonald polynomials

In what follows, we use the generating function

$D_{x}(u)=\sum_{r=0}^{n}(-u)^{r}D^{(r)}_{x}=\sum_{I\subseteq\left\\{1,\ldots,n\right\\}}(-u)^{I}A_{I}(x)T_{q,x}^{I}$
(3.7)

of the Macdonald-Ruijsenaars operators $D^{(r)}_{x}$ ($r=0,1,\ldots,n$), so
that

$D_{x}(u)P_{\lambda}(x)=\varepsilon(u)P_{\lambda}(x),\quad\varepsilon(u)=\sum_{r=0}^{n}(-u)^{r}e_{r}(t^{\rho}q^{\lambda})=\prod_{j=1}^{n}(1-ut^{n-j}q^{\lambda_{j}}),$
(3.8)

for each partition $\lambda$ with $\ell(\lambda)\leq n$. Also, using the
multi-index notation $T_{q,x}^{\mu}=T_{q,x_{1}}^{\mu_{1}}\cdots
T_{q,x_{n}}^{\mu_{n}}$, for a $q$-difference operator
$L=\sum_{\mu\in\mathbb{Z}^{n}}a_{\mu}(y)T_{q,y}^{\mu}$ (finite sum) in $y$
variables, we denote by
$L^{\ast}=\sum_{\mu\in\mathbb{Z}^{n}}T_{q,y}^{-\mu}a_{\mu}(y)$ the formal
adjoint of $L$ with respect to $d\omega_{n}(y)$, noting that
$(L_{1}L_{2})^{\ast}=L_{2}^{\ast}L_{1}^{\ast}$.

Recall that the Ruijsenaars-Macdonald operator $D_{x}(u)$ is formally self-
adjoint with respect to the weight function

$w^{\mathrm{sym}}(y)=\prod_{1\leq i<j\leq
n}\frac{(y_{i}/y_{j};q)_{\infty}(y_{j}/y_{i};q)_{\infty}}{(ty_{i}/y_{j};q)_{\infty}(ty_{j}/y_{i};q)_{\infty}}$
(3.9)

for the orthogonality of Macdonald polynomials; the superscript “sym” is used
to indicate that this weight function is symmetric with respect to
$y=(y_{1},\ldots,y_{n})$. Namely we have an identity

$D_{y}(u)^{\ast}=w^{\mathrm{sym}}(y)D_{y^{-1}}(u)w^{\mathrm{sym}}(y)^{-1}$
(3.10)

of $q$-difference operators. Also, the kernel function $K(x,y)$ satisfies the
kernel function identity

$D_{x}(u)K(x,y)=D_{y^{-1}}(u)K(x,y).$ (3.11)

(See for instance, [19] and [11].) Hence we have

$D_{x}(u)K(x,y)w^{\mathrm{sym}}(y)=w^{\mathrm{sym}}(y)D_{y^{-1}}K(x,y)=D_{y}(u)^{\ast}(K(x,y)w^{\mathrm{sym}}(y)).$
(3.12)

We use the idea of integral representation of [19] for our weight function
$w(y)$ defined above. For this purpose we rewrite $w^{\mathrm{sym}}(y)$ as
follows:

$\begin{split}w^{\mathrm{sym}}(y)&=\prod_{1\leq i<j\leq
n}\frac{(y_{j}/y_{i};q)_{\infty}(qy_{j}/ty_{i};q)_{\infty}}{(ty_{j}/y_{i};q)_{\infty}(qy_{j}/y_{i};q)_{\infty}}\frac{\theta(y_{i}/y_{j};q)}{\theta(ty_{i}/y_{j};q)}\\\
&=\prod_{1\leq i<j\leq
n}(1-y_{j}/y_{i})\frac{(qy_{j}/ty_{i};q)_{\infty}}{(ty_{j}/y_{i};q)_{\infty}}\cdot\prod_{1\leq
i<j\leq n}\frac{\theta(y_{i}/y_{j};q)}{\theta(ty_{i}/y_{j};q)}.\end{split}$
(3.13)

This means that

$w^{\mathrm{sym}}(y)=w(y)g(y),\quad g(y)=\prod_{1\leq i<j\leq
n}\frac{\theta(y_{i}/y_{j};q)}{\theta(ty_{i}/y_{j};q)}.$ (3.14)

Noting that

$T_{q,y_{i}}g(y)=t^{n-2i+1}g(y)\qquad(i=1,\ldots,n),$ (3.15)

we define the $n$-vector
$\kappa=(\kappa_{1},\ldots,\kappa_{n})\in\mathbb{C}^{n}$ by
$\kappa_{i}=(n-2i+1)\log t/\log q$ $(i=1,\ldots,n)$. Then we have

$T_{q,y}^{\mu}g(y)=q^{\left\langle\mu,\kappa\right\rangle}g(y)\qquad(\mu\in\mathbb{Z}^{n}),$
(3.16)

where $\left\langle\mu,\kappa\right\rangle=\sum_{i=1}^{n}\mu_{i}\kappa_{i}$.
This means that the ratio $g(y)/y^{\kappa}$ is a $q$-periodic function in each
variable $y_{i}$ ($i=1,\ldots,n$).

We introduce the generating function

$E_{x,s}(u)=\sum_{I\subseteq\left\\{1,\ldots,n\right\\}}(-u)^{|I|}s^{\epsilon_{I}}B_{I}(x)T_{q,x}^{\epsilon_{I}}$
(3.17)

for the modified Ruijsenaars-Macdonald operators $E^{(r)}_{x,s}$ of (2.37), so
that $D_{x}(u)=E_{x,t^{\rho}}(u)$. Then the Macdonald function $f(x;s)$ in the
asymptotic domain $|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|$ satisfies

$E_{x,s}(u)f(x;s)=\varepsilon(s;u)f(x;s),\quad\varepsilon(s;u)=\sum_{r=0}^{n}(-u)^{r}e_{r}(s)=\prod_{j=1}^{n}(1-us_{j}).$
(3.18)

It is known by [20] that $f(x;s)$ is holomorphic in the domain
$|x_{j}/x_{i}|<|t/q|$ ($1\leq i<j\leq n$) and $|s_{j}/s_{i}|<1/|q|$ ($1\leq
i<j\leq n$).

Returning to the kernel function, by (3.12) we obtain

$\begin{split}D_{x}(u)K(x,y)w(y)&=g(y)^{-1}D_{y}(u)^{\ast}\big{(}g(y)K(x,y)w(y)\big{)}\\\
&=(g(y)D_{y}(u)g(y)^{-1})^{\ast}\big{(}K(x,y)w(y)\big{)}\\\
&=(E_{y,t^{\rho}q^{-\kappa}})^{\ast}(K(x,y)w(y)).\end{split}$ (3.19)

Note that

$t^{\rho}q^{-\kappa}=(t^{n-1}\cdot t^{-n+1},t^{n-2}\cdot
t^{-n+3},\ldots,1\cdot t^{n-1})=(1,t,\ldots,t^{n-1}).$ (3.20)

Hereafter, for a vector $a=(a_{1},\ldots,a_{n})$, we denote by
$a^{\vee}=(a_{n},\ldots,a_{1})$ the reversal of $a$ (the action of the longest
element of $\mathfrak{S}_{n}$). In this notation of reversals, we obtain the
kernel function identity

$D_{x}(u)K(x,y)w(y)=(E_{y,{t^{\rho^{\vee}}}})^{\ast}(K(x,y)w(y)).$ (3.21)

###### Theorem 3.2

Let $\lambda$ be a partition with $l(\lambda)\leq n$. Setting
$s=t^{\rho}q^{\lambda}$, let $f(y;s^{\vee})$ be the Macdonald function in the
asymptotic domain $V_{\theta}$ with parameters
$s^{\vee}=t^{\rho^{\vee}}\\!q^{\lambda^{\vee}}$ with $\theta^{n-1}<|t/q|$ .
Then we have

$\int_{C_{r,\sigma}}K(x,y)w(y)\,y^{\lambda^{\vee}}\\!f(y;s^{\vee})d\omega_{n}(y)=b_{\lambda}P_{\lambda}(x).$
(3.22)

Proof : Taking $\sigma>0$ such that
$\sigma|q|<\min\left\\{1,\theta,1/|t|\right\\}$, we denote by
$\varphi(x)=\mathcal{K}(f(y;s^{\vee}))$ the left-hand side of (3.22). In this
setting, if $|x_{i}|<|q|\sigma^{n-1}r$ ($i=1,\ldots,n$), then the $n$-cycle
$C_{r,\sigma}$ is $q$-shift invariant with respect to the variables $y_{i}$
($i=1,\ldots,n$) by Cauchy’s theorem. Hence we obtain

$\begin{split}D_{x}(u)\varphi(x)&=\int_{C_{r,\sigma}}D_{x}(u)K(x,y)w(y)\,y^{\lambda^{\vee}}\\!f(y;s^{\vee})d\omega_{n}(y)\\\
&=\int_{C_{r,\sigma}}(E_{y,t^{\rho^{\vee}}})^{\ast}\big{(}K(x,y)w(y))\,y^{\lambda^{\vee}}\\!f(y;s^{\vee})d\omega_{n}(y)\\\
&=\int_{C_{r,\sigma}}K(x,y)w(y)E_{y,t^{\rho^{\vee}}}\big{(}y^{\lambda^{\vee}}\\!f(y;s^{\vee})\big{)}d\omega_{n}(y)\\\
&=\int_{C_{r,\sigma}}K(x,y)w(y)\,y^{\lambda^{\vee}}E_{y,t^{\rho^{\vee}}q^{\lambda^{\vee}}}\big{(}f(y;s^{\vee})\big{)}d\omega_{n}(y)\\\
\end{split}$ (3.23)

Here we compute

$E_{y,t^{\rho^{\vee}}}\big{(}y^{\lambda^{\vee}}\\!f(y;s^{\vee})\big{)}=y^{\lambda^{\vee}}E_{y,t^{\rho^{\vee}}q^{\lambda^{\vee}}}f(y;t^{\rho^{\vee}}q^{\lambda^{\vee}})=\varepsilon(t^{\rho^{\vee}}q^{\lambda^{\vee}};u)y^{\lambda^{\vee}}f(y;t^{\rho^{\vee}}q^{\lambda^{\vee}}).$
(3.24)

Since $\varepsilon(s^{\vee};u)=\varepsilon(s;u)=\prod_{i=1}^{n}(1-us_{i})$, we
obtain

$D_{x}(u)\varphi(x)=\varepsilon(t^{\rho}q^{\lambda};u)\varphi(x).$ (3.25)

Note that the coefficient of $u^{n}$ gives $t^{\binom{n}{2}}T_{q,x_{1}}\cdots
T_{q,x_{n}}\varphi(x)=t^{\binom{n}{2}}q^{|\lambda|}\varphi(x)$, which implies
$\varphi(x)$ is homogenous of degree $|\lambda|$ in $x$. Since $\varphi(x)$ is
a symmetric holomorphic function in $\mathbb{C}^{n}$, it is actually a
symmetric polynomial satisfying (3.25), and hence a constant multiple of the
Macdonald polynomial $P_{\lambda}(x)$. On the other hand, by (3.2) we have

$\varphi(x)=\sum_{\ell(\mu)\leq n}\ c_{\mu}b_{\mu}P_{\mu}(x),\qquad
c_{\mu}=\int_{C_{r,\sigma}}P_{\mu}(y^{-1})\,y^{\lambda^{\vee}}\\!f(y;s^{\vee})d\omega_{n}(y).$
(3.26)

Since $\varphi(x)$ is a constant multiple of $P_{\lambda}(x)$, we have
$c_{\mu}=0$ ($\mu\neq\lambda$) and
$\varphi(x)=c_{\lambda}b_{\lambda}P_{\lambda}(x)$. In fact we have
$c_{\lambda}=1$, since $P_{\lambda}(y^{-1})$ and
$y^{\lambda^{\vee}}f(y;s^{\vee})$ and $w(y)$ have leading terms
$y^{-\lambda^{\vee}}$, $y^{\lambda^{\vee}}$ and $1$, respectively, with
respect to the dominance order. This implies
$\varphi(x)=b_{\lambda}P_{\lambda}(x)$. $\square$

This theorem explains how the Macdonald polynomials are reconstructed by the
integral transform from the Macdonald functions $f(x;s)$ in the asymptotic
domain $|x_{1}|\gg|x_{2}|\gg\cdots\gg|x_{n}|$. The same idea will be used in
Section 5 for proving the convergence of elliptic deformations of the
Macdonald polynomials.

##  4 Existence of formal joint eigenfunctions

We begin this section by remarks on the condition on the parameter
$t\in\mathbb{C}^{\ast}$ that we impose for the separation of eigenvalues.
After giving proofs to Propositions 2.1 and 2.5, we establish the existence
theorems of formal joint eigenfunctions for the two cases of elliptic
deformations of Macdonald polynomials, and asymptotically free solutions in
the domain $|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$, respectively.

In order to clarify how the Ruijsenaars operators are related to the root
system of the Lie algebra $\mathfrak{gl}_{n}$, we recall here some standard
notations and terminologies relevant to our arguments. We denote by
$\mathsf{P}=\mathbb{Z}\epsilon_{1}\oplus\cdots\oplus\mathbb{Z}\epsilon_{n}$
the (integral) weight lattice of $\mathfrak{gl}_{n}$; it is a free
$\mathbb{Z}$-module with canonical basis $\epsilon_{1},\ldots,\epsilon_{n}$,
endowed with the the scalar product (symmetric bilinear form) $\left\langle\
,\,\right\rangle:\ \mathsf{P}\times\mathsf{P}\to\mathbb{Z}$ such that
$\left\langle\epsilon_{i},\epsilon_{j}\right\rangle=\delta_{i,j}$
($i,j\in\left\\{1,\ldots,n\right\\}$). We freely identify an integral weight
$\mu=\mu_{1}\epsilon_{1}+\cdots+\mu_{n}\epsilon_{n}\in\mathsf{P}$ with the
$n$-vector of integers $\mu=(\mu_{1},\ldots,\mu_{n})\in\mathbb{Z}^{n}$, and
denote by $x^{\mu}=x_{1}^{\mu_{1}}\cdots x_{n}^{\mu_{n}}$ the corresponding
monomial of the variables $x=(x_{1},\ldots,x_{n})$. Fixing the simple roots
$\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}\in\mathsf{P}$ ($i=1,\ldots,n-1$), we
denote by

$\mathsf{Q}=\mathbb{Z}\alpha_{1}\oplus\cdots\oplus\mathbb{Z}\alpha_{n-1}\subseteq\mathsf{P}$
(4.1)

the root lattice of $\mathfrak{gl}_{n}$, and by

$\mathsf{P}_{+}=\left\\{{\lambda\in\mathsf{P}}\big{|}{\
\left\langle\alpha_{i},\lambda\right\rangle\geq 0\ \
(i=1,\ldots,n-1)}\right\\}\subseteq\mathsf{P},$ (4.2)

the cone of dominant integral weights. Note that an integral weight
$\lambda=\lambda_{1}\epsilon_{1}+\cdots+\lambda_{n}\epsilon_{n}\in\mathsf{P}$,
or an integer vector
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Z}^{n}$, belongs to
$\mathsf{Q}$ if and only if $|\lambda|=\lambda_{1}+\cdots+\lambda_{n}=0$, and
to $\mathsf{P}_{+}$ if and only if $\lambda_{1}\geq\cdots\geq\lambda_{n}$. In
terms of the root system, the dominance order $\mu\leq\nu$ on $\mathsf{P}$ is
defined by the condition $\nu-\mu\in\mathsf{Q}_{+}=\mathbb{Z}_{\geq
0}\alpha_{1}\oplus\cdots\oplus\mathbb{Z}_{\geq 0}\alpha_{n-1}$. We denote by
$\Delta_{+}=\left\\{{\epsilon_{i}-\epsilon_{j}}\big{|}{1\leq i<j\leq
n}\right\\}\subseteq\mathsf{Q}_{+}$ the set of positive roots, so that

$\Delta(x)=\prod_{1\leq i<j\leq
n}(x_{i}-x_{j})=\prod_{i=1}^{n}x_{i}^{n-i}\prod_{1\leq i<j\leq
n}(1-x_{j}/x_{i})=x^{\rho}\prod_{\alpha\in\Delta_{+}}(1-x^{-\alpha}),$ (4.3)

where $\rho=\sum_{i=1}^{n}(n-i)\epsilon_{i}$.

###  4.1 Separation of eigenvalues

Introducing a parameter $u$, we define a generating function $D_{x}(u)$ of the
Macdonald-Ruijsenaars operators by

$D_{x}(u)=\sum_{r=0}^{n}(-u)^{r}D_{x}^{(r)}=\sum_{I\subseteq\left\\{1,\ldots,n\right\\}}(-u)^{|I|}A_{I}(x)T_{q,x}^{\epsilon_{I}},\quad
A_{I}(x)=\frac{T_{t,x}^{\epsilon_{I}}(\Delta(x))}{\Delta(x)}.$ (4.4)

Here we used the multi-index notation for
$T_{q,x}^{\mu}=T_{q,x_{1}}^{\mu_{1}}\cdots T_{q,x_{n}}^{\mu}$ for $q$-shift
operators, setting $\epsilon_{I}=\sum_{i\in I}\epsilon_{i}\in\mathsf{P}$ for
each subset $I\subseteq\left\\{1,\ldots,n\right\\}$. Then, the joint
eigenfunction equations (2.15) for the Macdonald polynomial $P_{\lambda}(x)$
($\lambda\in\mathsf{P}_{+}$) can be written as a single eigenfunction equation

$D_{x}(u)P_{\lambda}(x)=d_{\lambda}(u)P_{\lambda}(x),\quad
d_{\lambda}(u)=\sum_{r=0}^{n}(-u)^{r}e_{r}(t^{\rho}q^{\lambda})=\prod_{j=1}^{n}(1-ut^{n-j}q^{\lambda_{j}}),$
(4.5)

containing a parameter $u$. Note that, for a pair $\lambda,\mu\in\mathsf{P}$
of integral weights, $d_{\lambda}(u)=d_{\mu}(u)$ as polynomials in $u$ if and
only if there exists a permutation $\sigma\in\mathfrak{S}_{n}$ such that

$t^{n-j}q^{\mu_{j}}=t^{n-\sigma(j)}q^{\lambda_{\sigma(j)}}\quad(j=1,\ldots,n).$
(4.6)

For a subset $L\subseteq\mathsf{P}$, we say that the polynomials
$d_{\lambda}(u)$ separate $L$ if $d_{\lambda}(u)\neq d_{\mu}(u)$ for any
distinct pair $\lambda,\mu\in L$. The condition $t^{k}\notin
q^{\mathbb{Z}_{<0}}$ $(k=1,\ldots,n-1)$ that we imposed for the existence of
Macdonald polynomials is equivalent to saying that the polynomials
$d_{\lambda}(u)$ separate $\mathsf{P}_{+}$. In fact, under our assumption
$|q|<1$, we have

###### Lemma 4.1

$(1)$ The polynomials $d_{\lambda}(u)$ separate $\mathsf{P}$ if and only if
$t^{k}\notin q^{\mathbb{Z}}$ $(k=1,\ldots,n-1)$.
$(2)$ The polynomials $d_{\lambda}(u)$ separate $\mathsf{P}_{+}$ if and only
if $t^{k}\notin q^{\mathbb{Z}_{<0}}$ $(k=1,\ldots,n-1)$.

Proof : (1) If $d_{\lambda}(u)=d_{\mu}(u)$ for a distinct pair
$\lambda,\mu\in\mathsf{P}$, we have
$t^{n-j}q^{\mu_{j}}=t^{n-\sigma(j)}q^{\lambda_{\sigma(j)}}$ ($j=1,\ldots,n$)
for a permutation $\sigma\in\mathfrak{S}_{n}$. Since $\lambda\neq\mu$, we have
$\sigma\neq 1$, and hence there exists an index
$j\in\left\\{1,\ldots,n\right\\}$ such that $\sigma(j)\neq j$ and
$t^{\sigma(j)-j}=q^{\lambda_{\sigma(j)}-\mu_{j}}\in q^{\mathbb{Z}}$, which
means $t^{k}\in q^{\mathbb{Z}}$ for
$k=|\sigma(j)-j|\in\left\\{1,\ldots,n-1\right\\}$. Conversely, suppose that
$t^{n-i}=q^{-l}$ for some $i\in\left\\{1,\ldots,n-1\right\\}$ and
$l\in\mathbb{Z}$, setting $i=n-k$. Then we have

$\begin{split}(t^{n-1}q^{\lambda_{1}},\ldots,t^{n-i}q^{\lambda_{i}},\ldots,q^{\lambda_{n}})&=(t^{n-1}q^{\lambda_{1}},\ldots,q^{\lambda_{i}-l},\ldots,t^{n-i}q^{\lambda_{n}+l})\\\
&=(i,n).(t^{n-1}q^{\lambda_{1}},\ldots,t^{n-i}q^{\lambda_{n}+l},\ldots,q^{\lambda_{i}-l}),\end{split}$
(4.7)

where $(i,n)$ stands for the transposition of $i$ and $n$. This implies that
$d_{\lambda}(u)=d_{\mu}(u)$ for the pair

$\lambda=(\lambda_{1},\ldots,\lambda_{i},\ldots,\lambda_{n}),\quad\mu=(\lambda_{1},\ldots,\lambda_{n}+l,\ldots,\lambda_{i}-l)\in\mathsf{P}.$
(4.8)

(2) We first show that, if $|t|\leq 1$, then the polynomials $d_{\lambda}(u)$
separate $\lambda\in\mathsf{P}_{+}$. Under the assumption $|t|\leq 1$, the
sequence $|t^{n-j}q^{\lambda_{j}}|$ ($j=1,\ldots,n$) is non-decreasing for any
dominant $\lambda\in\mathsf{P}_{+}$. If $d_{\lambda}(u)=d_{\mu}(u)$ for a pair
$\lambda,\mu\in\mathsf{P}_{+}$, then we have
$|t^{n-j}q^{\lambda_{j}}|=|t^{n-j}q^{\mu_{j}}|$ and hence
$|q|^{\lambda_{j}}=|q|^{\mu_{j}}$ for $j=1,\ldots,n$. This implies
$\lambda_{j}=\mu_{j}$ ($j=1,\ldots,n$) (since $|q|<1$). Suppose now that
$d_{\lambda}(u)=d_{\mu}(u)$ for a distinct pair $\lambda,\mu\in\mathsf{P}$.
Then we have $|t|>1$, and also $t^{k}\in q^{\mathbb{Z}}$ for some
$k\in\left\\{1,\ldots,n-1\right\\}$ by (1). Hence $t^{k}\in
q^{\mathbb{Z}_{<0}}$ for some $k\in\left\\{1,\ldots,n-1\right\\}$. Conversely,
suppose that $t^{n-i}=q^{-l}$ for some $i\in\left\\{1,\ldots,n-1\right\\}$ and
$l\in\mathbb{Z}_{>0}$. Then $d_{\lambda}(u)=d_{\mu}(u)$ for
$\lambda=(m,\ldots,m,0,0\ldots,0)$ with $m$’s in the first $i-1$ places, and
$\mu=(m,\ldots,m,l,0,\ldots,-l)$, both in $\mathsf{P}_{+}$ if $m\geq l$.
$\square$

The following lemma is crucial in our proof of the existence theorem of formal
joint eigenfunctions for the elliptic Ruijsenaars operators.

###### Lemma 4.2

If the polynomials $d_{\lambda}(u)$ separate a subset $L\subseteq\mathsf{P}$,
then there exists a constant $c\in\mathbb{C}^{\ast}$, such that
$d_{\lambda}(c)\neq d_{\mu}(c)$ for any distinct pair $\lambda,\mu\in L$.

Proof : The set

$S=\left\\{{a\in\mathbb{C}^{\ast}}\big{|}{\ d_{\lambda}(a)=d_{\mu}(a)\ \
\mbox{for some distinct pair $\lambda,\mu\in
L$}}\right\\}\subseteq\mathbb{C}^{\ast}$ (4.9)

is countable, and hence the complement $\mathbb{C}^{\ast}\backslash S$ is non-
empty. This implies that any element $c\in\mathbb{C}^{\ast}\backslash S$
satisfies the condition $d_{\lambda}(c)\neq d_{\mu}(c)$ for any distinct pair
$\lambda,\mu\in\mathbb{C}^{\ast}$. $\square$

###  4.2 Elliptic Ruijsenaars operators as power series of $p$

In this subsection, we clarify the structure of $p$-expansion of the elliptic
Ruijsenaars operators, thereby proving Propositions 2.1 and 2.5.

Similarly to the trigonometric case, we introduce the generation function

$\mathcal{D}_{x}(p;u)=\sum_{r=0}^{n}(-u)^{r}\mathcal{D}^{(r)}_{x}(p)=\sum_{I\subseteq\left\\{1,\ldots,n\right\\}}(-u)^{|I|}\mathcal{A}_{I}(x;p)T_{q,x}^{\epsilon_{I}},\quad\mathcal{A}_{I}(x;p)=\frac{T_{t,x}^{\epsilon_{I}}(\Delta(x;p))}{\Delta(x;p)},$
(4.10)

for the elliptic Ruijsenaars operators, where
$\Delta(x;p)=x^{\rho}\prod_{1\leq i<j\leq n}\theta(x_{j}/x_{i};p)$. We express
the coefficients $\mathcal{A}_{I}(x;p)$ in two ways:

$\mathcal{A}_{I}(x;p)=t^{\left\langle\epsilon_{I},\rho\right\rangle}\mathcal{B}_{I}(x;p)=A_{I}(x)\mathcal{C}_{I}(x;p),$
(4.11)

where $A_{I}(x)=\mathcal{A}_{I}(x;0)$ are the coefficients of $D_{x}(u)$ in
(2.32) and

$\begin{split}\mathcal{B}_{I}(x;p)&=\prod_{\begin{subarray}{c}1\leq i<j\leq
n\\\ i\in I,\,j\notin
I\end{subarray}}\frac{\theta(x_{j}/tx_{i};p)}{\theta(x_{j}/x_{i};p)}\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\ i\notin I,\,j\in
I\end{subarray}}\frac{\theta(tx_{j}/x_{i};p)}{\theta(x_{j}/x_{i};p)},\\\
\mathcal{C}_{I}(x;p)&=\prod_{i\in I,\,j\notin
I}\frac{(ptx_{i}/x_{j};p)(px_{j}/tx_{i};p)}{(px_{i}/x_{j};p)(px_{j}/x_{i};p)}.\end{split}$
(4.12)

We remark here that, for each $I\subseteq\left\\{1,\ldots,n\right\\}$,
$\mathcal{B}_{I}(x;p)$ is a holomorphic function of the variables
$z_{0}=px_{1}/x_{n},z_{1}=x_{2}/x_{1},\ldots,z_{n-1}=x_{n}/x_{n-1}$ in the
polydisc $|z_{0}|<1,|z_{1}|<1,\ldots,|z_{n-1}|<1$. In fact, we have
$z_{0}z_{1}\cdots z_{n-1}=p$ and

$x_{j}/x_{i}=z_{i}\cdots z_{j-1},\quad px_{i}/x_{j}=z_{j}\cdots
z_{n-1}z_{0}\cdots z_{i-1}\quad(1\leq i<j\leq n).$ (4.13)

This implies in particular that
$\mathcal{B}_{I}(x;p)\in\mathbb{C}\left\\{px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}$,
and hence, $\mathcal{B}_{I}(x;p)$ is represented by a power series in $p$ of
the form

$\mathcal{B}_{I}(x;p)=\sum_{k=0}^{\infty}p^{k}\,\mathcal{B}_{I,k}(x),\quad\mathcal{B}_{I,k}(x)\in(x_{1}/x_{n})^{k}\mathbb{C}\left\\{x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}\quad(k=0,1,2,\ldots),$
(4.14)

which proves Proposition 2.5 for
$\mathcal{A}_{I}(x;p)=t^{\left\langle\epsilon_{I},\rho\right\rangle}\mathcal{B}_{I}(x;p)$.
On the other hand, from (4.12), $\mathcal{C}_{I}(x;p)$ is holomorphic in the
domain $|p|<|x_{j}/x_{i}|<|p|^{-1}$, and it can be represented there by a
power series of $p$ whose coefficients are Laurent polynomials in $x$:

$\mathcal{C}_{I}(x;p)=\sum_{k=0}^{p}p^{k}\mathcal{C}_{I,k}(x),\quad\mathcal{C}_{I,k}(x)\in\mathbb{C}[x^{\pm
1}]\quad(k=0,1,2,\ldots).$ (4.15)

Hence, each coefficients of $\mathcal{A}_{I}(x;p)$ of the elliptic Ruijsenaars
operators has a $p$-expansion of the form

$\mathcal{A}_{I}(x;p)=\sum_{k=0}^{\infty}p^{k}\,\mathcal{A}_{I,k}(x),\quad\mathcal{A}_{I,k}(x)=t^{\left\langle\epsilon_{I},\rho\right\rangle}\mathcal{B}_{I,k}(x)=A_{I}(x)\mathcal{C}_{I,k}(x)\quad(k=0,1,2,\ldots).$
(4.16)

Note that both expansions (4.14) and (4.15) make sense in the domain
$|p|^{\frac{1}{n-1}}<|x_{i+1}/x_{i}|<1$ ($i=1,\ldots,n-1$). Combining (4.14)
and (4.15), we have

$\mathcal{A}_{I,k}(x)\in\big{(}A_{I}(x)\mathbb{C}[x^{\pm
1}]\big{)}\cap\big{(}(x_{1}/x_{n})^{k}\mathbb{C}\left\\{x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}\big{)}$
(4.17)

for $k=0,1,2,\ldots$. For each $r=1,\ldots,n$, the elliptic Ruijsenaars
operator $\mathcal{D}_{x}^{(r)}(p)$ is a $\mathfrak{S}_{n}$-invariant
$q$-difference operator, and hence the coefficients
$\mathcal{D}^{(r)}_{x,k}(u)$ $(k=0,1,2\ldots)$ in the $p$-expansion

$\mathcal{D}^{(r)}_{x}(p)=\sum_{k=0}^{\infty}p^{k}\,\mathcal{D}^{(r)}_{x,k},\quad\mathcal{D}^{(r)}_{x,k}=\sum_{I\subseteq\left\\{1,\ldots,n\right\\};\
|I|=r}\mathcal{A}_{I,k}(x)T_{q,x}^{\epsilon_{I}}$ (4.18)

are also $\mathfrak{S}_{n}$-invariant. Since
$\Delta(x)\mathcal{D}^{(r)}_{x,k}$ $(k=0,1,2,\ldots)$ have Laurent polynomial
coefficients as well, they stabilize the ring $\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}$ of symmetric Laurent polynomials, namely
$\mathcal{D}^{(r)}_{x,k}\big{(}\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}\big{)}\subseteq\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}$. (Note that $\mathcal{D}^{(r)}_{x,k}$ do not stabilize
$\mathbb{C}[x^{\pm 1}]$.) Suppose that $\varphi(x)\in\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}_{\leq\mu}$ for some $\mu\in\mathsf{P}_{+}$. Since
$\varphi(x)\in
x^{\mu}\mathbb{C}\left\\{x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}$, by (4.17)
we obtain

$\psi(x)=\mathcal{D}^{(r)}_{x,k}\big{(}\varphi(x)\big{)}\in
x^{\mu+k\phi}\mathbb{C}\left\\{x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\},$
(4.19)

where $x^{\phi}=x_{1}/x_{n}$. Hence

$\psi(x)\in\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}\cap\big{(}x^{\mu+k\phi}\mathbb{C}\left\\{x_{2}/x_{1},\ldots,x_{n}/x_{n-1}\right\\}\big{)}=\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}_{\leq\mu+k\phi}.$ (4.20)

This means that $\mathcal{D}^{(r)}_{x,k}\big{(}\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}_{\leq\mu}\big{)}\subseteq\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}_{\leq\mu+k\phi}$ for all $k=0,1,2,\ldots$ and
$\mu\in\mathsf{P}_{+}$, which proves Proposition 2.1.

###  4.3 Elliptic deformation of Macdonald polynomials

In this subsection, we complete our proof of Theorem 2.2.

Denoting by $\mathcal{R}=\mathbb{C}[x^{\pm 1}]^{\mathfrak{S}_{n}}$ the ring of
symmetric Laurent polynomials in $x=(x_{1},\ldots,x_{n})$, we consider the
commuting family of $\mathbb{C}[[p]]$-linear operators

$\mathcal{D}^{(r)}_{x}(p):\ \
\mathcal{R}[[p]]\to\mathcal{R}[[p]]\qquad(r=1,\ldots,n)$ (4.21)

defined by the elliptic Ruijsenaars operators, and investigate the joint
eigenvalue problems

$\mathcal{D}^{(r)}_{x}(p)\psi(x;p)=\varepsilon^{(r)}(p)\psi(x;p)\quad(r=1,\ldots,n)$
(4.22)

where

$\psi(x;p)=\sum_{k=0}^{\infty}p^{k}\psi_{k}(x)\in\mathcal{R}[[p]],\quad\varepsilon^{(r)}(p)=\sum_{k=0}^{\infty}p^{k}\varepsilon^{(r)}_{k}\in\mathbb{C}[[p]].$
(4.23)

In terms of the generating function
$\mathcal{D}_{x}(p;u)=\sum_{r=0}^{n}(-u)^{r}\mathcal{D}^{(r)}_{x}(p)$, this
joint eigenvalue problem is expressed as an equation

$\mathcal{D}_{x}(u)\psi(x;p)=\varepsilon(p;u)\psi(x;p),\quad\varepsilon(p;u)=\sum_{r=0}^{n}(-u)^{r}\varepsilon^{(r)}(p)=\sum_{k=0}^{\infty}p^{k}\varepsilon_{k}(u)$
(4.24)

with a parameter $u$.

We assume hereafter that the parameter $t$ satisfies the condition
$t^{k}\notin q^{\mathbb{Z}_{<0}}$ ($k=1,\ldots,n-1$). Then, by Lemma 4.1, the
polynomials $d_{\lambda}(u)=\prod_{j=1}^{n}(1-ut^{n-j}q^{\lambda_{j}})$ in $u$
separate the cone $\mathsf{P}_{+}$ of dominant integral weights. We also fix
an arbitrary constant $c\in\mathbb{C}^{\ast}\backslash S$ with $S$ defined as
in (4.9) of Lemma 4.2, so that $d_{\lambda}(c)\neq d_{\mu}(c)$ for any
distinct $\lambda,\mu\in\mathsf{P}_{+}$. We first solve equation (4.24),
specializing the parameter $u$ to $c$, and then show the solution $\psi(x;p)$
gives a joint eigenfunction for all $\mathcal{D}^{(r)}_{x}(p)$
($r=1,\ldots,n$).

In terms of the expansion coefficients, the eigenfunction equation

$\mathcal{D}_{x}(p;c)\psi(x;p)=\varepsilon(p;c)\psi(x;p),\quad\psi(x;p)\in\mathcal{R}[[p]]$
(4.25)

is transferred to the recurrence relations

$\sum_{i=0}^{k}\mathcal{D}_{x,i}(c)\psi_{k-i}(x)=\sum_{i=0}^{k}\varepsilon_{i}(c)\psi_{k-i}(x)\qquad(k=0,1,2,\ldots).$
(4.26)

The conditions for $k=0$ is the eigenfunction equation

$D_{x}(c)\psi_{0}(x)=\varepsilon_{0}(c)\psi_{0}(x)$ (4.27)

for $\psi_{0}(x)\in\mathcal{R}=\mathbb{C}[x^{\pm 1}]^{\mathfrak{S}_{n}}$. If
$\psi_{0}(x)\neq 0$, it must be a constant multiple of the Macdonald
polynomial $P_{\lambda}(x)$ for some $\lambda\in\mathsf{P}_{+}$. Fixing a
dominant integral weight $\lambda\in\mathsf{P}_{+}$, we take

$\psi_{0}(x)=P_{\lambda}(x),\quad\varepsilon_{0}(c)=d_{\lambda}(c)$ (4.28)

for the initial data. We rewrite the recurrence relations for $k>0$ as

$\mathcal{D}_{x,0}(c)\psi_{k}(x)+\sum_{i=1}^{k}\mathcal{D}_{x,i}(c)\psi_{k-i}(x)=\varepsilon_{0}(c)\psi_{k}(x)+\sum_{i=1}^{k-1}\varepsilon_{i}(c)\psi_{k-i}(x)+\varepsilon_{k}(c)\psi_{0}(x),$
(4.29)

namely,

$\big{(}D_{x}(c)-d_{\lambda}(c)\big{)}\psi_{k}(x)-\varepsilon_{k}(c)P_{\lambda}(x)=-\sum_{i=1}^{k}\mathcal{D}_{x,i}(c)\psi_{k-i}(x)+\sum_{i=1}^{k-1}\varepsilon_{i}(c)\psi_{k-i}(x).$
(4.30)

At this step, regarding the right-hand side as known, we need to solve this
equation for $\psi_{k}(x)\in\mathcal{R}$ and
$\varepsilon_{k}(c)\in\mathbb{C}$. Under the genericity assumption on $c$, we
verify that $\psi_{k}(x)$ and $\varepsilon_{k}(c)$ are determined inductively
from $\psi_{i}(x)$ and $\varepsilon_{i}(c)$ ($i=0,1,\ldots,k-1$). For that
purpose, we use the decomposition

$\mathcal{R}=\mathbb{C}P_{\lambda}(x)\oplus\mathcal{S}_{\lambda},\quad\mathcal{S}_{\lambda}=\bigoplus_{\mu\in
P_{+};\,\mu\neq\lambda}\mathbb{C}P_{\mu}(x).$ (4.31)

of $\mathcal{R}$ by means of the Macdonald basis. Note that

$\big{(}D_{x}(c)-d_{\lambda}(c)\big{)}P_{\mu}(x)=\big{(}d_{\mu}(c)-d_{\lambda}(c))P_{\mu}(x)$
(4.32)

and $d_{\mu}(c)\neq d_{\lambda}(c)$ for any $\mu\in P_{+}$ with
$\mu\neq\lambda$. This implies that
$\mathrm{Ker}(D_{x}(c)-d_{\lambda}(c):\mathcal{R}\to\mathcal{R})=\mathbb{C}P_{\lambda}(x)$
and that $D_{x}(c)-d_{\lambda}(c)$ induces a $\mathbb{C}$-isomorphism
$\mathcal{S}_{\lambda}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{S}_{\lambda}$.
In the recurrence relation (4.30), decompose the right-hand side into the form
$a_{k}P_{\lambda}(x)+\varphi_{k}(x)$ with $a_{k}\in\mathbb{C}$ and
$\varphi_{k}(x)\in\mathcal{S}_{\lambda}$. Then $\psi_{k}(x)$ and
$\varepsilon_{k}(c)$ are determined by

$\varepsilon_{k}(c)=-a_{k},\quad(D_{x}(c)-d_{\lambda}(c))\psi_{k}(x)=\varphi_{k}(x).$
(4.33)

If we assume $\psi_{k}(x)\in\mathcal{S}_{\lambda}$ ($k>0$), then $\psi_{k}(x)$
and $\varepsilon_{k}(c)$ are determined uniquely by the recurrence. Otherwise,
$\psi_{k}(x)$ is determined with freedom of adding a multiple of
$P_{\lambda}(x)$ at each $k$th step. (As we will see below, this ambiguity
corresponds to the multiplication by a power series
$\gamma(p)\in\mathbb{C}[[p]]$ with $\gamma(0)=1$.) Also, it is verified
inductively by the recurrence (4.30) that
$\psi_{k}(x)\in\mathcal{R}_{\leq\lambda+k\phi}$, since
$\psi_{\lambda,k-i}(x)\in\mathcal{R}_{\leq\lambda+(k-i)\phi}$ and
$\mathcal{D}_{x,i}(c)(\mathcal{R}_{\leq\lambda+(k-i)\phi})\subseteq\mathcal{R}_{\leq\lambda+k\phi}$
by Proposition 2.1.

From now on, we denote by

$\varphi_{\lambda}(x;p)=\sum_{k=0}^{\infty}p^{k}\,\varphi_{\lambda;k}(x)\in\mathcal{R}[[p]]$
(4.34)

the unique formal solution of the eigenfunction equation (4.25) determined by
the condition that $\varphi_{\lambda;0}(x)=P_{\lambda}(x)$, and
$\varphi_{\lambda;k}(x)\in\mathcal{S}_{\lambda}$ ($k>0$). (This normalization
of $\varphi_{\lambda}(x;p)$ is different from the one we adopted below Theorem
2.2.) Also, we denote by $\varepsilon_{\lambda}(p;c)$ the corresponding formal
eigenvalue. At this stage, $\varphi_{\lambda}(x;p)$ may depend on the constant
$c\in\mathbb{C}^{\ast}\backslash S$ that we started with, but it is not the
case as it will turn out later. For this reason, we are suppressing the
possible dependence on $c$ in the notation of $\varphi_{\lambda}(x;p)$. We
also remark that any formal solution of the eigenfunction equation (4.25) with
initial condition $\psi(x;0)=P_{\lambda}(x)$ is expressed as
$\psi(x;p)=\gamma(p)\varphi_{\lambda}(x;p)$ for some
$\gamma(p)\in\mathbb{C}[[p]]$ with $\gamma(0)=1$. In fact, express $\psi(x;p)$
as

$\psi(x;p)=\sum_{k=0}^{\infty}p^{k}\psi_{k}(x),\quad\psi_{k}(x)=\sum_{\mu\in
P_{+}}\psi_{k,\mu}P_{\mu}(x)\quad(k=0,1,2,\ldots)$ (4.35)

with $\psi_{0}(x)=P_{\lambda}(x)$ and $\psi_{0,\mu}=\delta_{\mu,\lambda}$. If
we set $\gamma(p)=\sum_{k=0}^{\infty}p^{k}\psi_{k,\lambda}$, it is easy to
show that $\gamma(p)^{-1}\psi(x;p)$ satisfies the normalization condition for
$\varphi_{\lambda}(x;p)$.

We next show that the formal solution $\varphi_{\lambda}(x;p)$ is in fact a
joint eigenfunction for the elliptic Ruijsenaars operators
$\mathcal{D}^{(r)}_{x}(p)$ ($r=1,\ldots,n$) acting on $\mathcal{R}[[p]]$.
Since each $\mathcal{D}_{x}^{(r)}(p)$ commute with $\mathcal{D}_{x}(p;c)$, we
have

$\mathcal{D}_{x}(p;c)\mathcal{D}_{x}^{(r)}(p)\varphi_{\lambda}(x;p)=\mathcal{D}_{x}^{(r)}(p)\mathcal{D}_{x}(p;c)\varphi_{\lambda}(x;p)=\varepsilon_{\lambda}(p;c)\mathcal{D}^{(r)}_{x}(p)\varphi_{\lambda}(x;p).$
(4.36)

This means that $\mathcal{D}^{(r)}_{x}(p)\varphi_{\lambda}(x;p)$ is a formal
solution of the eigenfunction equation (4.25) with initial condition

$\mathcal{D}^{(r)}_{x}(p)\varphi_{\lambda}(x;p)\big{|}_{p=0}=D^{(r)}_{x}P_{\lambda}(x)=e_{r}(t^{\rho}q^{\lambda})P_{\lambda}(x).$
(4.37)

Hence, $\mathcal{D}^{(r)}_{x}(p)\varphi_{\lambda}(x;p)$ is expressed as
$\varepsilon^{(r)}_{\lambda}(p)\varphi_{\lambda}(x;p)$ with some
$\varepsilon^{(r)}_{\lambda}(p)\in\mathbb{C}[[p]]$ such that
$\varepsilon^{(r)}_{\lambda}(0)=e_{r}(t^{\rho}q^{\lambda})$. Namely, we have

$\mathcal{D}^{(r)}_{x}(p)\varphi_{\lambda}(x;p)=\varepsilon^{(r)}_{\lambda}(p)\varphi_{\lambda}(x;p),\quad\varepsilon^{(r)}_{\lambda}(0)=e_{r}(t^{\rho}q^{\lambda})\quad(r=0,1,\ldots,n),$
(4.38)

as desired. In particular, we have

$\mathcal{D}_{x}(p;u)\varphi_{\lambda}(x;p)=\varepsilon_{\lambda}(p;u)\varphi_{\lambda}(x;p),\quad\varepsilon_{\lambda}(p;u)=\sum_{r=0}^{n}(-1)^{r}\varepsilon^{(r)}_{\lambda}(p)$
(4.39)

for arbitrary $u\in\mathbb{C}$. This implies also that
$\varphi_{\lambda}(x;p)$ does not depend on the constant
$c\in\mathbb{C}^{\ast}\backslash S$ that we have chosen for the construction
of $\varphi_{\lambda}(x;p)$. This completes the proof of Theorem 2.2.

###### Remark 4.3

If we regard $q$ and $t$ as indeterminates, the formal solution
$\varphi_{\lambda}(x;p)$ mentioned above, as well as the normalized formal
solution in the sense of (2.21), is uniquely determined as a formal power
series of $p$ with coefficients in $\mathbb{C}(q,t)[x^{\pm
1}]^{\mathfrak{S}_{n}}$.

###  4.4 Asymptotically free eigenfunctions in the domain
$|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$

In this subsection, we complete our proof of Theorem 2.6.

As we remarked in Subsection 4.2, the coefficients $\mathcal{A}_{I}(x;p)$ of
the elliptic Ruijsenaars operators define formal power series in

$\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]],$ (4.40)

which is the set of all formal power series of the form

$\varphi(x;p)=\sum_{l_{0},l_{1},\ldots,l_{n-1}\in\mathbb{Z}_{\geq
0}}\varphi_{l_{0},l_{1},\dots,l_{n-1}}(px_{1}/x_{n})^{l_{0}}(x_{2}/x_{1})^{l_{1}}\cdots(x_{n}/x_{n-1})^{l_{n-1}}.$
(4.41)

In order to simplify the presentation, we use below the notation of monomials
attached to the root lattice $\mathsf{Q}^{\mathrm{af}}$ of the affine Lie
algebra $\widehat{\mathfrak{gl}}_{n}$ [9]. Adding the null root $\delta$ to
the basis, we extend the weight lattice $\mathsf{P}$ of $\mathfrak{gl}_{n}$ to

$\widetilde{\mathsf{P}}=\mathbb{Z}\delta\oplus\mathsf{P}=\mathbb{Z}\delta\oplus\mathbb{Z}\epsilon_{1}\oplus\cdots\oplus\mathbb{Z}\epsilon_{n},$
(4.42)

together with the scalar product
$\widetilde{\mathsf{P}}\times\widetilde{\mathsf{P}}\to\mathbb{Z}$ extended by
$\left\langle\mu,\delta\right\rangle=\left\langle\delta,\mu\right\rangle=0$
($\mu\in\mathsf{P}$) and $\left\langle\delta,\delta\right\rangle=0$. For each
element $\lambda=k\delta+\mu\in\widetilde{\mathsf{P}}$ $(k\in\mathbb{Z},\
\mu\in\mathsf{P})$, we attach the monomial

$x^{\lambda}=p^{-k}x^{\mu}=p^{-k}x_{1}^{\mu_{1}}\cdots x_{n}^{\mu_{n}}$ (4.43)

with the convention $x^{\delta}=p^{-1}$. Introducing the 0th simple affine
root $\alpha_{0}=\delta-\phi=\delta-\epsilon_{1}+\epsilon_{n}$, we denote by

$\mathsf{Q}^{{\mathrm{af}}}=\mathbb{Z}\alpha_{0}\oplus\mathbb{Z}\alpha_{1}\oplus\cdots\oplus\mathbb{Z}\alpha_{n-1}=\mathbb{Z}\delta\oplus\mathsf{Q}\subseteq\widetilde{\mathsf{P}}$
(4.44)

the root lattice of the affine Lie algebra $\widehat{\mathfrak{gl}}_{n}$. Note
here that $\alpha_{0}+\alpha_{1}+\cdots+\alpha_{n-1}=\delta$. We also set
$\mathsf{Q}^{\mathrm{af}}_{+}=\mathbb{Z}_{\geq
0}\alpha_{0}\oplus\mathbb{Z}_{\geq
0}\alpha_{1}\oplus\cdots\oplus\mathbb{Z}_{\geq 0}\alpha_{n-1}.$ Noting that

$x^{-\alpha_{0}}=px_{1}/x_{n},\ \ x^{-\alpha_{1}}=x_{2}/x_{1},\ \ \ldots,\ \
x^{-\alpha_{n-1}}=x_{n}/x_{n-1},$ (4.45)

we rewrite the monomials in
$\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]]$ as

$(px_{1}/x_{n})^{l_{0}}(x_{2}/x_{1})^{l_{1}}\cdots(x_{n}/x_{n-1})^{l_{n-1}}=x^{-\beta},\quad\beta=l_{0}\alpha_{0}+l_{1}\alpha_{1}+\cdots+l_{n-1}\alpha_{n-1},$
(4.46)

and call

$\mathrm{ht}(\beta)=l_{0}+l_{1}+\cdots+l_{n-1}\in\mathbb{Z}_{\geq 0}$ (4.47)

the height of $\beta\in\mathsf{Q}^{\mathrm{af}}_{+}$. In this notation, the
formal power series in
$\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]]$ are simply
expressed as

$\varphi(x;p)=\sum_{\beta\in\mathsf{Q}^{\mathrm{af}}_{+}}\varphi_{\beta}x^{-\beta},\quad\varphi_{\beta}\in\mathbb{C}\
\ (\beta\in\mathsf{Q}^{\mathrm{af}}_{+}).$ (4.48)

We recall the definition of the modified elliptic Ruijsenaars operators
$\mathcal{E}^{(r)}_{x,s}(p)$ that we introduced in Subsection 2.3:

$\mathcal{E}^{(r)}_{x,s}(p)=\sum_{I\subseteq\left\\{1,\ldots,n\right\\};\
|I|=r}s^{\epsilon_{I}}\mathcal{B}_{I}(x;p)T_{q,x}^{\epsilon_{I}}\qquad(r=0,1,\ldots,n),$
(4.49)

where

$\mathcal{B}_{I}(x;p)=\prod_{\begin{subarray}{c}1\leq i<j\leq n\\\ i\in
I;\,j\notin
I\end{subarray}}\frac{\theta(x_{j}/tx_{i};p)}{\theta(x_{j}/x_{i};p)}\prod_{\begin{subarray}{c}1\leq
i<j\leq n\\\ i\notin I;\,j\in
I\end{subarray}}\frac{\theta(tx_{j}/x_{i};p)}{\theta(x_{j}/x_{i};p)}\qquad(I\subseteq\left\\{1,\ldots,n\right\\}).$
(4.50)

Note that the coefficients $\mathcal{B}_{I}(x;p)$ are expressed as

$\mathcal{B}_{I}(x;p)=\sum_{\beta\in\mathsf{Q}^{{\mathrm{af}}}_{+}}x^{-\beta}\,b^{I}_{\beta}\in\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]],\quad
b^{I}_{0}=1,$ (4.51)

through the expansion in the domain $|px_{1}/x_{n}|<1$, $|x_{2}/x_{1}|<1$,
$\ldots$, $|x_{n}/x_{n-1}|<1$.

In what follows, we regard $s=(s_{1},\ldots,s_{n})\in(\mathbb{C}^{\ast})^{n}$
as fixed constants, supposing that the genericity condition $s_{j}/s_{i}\notin
q^{\mathbb{Z}}$ ($1\leq i<j\leq n$) is satisfied. In this setting, we
investigate the joint eigenvalue problem

$\mathcal{E}^{(r)}_{x,s}(p)\varphi(x;p)=\varepsilon^{(r)}(p)\varphi(x;p),\quad\varepsilon^{(r)}(0)=e_{r}(s)\qquad(r=1,\ldots,n)$
(4.52)

for a formal power series

$\varphi(x;p)=\sum_{\mu\in\beta\in\mathsf{Q}^{\mathrm{af}}_{+}}\varphi_{\beta}x^{-\beta}\in\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]]$
(4.53)

with leading term $\varphi_{0}=1$. Here we have suppressed the dependence on
$s$ in the notation of $\varphi(x;p)$ and $\varepsilon^{(r)}(p)$. We also use
the generation functions

$\mathcal{E}_{x,s}(p;u)=\sum_{r=0}^{n}(-u)^{r}\mathcal{E}^{(r)}_{x,s}(p),\quad\varepsilon(p;u)=\sum_{r=0}^{n}(-u)^{r}\varepsilon^{(r)}(p)$
(4.54)

and consider the eigenfunction equation

$\mathcal{E}_{x,s}(p;u)\varphi(x;p)=\varepsilon(p;u)\varphi(x;p),\quad\varepsilon(0;u)=\prod_{i=1}^{n}(1-s_{i}u),$
(4.55)

with a parameter $u$.

In view of the expansion (4.51), we express the operator
$\mathcal{E}_{x,s}(p;u)$ as follows:

$\begin{split}\mathcal{E}_{x,s}(p;u)&=\sum_{\beta\in\mathsf{Q}_{+}^{\mathrm{af}}}x^{-\beta}\,b_{\beta}(T_{q,x};u),\quad
b_{\beta}(\xi;u)=\sum_{I\subseteq\left\\{1,\ldots,n\right\\}}(-u)^{|I|}b^{I}_{\beta}s^{\epsilon_{I}}\xi^{\epsilon_{I}},\end{split}$
(4.56)

with symbols $\xi=(\xi_{1},\ldots,\xi_{n})$ attached to
$T_{q,x}=(T_{q,x_{1}},\ldots,T_{q,x_{n}})$. Since $b^{I}_{0}=1$, we have

$b_{0}(\xi;u)=\prod_{j=1}^{n}(1-us_{j}\xi_{j}).$ (4.57)

###### Lemma 4.4

Suppose that $s_{j}/s_{i}\notin q^{\mathbb{Z}^{\ast}}$ $(1\leq i<j\leq n)$,
where $\mathbb{Z}^{\ast}=\mathbb{Z}\backslash\left\\{0\right\\}$. Then, there
exists a countable set $S\subseteq\mathbb{C}^{\ast}$ such that, if
$c\in\mathbb{C}^{\ast}\backslash S$,

$\prod_{j=1}^{n}(1-cs_{j}q^{m_{j}})\neq\prod_{j=1}^{n}(1-cs_{j})$ (4.58)

for any nonzero $m=(m_{1},\ldots,m_{n})\in\mathbb{Z}^{n}$.

Proof : For each $m\in\mathbb{Z}^{n}\backslash\left\\{0\right\\}$,
$g_{m}(u)=\prod_{j=1}^{m}(1-us_{j}q^{m_{j}})-\prod_{j=1}^{m}(1-us_{j})$ is a
nonzero polynomial in $u$ by the assumption $s_{j}/s_{i}\notin
q^{\mathbb{Z}^{\ast}}$ ($1\leq i<j\leq n)$. (In fact, suppose that there
exists a nonzero $m\in\mathbb{Z}^{n}$ and a permutation
$\sigma\in\mathfrak{S}_{n}$ such $s_{j}q^{m_{j}}=s_{\sigma(j)}$ for
$j=1,\ldots,m$. Then, under our assumption that $|q|<1$, for each index
$i\in\left\\{1,\ldots,n\right\\}$ such that $m_{i}\neq 0$, we have
$\sigma(i)\neq i$ and $s_{\sigma(i)}/s_{i}\in q^{\mathbb{Z}^{\ast}}$, leading
to a contradiction.) Then, the set

$S=\left\\{{a\in\mathbb{C}^{\ast}}\big{|}{\ g_{m}(a)=0\ \mbox{for some}\
m\in\mathbb{Z}^{m}\backslash\left\\{0\right\\}}\right\\}$ (4.59)

is countable, and fulfills the condition of Lemma. $\square$

Fixing a generic constant $c\in\mathbb{C}^{\ast}\backslash S$ in this sense,
we investigate the eigenfunction equation

$\mathcal{E}_{x,s}(p;c)f(x;p)=\varepsilon(p;c)f(x;p),\quad\varepsilon(p;c)=\sum_{k=0}^{\infty}p^{k}\varepsilon_{k}(c)=\sum_{k=0}^{\infty}x^{-k\delta}\varepsilon_{k}(c),$
(4.60)

where

$f(x;p)=\sum_{\mu\in\mathsf{Q}^{{\mathrm{af}}}_{+}}x^{-\mu}\,f_{\mu}\in\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]].$
(4.61)

We have suppressed the dependence of $f(x;p)$ and $\varepsilon(p;u)$ on the
$s$ parameters in order to simplify the notation. The eigenfunction equation
(4.60) is then equivalent to the recurrence relations

$\sum_{\beta+\nu=\mu}b_{\beta}(q^{-\nu};c)f_{\nu}=\sum_{k\delta+\nu=\mu}\varepsilon_{k}(c)f_{\nu}\qquad(\mu\in\mathsf{Q}^{{\mathrm{af}}}_{+})$
(4.62)

for the coefficients of $f(x;p)$. In view of condition

$b_{0}(1;c)f_{0}=\varepsilon_{0}(c)f_{0},$ (4.63)

of the case $\mu=0$, we start with

$f_{0}=1,\quad\varepsilon_{0}(c)=b_{0}(1;c)=\prod_{j=1}^{n}(1-cs_{j}).$ (4.64)

For $\mu>0$ (i.e. $\mu\in\mathsf{Q}^{\mathrm{af}}_{+}$, $\mu\neq 0$), we have

$(b_{0}(q^{-\mu};c)-\varepsilon_{0}(c))f_{\mu}+\sum_{\beta+\nu=\mu;\,\beta>0}b_{\beta}(q^{-\nu};c)f_{\nu}=\sum_{k\delta+\nu=\mu;\,k>0}\varepsilon_{k}(c)f_{\nu}.$
(4.65)

Note here that, if
$\mu=k_{0}\alpha_{0}+k_{1}\alpha_{1}+\cdots+k_{n-1}\alpha_{n-1}\in\mathsf{Q}^{\mathrm{af}}_{+}$,
we have

$\left\langle\epsilon_{j},\mu\right\rangle=k_{j}-k_{j-1}\quad(j=1,\ldots,n),$
(4.66)

with the convention $k_{n}=k_{0}$, and hence

$b_{0}(q^{-\mu};c)=\prod_{j=1}^{n}(1-cs_{j}q^{-k_{j}+k_{j-1}}).$ (4.67)

If $\mu\notin\mathbb{Z}_{>0}\delta$, there exists an index
$i\in\left\\{1,\ldots,n\right\\}$ such that $k_{i}-k_{i-1}\neq 0$, and hence

$\prod_{j=1}^{n}(1-cs_{j}q^{-k_{j}+k_{j-1}})\neq\prod_{j=1}^{n}(1-cs_{j})$
(4.68)

by the genericity assumption of $c$. Since
$b_{0}(q^{-\mu};c)\neq\varepsilon_{0}(c)$, by the recurrence relation (4.65)
$f_{\mu}$ is determined uniquely from $f_{\nu}$ ($\nu<\mu$) and
$\varepsilon_{k}(c)$ with $\mathrm{ht}(k\delta)=kn<\mathrm{ht}(\mu)$.

When $\mu=l\delta$ ($l>0$), we have
$b_{0}(q^{-l\delta};u)=b_{0}(1;c)=\varepsilon_{0}(c)$. Hence the recurrence
relation is written as

$\sum_{\beta+\nu=l\delta;\,\beta>0}b_{\beta}(q^{-\nu};c)f_{\nu}=\varepsilon_{l}(c)+\sum_{k\delta+\nu=l\delta;\,0<k<l}\varepsilon_{k}(c)f_{\nu}.$
(4.69)

This implies that $\varepsilon_{l}(c)$ is determined uniquely from $f_{\nu}$
($\nu<l\delta$) and $\varepsilon_{k}(c)$ ($k<l$), and $f_{l\delta}$ can be
fixed arbitrarily. Since this ambiguity of $f_{k\delta}$ ($k>0$) corresponds
to the multiplication by an element $\gamma(p)\in\mathbb{C}[[p]]$ with
$\gamma(0)=1$, we adopt here the normalization

$f_{0}=1,\quad f_{k\delta}=0\quad\mbox{for all}\quad k>0.$ (4.70)

Under this normalization, the recurrence relation for $\varepsilon_{l}(u)$
$(l>0)$ simplifies as

$\varepsilon_{l}(c)=\sum_{\beta+\nu=l\delta;\,\beta>0}b_{\beta}(q^{-\nu};c)f_{\nu}.$
(4.71)

This procedure implies the unique existence of a normalized formal solution
$f(x;p)$. Also, one can verify that any eigenfunction is expressed as
$\gamma(p)f(x;p)$ with some $\gamma(p)\in\mathbb{C}[[p]]$, by a similar
argument to the one we used in Subsection 4.3.

We now prove that $f(x;p)$ is a joint eigenfunction for all
$\mathcal{E}_{x,s}^{(r)}(p)$ ($r=0,1,\ldots,n$). Since
$\mathcal{E}_{x,s}^{(r)}(p)$ commutes with $\mathcal{E}_{x,s}(p;c)$, we have

$\mathcal{E}_{x,s}(p;c)\mathcal{E}_{x,s}^{(r)}(p)f(x;p)=\mathcal{E}_{x,s}^{(r)}(p)\mathcal{E}_{x,s}(p;c)f(x;p)=\varepsilon(p;c)\mathcal{E}_{x,s}^{(r)}(p)f(x;p).$
(4.72)

This means that $g_{r}(x;p)=\mathcal{E}_{x,s}^{(r)}(p)f(x;p)$ is a formal
eigenfunction of $\mathcal{E}_{x,s}(p;c)$, and hence
$g_{r}(x;p)=\varepsilon^{(r)}(p)f(x;p)$ for some
$\varepsilon^{(r)}(p)\in\mathbb{C}[[p]]$, namely,
$\mathcal{E}_{x,s}^{(r)}(p)f(x;p)=\varepsilon^{(r)}(p)f(x;p)$ for
$r=0,1,\ldots,n$. This also implies that the normalized formal solution
$f(x;p)$ satisfies

$\mathcal{E}_{x,s}(p;u)f(x;p)=\varepsilon(p;u)f(x;p),\quad\varepsilon(p;u)=\sum_{r=0}^{n}(-u)^{r}\varepsilon^{(r)}(p)$
(4.73)

for arbitrary $u\in\mathbb{C}$. In particular, $f(x;p)$ does not depend on the
choice of $c\in\mathbb{C}^{\ast}\backslash S$ that we started with. This
completes the proof of Theorem 2.6.

###### Remark 4.5

If we regard as $q$ and $t$ as indeterminates, the coefficients $f_{\beta}$
$(\beta\in\mathsf{Q}^{\mathrm{af}}_{+}$) of the normalized formal solution
$f(x;p)$ are determined uniquely as rational functions in $(q,t)$.

##  5 Convergence of perturbative eigenfunctions

In this section, we first prove the convergence of the formal joint
eigenfunctions $f(x;s;p)$ of the modified elliptic Ruijsenaars operators in
the asymptotic domain $|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$, and then apply
an integral transform to them for establishing the convergence of symmetric
formal joint eigenfunctions $\mathcal{P}_{\lambda}(x;p)$
$(\lambda\in\mathsf{P}_{+})$ of the elliptic Ruijsenaars operators around of
the torus $\mathbb{T}^{n}$. We also include a proof of Theorem 2.4 on the
orthogonality of elliptic deformations $\mathcal{P}_{\lambda}(x;p)$ of
Macdonald polynomials.

###  5.1 Ruijsenaars functions in the asymptotic domain
$|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$

In what follows, we fix the parameters
$s=(s_{1},\ldots,s_{n})\in(\mathbb{C}^{\ast})^{n}$, assuming that they satisfy
the genericity condition $s_{j}/s_{i}\notin q^{\mathbb{Z}}$ ($1\leq i<j\leq
n$). Keeping the notation of Subsection 4.4, we denote by $f(x;p)$ of the
normalized formal solution of the eigenfunction equation

$\mathcal{E}_{x,s}(p;u)f(x;p)=\varepsilon(p;u)f(x;p),\quad
f(x;p)\in\mathbb{C}[[px_{1}/x_{n},x_{2}/x_{1},\ldots,x_{n}/x_{n-1}]]$ (5.1)

in the asymptotic domain $|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$. (We write
$f(x;p)=f(x;s;p)$ and $\varepsilon(p;u)=\varepsilon(s;p;u)$ when we need to
make the dependence on $s$ manifest.) As before, we expand the modified
elliptic Ruijsenaars operator in the form

$\mathcal{E}_{x,s}(p;u)=\sum_{I\subseteq\left\\{1,\ldots,n\right\\}}(-u)^{|I|}s^{\epsilon_{I}}\mathcal{B}_{I}(x;p)T_{q,x}^{\epsilon_{I}}=\sum_{\beta\in\mathsf{Q}^{\mathrm{af}}_{+}}x^{-\beta}b_{\beta}(T_{q,x};u),$
(5.2)

where $b_{\beta}(\xi;u)$ are the symbols of $q$-difference operators defined
by (4.56). Then the formal power series

$f(x;p)=\sum_{\mu\in\mathsf{Q}^{\mathrm{af}}_{+}}x^{-\mu}f_{\mu},\quad\varepsilon(p;u)=\sum_{k=0}^{\infty}p^{k}\varepsilon_{k}(u)$
(5.3)

satisfy the initial condition

$f_{0}=1,\quad\varepsilon_{0}(u)=b_{0}(1;u)=\prod_{j=1}^{n}(1-us_{i})$ (5.4)

and the recurrence formulas

$(b_{0}(q^{-\mu};u)-\varepsilon_{0}(u))f_{\mu}=-\sum_{\beta+\nu=\mu;\,\beta>0}b_{\beta}(q^{-\nu};u)f_{\nu}+\sum_{k\delta+\nu=\mu;\,k>0}\varepsilon_{k}(u)f_{\nu}\quad(\mu\in\mathsf{Q}^{{\mathrm{af}}}_{+}\backslash\mathbb{Z}\delta),$
(5.5)

and

$f_{l\delta}=0,\quad\varepsilon_{l}(u)=\sum_{\beta+\nu=l\delta;\,\beta>0}b_{\beta}(q^{-\nu};u)f_{\nu}\qquad(l=1,2,\ldots).$
(5.6)

Our goal is to establish estimates for the coefficients $f_{\mu}$ and
$\varepsilon_{k}(u)$ satisfying these recurrence formulas, so as to guarantee
the absolute convergence of $f(x;p)$ and $\varepsilon(p;u)$. As far as the
condition $s_{j}/s_{i}\notin q^{\mathbb{Z}}$ ($1\leq i<j\leq n$) is satisfied,
our estimates for $f_{\mu}$ will be uniform on every compact subset with
respect to the parameters $(s;q,t)$.

Setting $U=\left\\{{q\in\mathbb{C}^{\ast}}\big{|}{\ |q|<1}\right\\}$, we
consider the following subset $D$ of the space of parameters
$(s;q)=(s_{1},\ldots,s_{n};q)\in(\mathbb{C}^{\ast})^{n}\times U$:

$D=\left\\{{(s;q)\in(\mathbb{C}^{\ast})^{n}\times U}\big{|}{\
1-q^{l}s_{j}/s_{i}\neq 0\ \ \mbox{for any distinct
$i,j\in\left\\{1,\ldots,n\right\\}$ and $l\in\mathbb{Z}$}}\right\\}.$ (5.7)

Note that $D$ is a locally finite intersection of complements of
hypersurfaces, and hence an open subset of $(\mathbb{C}^{\ast})^{n}\times U$.

###### Lemma 5.1

For any compact subset $K\subseteq D$, there exist real constants
$\tau_{0}\in(0,1]$ and $\tau_{1}>0$ such that

$\tau_{0}\leq|s_{j}/s_{i}|\leq\tau_{0}^{-1},\quad|1-q^{l}s_{j}/s_{i}|\geq\tau_{1}\quad(l\in\mathbb{Z}_{\geq
0})$ (5.8)

for any distinct $i,j\in\left\\{1,\ldots,n\right\\}$.

Proof : We first take $\tau_{0}\in(0,1]$ and $\rho\in(0,1)$ so that
$\tau_{0}\leq|s_{j}/s_{i}|\leq\tau_{0}^{-1}$
$(i,j\in\left\\{1,\ldots,n\right\\};\ i\neq j)$ and $|q|\leq\rho$ for all
$(s;q)\in K$. Let $l_{0}\in\mathbb{Z}_{\geq 0}$ be sufficient large so that
$\rho^{l_{0}}\leq\tau_{0}$. Then, for $l>l_{0}$, we have
$|q^{l}s_{j}/s_{i}|\leq\rho^{l_{0}+1}\tau_{0}^{-1}\leq\rho$ hence
$|1-q^{l}s_{j}/s_{i}|\geq 1-\rho>0$ on $K$, for any distinct $i,j$. The
condition of Lemma is fulfilled if we let $\tau_{1}>0$ sufficiently small so
that $\tau_{1}\leq 1-\rho$ and $|1-q^{l}s_{j}/s_{i}|\geq\tau_{1}$ $(0\leq
l\leq l_{0})$ on $K$, for any distinct $i,j$. $\square$

From now on, we assume that the parameters $(s;q,t)$ belong to an arbitrary
compact subset of $\in D\times\mathbb{C}^{\ast}$, and fix two real constants
$\tau_{0}\in(0,1]$ and $\tau_{1}>0$ as in Lemma 5.1.

As we already remarked in Subsection 2.2, the coefficients
$\mathcal{B}_{I}(x;p)$ of $\mathcal{E}_{x,s}(p;u)$ are holomorphic functions
in the variables $z=(z_{0},z_{1},\ldots,z_{n-1})$, defined by

$z_{0}=px_{1}/x_{n},\ z_{1}=x_{2}/x_{1},\ \ldots,\ z_{n-1}=x_{n}/x_{n-1},$
(5.9)

in the polydisc $|z_{0}|<1$, $|z_{1}|<1$, $\ldots$, $|z_{n-1}|<1$. Hence,
their Taylor expansions

$\mathcal{B}_{I}(x;p)=\sum_{k_{0},k_{1},\ldots,k_{n-1}\geq
0}z_{0}^{k_{0}}z_{1}^{k_{1}}\ldots
z_{n-1}^{k_{n-1}}\,b^{I}_{k_{0},k_{1},\ldots,k_{n-1}}\in\mathbb{C}[[z_{0},z_{1},\ldots,z_{n-1}]]$
(5.10)

satisfy the Cauchy estimate: For each $r\in(0,1)$, there exists a constant
$M_{r}>0$ such that

$|b^{I}_{k_{0},k_{1},\ldots,k_{n-1}}|\leq\frac{M_{r}}{r^{k_{0}+k_{1}+\cdots+k_{n-1}}}\qquad(I\subseteq\left\\{1,\ldots,n\right\\};k_{0},k_{1},\ldots,k_{n-1}\geq
0).$ (5.11)

By the identification
$\beta=k_{0}\alpha_{0}+k_{1}\alpha_{1}+\cdots+k_{n-1}\alpha_{n-1}$, the Cauchy
estimate can be rewritten as

$|b^{I}_{\beta}|\leq\frac{M_{r}}{r^{\mathrm{ht}(\beta)}}\qquad(I\subseteq\left\\{1,\ldots,n\right\\};\beta\in\mathsf{Q}^{{\mathrm{af}}}_{+}),$
(5.12)

where $\mathrm{ht}(\beta)=k_{0}+k_{1}+\cdots+k_{n-1}$ denotes the height of
$\beta\in\mathsf{Q}^{{\mathrm{af}}}_{+}$.

This Cauchy estimate (5.12) implies the estimate

$|b_{\beta}(\xi;u)|\leq\frac{M_{r}}{r^{\mathrm{ht}(\beta)}}\sum_{I\subseteq\left\\{1,\ldots,n\right\\}}|u||s^{\epsilon_{I}}||\xi^{\epsilon_{I}}|=\frac{M_{r}}{r^{\mathrm{ht}(\beta)}}\prod_{j=1}^{n}(1+|u||s_{j}||\xi_{j}|)$
(5.13)

for the coefficients $b_{\beta}(\xi;u)$
$(\beta\in\mathsf{Q}^{{\mathrm{af}}}_{+})$ of (4.56) appearing in the
recurrence relations. If
$\nu=l_{0}\alpha_{0}+l_{1}\alpha_{1}+\cdots+l_{n-1}\alpha_{n-1}\in\mathsf{Q}_{+}^{{\mathrm{af}}}$,
we have

$|b_{\beta}(q^{-\nu};u)|\leq\frac{M_{r}}{r^{\mathrm{ht}(\beta)}}\prod_{j=1}^{n}(1+|u||s_{j}||q|^{-l_{j}+l_{j-1}}),$
(5.14)

with $l_{n}=l_{0}$ as before. With the notation
$|x|_{+}=\max\left\\{x,0\right\\}$, we obtain

$\begin{split}\prod_{j=1}^{n}(1+|u||s_{j}||q|^{-l_{j}+l_{j-1}})&\leq\prod_{j=1}^{n}(1+|u||s_{j}||q|^{-|l_{j}-l_{j-1}|_{+}})\\\
&\leq|q|^{-\sum_{j=1}^{n}|l_{j}-l_{j-1}|_{+}}\prod_{i=1}^{n}(1+|u||s_{j}|).\end{split}$
(5.15)

In view of this estimate, we define

$\mathrm{d}(\nu)=\sum_{j=1}^{n}|l_{j}-l_{j-1}|_{+}=\sum_{1\leq j\leq n;\
l_{j}>l_{j-1}}(l_{j}-l_{j-1})\in\mathbb{Z}_{\geq 0}.$ (5.16)

Note also that $\sum_{j=1}^{n}|l_{j}-l_{j-1}|=2\mathrm{d}(\nu)$, since
$\sum_{j=1}^{n}(l_{j}-l_{j-1})=0$. Then we have

###### Lemma 5.2

For any $\beta\in\mathsf{Q}_{+}^{{\mathrm{af}}}$ with $\beta>0$,

$|b_{\beta}(q^{-\nu};u)|\leq\frac{M_{r}}{r^{\mathrm{ht}(\beta)}|q|^{\mathrm{d}(\nu)}}\prod_{j=1}^{n}(1+|u||s_{j}|)\qquad(\nu\in\mathsf{Q}_{+}^{\mathrm{af}}).\vspace{-12pt}$
(5.17)

$\square$

In estimating the leading factor in the recurrence relation (5.5),

$b_{0}(q^{-\mu};u)-\varepsilon_{0}(u)=\prod_{j=1}^{n}(1-us_{j}q^{-k_{j}+k_{j-1}})-\prod_{j=1}^{n}(1-us_{j})$
(5.18)

for $\mu\in\mathsf{Q}^{{\mathrm{af}}}_{+}\backslash\mathbb{Z}\delta$ with
$\mu=\sum_{j=0}^{n-1}k_{j}\alpha_{j}$, we choose an index
$i\in\left\\{1,\ldots,n\right\\}$ such that
$\left\langle\epsilon_{i},\mu\right\rangle=k_{i}-k_{i-1}\neq 0$, and make the
substitution $u=s_{i}^{-1}$ to obtain

$b_{0}(q^{-\mu};s_{i}^{-1})-\varepsilon_{0}(s_{i}^{-1})=b_{0}(q^{-\mu};s_{i}^{-1})=(1-q^{-k_{i}+k_{i-1}})\prod_{j\neq
i}(1-q^{-k_{j}+k_{j-1}}s_{j}/s_{i}).$ (5.19)

###### Lemma 5.3

With the real constants $\tau_{0},\tau_{1}$ as in (5.8), we have

$|b_{0}(q^{-\mu};s_{i}^{-1})|\geq\tau|q|^{-d(\mu)},\quad\tau=|1-q|(\tau_{0}\tau_{1})^{n-1},$
(5.20)

for any $\mu\in\mathsf{Q}^{\mathrm{af}}_{+}\backslash\mathbb{Z}\delta$, and
for any $i\in\left\\{1,\ldots,n\right\\}$ such that
$\left\langle\epsilon_{i},\mu\right\rangle\neq 0$.

Proof : For $l\in\mathbb{Z}$, we have

$|1-q^{-l}|\geq|1-q|\quad(l<0),\qquad|1-q^{-l}|\geq|q|^{-l}|1-q|\quad(l>0),$
(5.21)

and hence

$|1-q^{-l}|\geq|q|^{-|l|_{+}}|1-q|\quad(l\in\mathbb{Z},\ l\neq 0).$ (5.22)

Similarly, for any distinct $i,j\in\left\\{1,\ldots,n\right\\}$ we have

$\begin{split}&|1-q^{-l}s_{j}/s_{i}|\geq\tau_{1}\quad(l\leq 0),\quad\\\
&|1-q^{-l}s_{j}/s_{i}|=|q|^{-l}|s_{j}/s_{i}||1-q^{l}s_{i}/s_{j}|\geq|q|^{-l}\tau_{0}\tau_{1}\quad(l\geq
0),\end{split}$ (5.23)

and hence

$|1-q^{-l}s_{j}/s_{i}|\geq|q|^{-|l|_{+}}\tau_{0}\tau_{1}\quad(l\in\mathbb{Z}).$
(5.24)

These inequalities imply

$\begin{split}|b_{0}(q^{-\mu};s_{i}^{-1})|&\geq|q|^{-|k_{i}-k_{i-1}|_{+}}|1-q|\prod_{j\neq
i}\big{(}|q|^{-|k_{j}-k_{j-1}|}\tau_{0}\tau_{1}\big{)}\\\
&\geq|q|^{-\mathrm{d}(\mu)}|1-q|(\tau_{0}\tau_{1})^{n-1}\end{split}$ (5.25)

for any $i\in\left\\{1,\ldots,n\right\\}$ such that
$\left\langle\epsilon_{i},\mu\right\rangle=k_{i}-k_{i-1}\neq 0$. $\square$

We give a proof of convergence of $f(x;p)$ on the basis of Lemma 5.2 and Lemma
5.3. In view of the recurrence relation, for each
$\mu\in\mathsf{Q}^{{\mathrm{af}}}_{+}$ we set

$g_{\mu}(u)=\sum_{\beta+\nu=\mu;\,\beta>0}b_{\beta}(q^{-\nu};u)f_{\nu},$
(5.26)

so that $\varepsilon_{l}(u)=g_{l\delta}(u)$ ($l=1,2,\ldots$) and

$(b_{0}(q^{-\mu};u)-\varepsilon_{0}(u))f_{\mu}=-g_{\mu}(u)+\sum_{k\delta+\nu=\mu;\,k>0}g_{k\delta}(u)f_{\nu}.\quad$
(5.27)

We apply Lemma 5.2 to the estimate of $g_{\mu}(s_{i}^{-1})$ ($i=1,\ldots,n$).
By Lemma 5.2 we have

$|b_{\beta}(q^{-\nu};s_{i}^{-1})|\leq\frac{M_{r}}{r^{\mathrm{ht}(\beta)}|q|^{\mathrm{d}(\nu)}}\prod_{j=1}^{n}(1+|s_{j}/s_{i}|)\quad(i=1,\ldots,n)$
(5.28)

for any $\beta>0$ and $\nu\in\mathsf{Q}^{{\mathrm{af}}}_{+}$. With the
constant $\tau_{0}$ of (5.8), we have $\prod_{j=1}^{n}(1+|s_{j}/s_{i}|)\leq
2(1+\tau_{0}^{-1})^{n-1}$, and hence

$|b_{\beta}(q^{-\nu};s_{i}^{-1})|\leq\frac{M_{r}L}{r^{\mathrm{ht}(\beta)}|q|^{\mathrm{d}(\nu)}}\quad(i=1,\ldots,n),\quad
L=2(1+\tau_{0}^{-1})^{n-1}.$ (5.29)

We apply this estimate to $g_{\mu}(u)$ to obtain

$|g_{\mu}(s_{i}^{-1})|\leq\sum_{\beta+\nu=\mu;\,\beta>0}\frac{M_{r}L}{r^{\mathrm{ht}(\beta)}|q|^{\mathrm{d}(\nu)}}|f_{\nu}|,$
(5.30)

or equivalently,

$r^{\mathrm{ht}(\mu)}|g_{\mu}(s_{i}^{-1})|\leq
M_{r}L\sum_{\beta+\nu=\mu;\,\beta>0}\frac{r^{\mathrm{ht}(\nu)}}{|q|^{\mathrm{d}(\nu)}}|f_{\nu}|\quad(i=1,\ldots,n).$
(5.31)

Next we apply Lemmas 5.2 and 5.3 to the estimate of $f_{\mu}$. As to the
recurrence relation for $\mu>0$ with $\mu\notin\mathbb{Z}_{>0}\delta$, we
choose an index $i\in\left\\{1,\ldots,n\right\\}$ with
$\left\langle\epsilon_{i},\mu\right\rangle\neq 0$, and specialize it by the
substitution $u=s_{i}^{-1}$ to obtain

$b_{0}(q^{-\mu};s_{i}^{-1})f_{\mu}=-g_{\mu}(s_{i}^{-1})+\sum_{k\delta+\nu=\mu;\,k>0}g_{k\delta}(s_{i}^{-1})f_{\nu}.$
(5.32)

Then, by Lemma 5.2 and Lemma 5.3, we have the estimate

$\tau|q|^{-\mathrm{d}(\mu)}|f_{\mu}|\leq|g_{\mu}(s_{i}^{-1})|+\sum_{k\delta+\nu=\mu;\,k>0}|g_{k\delta}(s_{i}^{-1})||f_{\nu}|,$
(5.33)

which we rewrite as

$\tau\frac{r^{\mathrm{ht}(\mu)}}{|q|^{\mathrm{d}(\mu)}}|f_{\mu}|\leq
r^{\mathrm{ht}(\mu)}|g_{\mu}(s_{i}^{-1})|+\sum_{k\delta+\nu=\mu;\,k>0}r^{\mathrm{ht}(k\delta)}|g_{k\delta}(s_{i}^{-1})|r^{\mathrm{ht}(\nu)}|f_{\nu}|.$
(5.34)

Having the two estimates (5.31) and (5.34), we propose to define two families
of positive constants $\mathsf{a}_{\mu}$ $(\mu>0)$ and $\mathsf{b}_{\mu}$
($\mu\geq 0$) (majorants) such that

$r^{\mathrm{ht}(\mu)}|g_{\mu}(s_{i}^{-1})|\leq\mathsf{a}_{\mu}\quad(\mu>0;\
i=1,\ldots,n),\qquad\frac{r^{\mathrm{ht}(\mu)}}{|q|^{\mathrm{d}(\mu)}}|f_{\mu}|\leq\mathsf{b}_{\mu}\quad(\mu\geq
0).$ (5.35)

We specify these constants by the recurrence relations

$\mathsf{a}_{\mu}=\mathsf{s}\sum_{\begin{subarray}{c}\beta+\nu=\mu\\\
\beta>0\end{subarray}}\mathsf{b}_{\nu},\qquad\mathsf{b}_{\mu}=\mathsf{t}\sum_{\beta+\nu=\mu;\,\beta>0}\mathsf{a}_{\beta}\mathsf{b}_{\nu}\qquad(\mu>0)$
(5.36)

with the initial condition $\mathsf{b}_{0}=1$ so that
$|f_{0}|=\mathsf{b}_{0}$, where $\mathsf{s}=M_{r}L$ and
$\mathsf{t}=\tau^{-1}$. We recursively verify that these constants fulfill the
estimates of (5.35). In fact, by (5.31) we have

$\begin{split}r^{\mathrm{ht}(\mu)}|g_{\mu}(s_{i}^{-1})|&\leq\mathsf{s}\sum_{\beta+\nu=\mu;\,\beta>0}\mathsf{b}_{\nu}=\mathsf{a}_{\mu}.\end{split}$
(5.37)

for $\mu>0$, and also, by (5.34) we obtain

$\begin{split}\frac{r^{\mathrm{ht}(\mu)}}{|q|^{\mathrm{d}(\mu)}}|f_{\mu}|&\leq\mathsf{t}\Big{(}\mathsf{a}_{\mu}+\sum_{k\delta+\nu=\mu;\,k>0}\mathsf{a}_{k\delta}|q|^{\mathrm{d}(\nu)}\mathsf{b}_{\nu}\Big{)}\leq\mathsf{t}\sum_{\beta+\nu=\mu;\,\beta>0}\mathsf{a}_{\beta}\mathsf{b}_{\nu}=\mathsf{b}_{\mu}\end{split}$
(5.38)

for $\mu>0$ with $\mu\notin\mathbb{Z}\delta$.

Finally we establish Cauchy estimates for $\mathsf{a}_{\mu}$ $(\mu>0)$ and
$\mathsf{b}_{\mu}$ $(\mu\geq 0)$ by means of the generating functions

$\mathsf{a}(z)=\sum_{\mu>0}\mathsf{a}_{\mu}z_{0}^{k_{0}}z_{1}^{k_{1}}\cdots
z_{n-1}^{k_{n-1}},\qquad\mathsf{b}(z)=\sum_{\mu\geq
0}\mathsf{b}_{\mu}z_{0}^{k_{0}}z_{1}^{k_{1}}\cdots z_{n-1}^{k_{n-1}},$ (5.39)

in $n$ variables $z=(z_{0},z_{1},\ldots,z_{n-1})$, with the parametrization
$\mu=\sum_{j=0}^{n-1}k_{j}\alpha_{j}\in\mathsf{Q}^{{\mathrm{af}}}_{+}$ as
before. Note that these formal power series satisfy $\mathsf{a}(0)=0$ and
$\mathsf{b}(0)=1$. The recurrence relations for $\mathsf{a}_{\mu}$ and
$\mathsf{b}_{\mu}$ are expressed as the algebraic equations

$\mathsf{a}(z)=\mathsf{s}\,\mathsf{c}(z)\mathsf{b}(z),\quad\mathsf{b}(z)-1=\mathsf{t}\,\mathsf{a}(z)\mathsf{b}(z)$
(5.40)

for $\mathsf{a}(z)$, $\mathsf{b}(z)$, where

$\mathsf{c}(z)=\prod_{i=0}^{n-1}(1-z_{i})^{-1}-1.$ (5.41)

Eliminating $\mathsf{b}(z)$ from (5.40), we obtain a quadratic equation

$\mathsf{t}\,\mathsf{a}(z)^{2}-\mathsf{a}(z)+\mathsf{s}\,\mathsf{c}(z)=0$
(5.42)

for $\mathsf{a}(z)$, and hence, by $\mathsf{a}(0)=0$ we obtain

$\mathsf{a}(z)=\frac{1}{2\,\mathsf{t}}\Big{(}1-\sqrt{1-4\,\mathsf{s}\mathsf{t}\,\mathsf{c}(z)}\Big{)}.$
(5.43)

Similarly, eliminating $\mathsf{a}(z)$ from (5.40), we obtain a quadratic
equation

$\mathsf{s}\mathsf{t}\,\mathsf{c}(z)\mathsf{b}(z)^{2}-\mathsf{b}(z)+1=0$
(5.44)

for $\mathsf{b}(z)$, and hence, by $\mathsf{b}(0)=1$ we obtain

$\mathsf{b}(z)=\frac{2}{1+\sqrt{1-4\,\mathsf{s}\mathsf{t}\,\mathsf{c}(z)}}.$
(5.45)

Note that $\mathsf{t}=\tau^{-1}$ and $\mathsf{s}\mathsf{t}=M_{r}L\tau^{-1}$.
Since $\mathsf{c}(z)$ is a holomorphic function in the polydisc
$\mathcal{U}=\left\\{\,|z_{i}|<1\ (i=0,\ldots,n-1)\right\\}$ with
$\mathsf{c}(0)=0$, the formal power series $\mathsf{a}(z)$ and $\mathsf{b}(z)$
in $z$ represent holomorphic functions on the domain
$\left\\{{z\in\mathcal{U}}\big{|}{\,|\mathsf{c}(z)|<1/4\,\mathsf{s}\mathsf{t}}\right\\}$
containing 0. By the Cauchy estimate, this implies that there exist positive
constants $\rho>0$ and $K>0$, depending on $r$ and $\tau_{0}$, $\tau_{1}$,
such that

$\mathsf{a}_{\mu}\leq\frac{K}{{\rho}^{\mathrm{ht}(\mu)}},\quad\mathsf{b}_{\mu}\leq\frac{K}{{\rho}^{\mathrm{ht}(\mu)}}\quad(\mu>0).$
(5.46)

Returning to (5.35), we obtain

$|g_{\mu}(s_{i}^{-1})|\leq\frac{K}{(r\rho)^{\mathrm{ht}(\mu)}},\quad|f_{\mu}|\leq|q|^{\mathrm{d}(\mu)}\frac{K}{(r\rho)^{\mathrm{ht}(\mu)}}\leq\frac{K}{(r\rho)^{\mathrm{ht}(\mu)}}$
(5.47)

for all $\mu>0$. This means that the formal power series

$f(x;p)=\sum_{\mu\in\mathsf{Q}^{{\mathrm{af}}}_{+}}x^{-\mu}f_{\mu}=\sum_{\mu\in\mathsf{Q}^{{\mathrm{af}}}_{+}}z_{0}^{k_{0}}z_{1}^{k_{1}}\cdots
z_{n-1}^{k_{n-1}}f_{\mu}$ (5.48)

in $z=(z_{0},z_{1},\ldots,z_{n-1})$ converges in the domain
$|z_{i}|<\sigma=r\rho$ ($i=0,1,\ldots,n-1$). Also, noting that
$\varepsilon_{k}(u)=g_{k\delta}(u)$ ($k=1,2,\ldots$), for each $i=1,\ldots,n$
we have

$|\varepsilon_{k}(s_{i}^{-1})|\leq\frac{K}{\sigma^{\mathrm{ht}(k\delta)}}=\frac{K}{(\sigma^{n})^{k}}\quad(k=1,2,\ldots).$
(5.49)

Hence
$\varepsilon(p;s_{i}^{-1})=\sum_{k=0}^{\infty}p^{k}\varepsilon_{k}(s_{i}^{-1})$
converges absolutely in the disc $|p|<\sigma^{n}$. Since
$\varepsilon(p;s_{i}^{-1})=\sum_{j=0}^{n}(-s_{i}^{-1})^{j}\varepsilon^{(j)}(p)$,
we have

$1-s_{i}^{-1}\varepsilon^{(1)}(p)+s_{i}^{-2}\varepsilon^{(2)}(p)+\cdots+(-s_{i}^{-1})^{n}\varepsilon^{(n)}(p)=\varepsilon(p;s_{i}^{-1})\quad(i=1,\ldots,n).$
(5.50)

Since $s_{i}\neq s_{j}$ ($1\leq i<j\leq n$), solving this system of linear
equations for $\varepsilon^{(i)}(p)$ ($j=1,\ldots,n$), we see that the formal
eigenvalues $\varepsilon^{(j)}(p)$ ($j=1,\ldots,n$) are also convergent in the
same disc $|p|<\sigma^{n}$. This completes the proof of Theorem 2.7.

When we make the dependence on $s=(s_{1},\ldots,s_{n})$ and $q,t$ explicit, we
write $f(x;s;p|q,t)$ for $f(x;p)$, under the condition that $s_{j}/s_{i}\notin
q^{\mathbb{Z}}$ $(1\leq i<j\leq n)$. We remark again that the convergence
established above is uniform on every compact subsets with respect to
$(s;q,t)\in D\times\mathbb{C}^{\ast}$. Hence, $f(x;s;p|q,t)$ depends
homomorphically on the parameters $(s;q,t)\in D\times\mathbb{C}^{\ast}$.

We refer to this normalized joint eigenfunction as the $($stationary$)$
Ruijsenaars function in the asymptotic domain
$|x_{1}|\gg\cdots\gg|x_{n}|\gg|px_{1}|$.

###### Remark 5.4

Suppose that $s_{i}\neq s_{j}$ for any distinct
$i,j\in\left\\{1,\ldots,s\right\\}$. Then the assumption $s_{j}/s_{i}\notin
q^{\mathbb{Z}}$ ($1\leq i<j\leq n$) of Theorem 2.7 is fulfilled automatically
for sufficiently small $q$. Let $\tau_{0}\in(0,1]$ satisfy
$\tau_{0}\leq|s_{j}/s_{i}|\leq\tau_{0}^{-1}$ $(1\leq i<j\leq n)$, and for any
$\theta$ with $0<\theta<\tau_{0}$ set

$\tau_{1}=\min\left\\{1-\theta/\tau_{0},|1-s_{j}/s_{i}|\ \
(i,j\in\left\\{1,\ldots,n\right\\};\ i\neq j)\right\\}.$ (5.51)

Then, for any $q\in\mathbb{C}^{\ast}$ with $|q|\leq\theta$, we have
$|1-q^{l}s_{j}/s_{i}|\geq\tau_{1}$ for all distinct
$i,j\in\left\\{1,\ldots,n\right\\}$ and $l\in\mathbb{Z}_{\geq 0}$. Hence, the
constants $K$ and $\rho$ in (5.47) do not depend on $q$ with $|q|\leq\theta$.
This implies that $f(x;s;p)=f(x;s;p|q,t)$ is also a holomorphic function on
$q\in\mathbb{C}$ with $|q|<\tau_{0}$ and $t\in\mathbb{C}^{\ast}$, including
$q=0$. Also, we have $f(x;s;p|0,t)=1$ by (5.47).

###  5.2 Elliptic deformation of Macdonald polynomials

We now proceed to the problem of symmetric joint eigenfunctions around the
torus $\mathbb{T}^{n}$. We apply an idea of integral transforms, similar to
that we already presented in Section 3, to construct elliptic deformations of
Macdonald polynomials from Ruijsenaars functions in the asymptotic domain
$|y_{1}|\gg\cdots\gg|y_{n}|\gg|py_{1}|$.

Recall that the Ruijsenaars elliptic gamma function is defined by

$\Gamma(z;p,q)=\frac{(pq/z;p,q)_{\infty}}{(z;p,q)_{\infty}},\quad(z;p,q)_{\infty}=\prod_{i,j=0}^{\infty}(1-p^{i}q^{j}z).$
(5.52)

Note that $\Gamma(z;p,q)$ is a meromorphic function on $\mathbb{C}^{\ast}$,
and satisfies the functional equations

$\Gamma(qz;p,q)=\theta(z;p)\Gamma(z;p,q),\ \
\Gamma(pz;p,q)=\theta(z;q)\Gamma(z;p,q),\ \
\Gamma(pq/z;p,q)=\Gamma(z;p,q)^{-1}.$ (5.53)

In order to formulate an integral transform in the elliptic case, we use the
following elliptic deformation of $K(x,y)$ and $w(y)$:

$K(x,y;p)=\prod_{i=1}^{n}\prod_{j=1}^{n}\frac{\Gamma(x_{i}/y_{j};p,q)}{\Gamma(tx_{i}/y_{j};p,q)},\quad
w(y;p)=\prod_{1\leq i<j\leq
n}\theta(y_{j}/y_{i};p)\frac{\Gamma(ty_{j}/y_{i};p,q)}{\Gamma(qy_{j}/ty_{i};p,q)}.$
(5.54)

The weight function $w(y;p)$ is holomorphic in the domain

$|pt|<|y_{j}/y_{i}|<|t|^{-1}\qquad(1\leq i<j\leq n).$ (5.55)

Also, the kernel function $K(x,y;p)$ is holomorphic in the domain

$|pq/t|<|x_{i}/y_{j}|<1\qquad(i,j=1,\ldots,n).$ (5.56)

Supposing that $0<\theta\leq\min\left\\{1,1/|t|,|t/q|\right\\}$ and
$|p|<\theta^{n}$, we define the domain
$\mathcal{V}_{\theta}\subset(\mathbb{C}^{\ast})^{n}$ by

$\mathcal{V}_{\theta}=\left\\{{y=(y_{1},\ldots,y_{n})\in(\mathbb{C}^{\ast})^{n}\,}\big{|}{\
|y_{2}/y_{1}|<\theta,\ \ldots,\ |y_{n}/y_{n-1}|<\theta,\
|py_{1}/y_{n}|<\theta}\right\\}.$ (5.57)

For any $r>0$ and $\sigma>0$ such that $|p|\leq\sigma^{n}<\theta^{n}$, we take
the $n$-cycle

$C_{r,\sigma}=\left\\{{y=(y_{1},\ldots,y_{n})\in(\mathbb{C}^{\ast})^{n}}\big{|}{\
|y_{i}|=\sigma^{i-1}r\ \
(i=1,\ldots,n)}\right\\}\subset(\mathbb{C}^{\ast})^{n}$ (5.58)

as in the trigonometric case. Then $C_{r,\sigma}$ is contained in
$\mathcal{V}_{\theta}$. Also, since $|pt|\leq\sigma^{n}|t|<\sigma^{n-1}$,
$w(y;p)$ is holomorphic in an neighborhood of $C_{r,\sigma}$. On the other
hand, if $y\in C_{r,\sigma}$, $K(x,y;p)$ is holomorphic in the domain

$\mathcal{U}_{r,\sigma}=\left\\{{x=(x_{1},\ldots,x_{n})\in(\mathbb{C}^{\ast})^{n}}\big{|}{\
|pq/t|r<|x_{i}|<\sigma^{n-1}r\ \ (i=1,\ldots,n)}\right\\}.$ (5.59)

Since $|pq/t|<\sigma^{n-1}$, $\mathcal{U}_{r,\sigma}$ is not empty. Note also
that, if $x\in\mathcal{U}_{r,\sigma}$, then
$|pq/t|\sigma^{-n+1}<|x_{j}/x_{i}|<|t/pq|\sigma^{n-1}$. In view of this fact,
we set

$\mathcal{U}_{\sigma}=\left\\{{x=(x_{1},\ldots,x_{n})\in(\mathbb{C}^{\ast})^{n}}\big{|}{\
|pq/t|\sigma^{-n+1}<|x_{j}/x_{i}|<|t/pq|\sigma^{n-1}\ \ (1\leq i<j\leq
n)}\right\\},$ (5.60)

so that $\mathcal{U}_{r,\sigma}\subseteq\mathcal{U}_{\sigma}$ for any $r>0$.

###### Lemma 5.5

Let $\theta$ be a positive constant such that
$\theta\leq\min\left\\{1,1/|t|,|t/q|\right\\}$ and suppose that
$|p|<\theta^{n}$. For each holomorphic function $\psi(y;p)$ in the domain
$\mathcal{V}_{\theta}$, consider the holomorphic function $\varphi(x;p)$ in
the domain $\mathcal{U}_{r,\sigma}$ defined by the integral transform

$\varphi(x;p)=\int_{C_{r,\sigma}}K(x,y;p)w(y;p)\psi(y;p)d\omega_{n}(y),$
(5.61)

taking $r>0$ and $\sigma>0$ such that $\sigma<\theta$ and $|p|\leq\sigma^{n}$.
Then $\varphi(x;p)$ is continued to a symmetric holomorphic function in the
domain $\mathcal{U}_{\theta}$, which does not depend on the choice of $r$,
$\sigma$.

Proof : We verify that

$\mathcal{U}_{\sigma}=\bigcup_{r>0}\ \mathcal{U}_{r,\sigma}.$ (5.62)

For $x\in\mathcal{U}_{\sigma}$ given, take two indices
$k,l\in\left\\{1,\ldots,n\right\\}$ such that
$|x_{k}|=\max\left\\{|x_{1}|,\ldots,|x_{n}|\right\\}$ and
$|x_{l}|=\min\left\\{|x_{1}|,\ldots,|x_{n}|\right\\}$. Then we have
$|pq/t|\sigma^{-n+1}<|x_{l}/x_{k}|\leq 1$. Since
$|x_{k}|/\sigma^{n-1}<|x_{l}|/|pq/t|$, we can find an $r>0$ such that
$|x_{k}|/\sigma^{n-1}<r<|x_{l}|/|pq/t|$. Then we have
$|pq/t|r<|x_{l}|\leq|x_{k}|<\sigma^{n-1}r$. This means that
$|pq/t|r<|x_{i}|<\sigma^{n-1}r$ $(i=1,\ldots,n)$, namely
$x\in\mathcal{U}_{r,\sigma}$. Since the integral transforms defined in
$\mathcal{U}_{r,\sigma}$ do not depend on $r>0$, they define a holomorphic
function $\varphi(x;p)$ in $\mathcal{U}_{\sigma}$. Since $\sigma>0$ can be
taken arbitrarily as far as $|p|^{\frac{1}{n}}\leq\sigma<\theta$,
$\varphi(x;p)$ is continued to $\mathcal{U}_{\theta}$. $\square$

We remark that a natural elliptic counterpart of the symmetric weight function
$w^{\mathrm{sym}}(y)$ of (3.9) is given by

$w^{\mathrm{sym}}(y;p)=\prod_{1\leq i<j\leq
n}\frac{\Gamma(ty_{i}/y_{j};p,q)\Gamma(ty_{j}/y_{i};p,q)}{\Gamma(y_{i}/y_{j};p,q)\Gamma(y_{j}/y_{i};p,q)}.$
(5.63)

Similarly to the trigonometric case, the elliptic Ruijsenaars operator
$\mathcal{D}_{x}(p;u)$ is formally self-adjoint with respect to this weight
function $w^{\mathrm{sym}}(y;p)$ in the following sense. (For the definition
of the formal adjoint of a $q$-difference operator, see the explanation below
formula (3.8).)

###### Lemma 5.6

$\mathcal{D}_{y}(p;u)^{\ast}=w^{\mathrm{sym}}(y;p)\mathcal{D}_{y^{-1}}(p;u)w^{\mathrm{sym}}(y;p)^{-1}.$
(5.64)

Proof : The elliptic Ruijsenaars operator of $r$th order is expressed as

$\mathcal{D}^{(r)}_{y}(p)=\sum_{\begin{subarray}{c}I\subseteq\left\\{1,\ldots,n\right\\}\\\
|I|=r|\end{subarray}}\mathcal{A}_{I}(y;p)T_{q,y}^{\epsilon_{I}},\quad\mathcal{A}_{I}(y;p)=t^{\binom{r}{2}}\prod_{i\in
I;\,j\notin I}\frac{\theta(ty_{i}/y_{j};p)}{\theta(y_{i}/y_{j};p)}.$ (5.65)

We decompose $w^{\mathrm{sym}}(y;p)$ as

$w^{\mathrm{sym}}(y;p)=w^{\mathrm{sym}}_{+}(y;p)w^{\mathrm{sym}}_{+}(y^{-1};p),\quad
w^{\mathrm{sym}}_{+}(y;p)=\prod_{1\leq i<j\leq
n}\frac{\Gamma(ty_{i}/y_{j};p,q)}{\Gamma(y_{i}/y_{j};p,q))},$ (5.66)

and look at $I=\left\\{1,\ldots,r\right\\}$:

$\frac{T_{q,y}^{\epsilon_{\left\\{1,\ldots,r\right\\}}}(w^{\mathrm{sym}}_{+}(y;p))}{w^{\mathrm{sym}}_{+}(y;p)}=\prod_{i=1}^{r}\prod_{j=r+1}^{n}\frac{\theta(ty_{i}/y_{j};p)}{\theta(y_{i}/y_{j};p)}=\prod_{i\in\left\\{1,\ldots,r\right\\},\,j\notin\left\\{1,\ldots,r\right\\}}\frac{\theta(ty_{i}/y_{j};p)}{\theta(y_{i}/y_{j};p)}.$
(5.67)

For each $I\subseteq\left\\{1,\ldots,n\right\\}$ with $|I|=r$, choose a
permutation $\sigma_{I}\in\mathfrak{S}_{n}$ such that
$\sigma_{I}(\left\\{1,\ldots,r\right\\})=I$. Then, applying $\sigma_{I}$ to
(5.67) we obtain

$\frac{T_{q,y}^{\epsilon_{I}}(\sigma_{I}w^{\mathrm{sym}}_{+}(y;p))}{\sigma_{I}w^{\mathrm{sym}}_{+}(y;p)}=\prod_{i\in
I,\,j\notin I}\frac{\theta(ty_{i}/y_{j};p)}{\theta(y_{i}/y_{j};p)},$ (5.68)

namely,

$\mathcal{A}_{I}(y;p)=t^{\binom{r}{2}}\frac{T_{q,y}^{\epsilon_{I}}(\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p))}{\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p)}=t^{\binom{r}{2}}\frac{1}{\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p)}\,T_{q,y}^{\epsilon_{I}}\,\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p).$
(5.69)

Hence $\mathcal{D}^{(r)}_{y}(p)$ and its formal adjoint
$\mathcal{D}^{(r)}_{y}(p)^{\ast}$ are expressed as

$\mathcal{D}_{y}^{(r)}(p)=t^{\binom{r}{2}}\sum_{\begin{subarray}{c}I\subseteq\left\\{1,\ldots,n\right\\}\\\
|I|=r\end{subarray}}\frac{1}{\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p)}\,T_{q,y}^{\epsilon_{I}}\,\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p),$
(5.70)

and

$\mathcal{D}_{y}^{(r)}(p)^{\ast}=t^{\binom{r}{2}}\sum_{\begin{subarray}{c}I\subseteq\left\\{1,\ldots,n\right\\}\\\
|I|=r\end{subarray}}\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p)\,T_{q,y}^{-\epsilon_{I}}\,\frac{1}{\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p)}.$
(5.71)

Note that $\mathfrak{S}_{n}$-invariance of $w^{\mathrm{sym}}(y;p)$ implies

$w^{\mathrm{sym}}(y;p)=\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p)\,\sigma_{I}.w^{\mathrm{sym}}_{+}(y^{-1};p).$
(5.72)

Hence we have

$\begin{split}&w^{\mathrm{sym}}(y;p)^{-1}\mathcal{D}_{y}^{(r)}(p)^{\ast}\,w^{\mathrm{sym}}(y;p)\\\
&=t^{\binom{r}{2}}\sum_{\begin{subarray}{c}I\subseteq\left\\{1,\ldots,n\right\\}\\\
|I|=r\end{subarray}}\frac{\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p)}{w^{\mathrm{sym}}(y;p)}\,T_{q,y}^{-\epsilon_{I}}\,\frac{w^{\mathrm{sym}}(y;p)}{\sigma_{I}.w^{\mathrm{sym}}_{+}(y;p)}\\\
&=t^{\binom{r}{2}}\sum_{\begin{subarray}{c}I\subseteq\left\\{1,\ldots,n\right\\}\\\
|I|=r\end{subarray}}\frac{1}{\sigma_{I}.w^{\mathrm{sym}}_{+}(y^{-1};p)}T_{q,y}^{-\epsilon_{I}}\,\sigma_{I}.w^{\mathrm{sym}}_{+}(y^{-1};p)=\mathcal{D}^{(r)}_{y^{-1}}(p),\end{split}$
(5.73)

as desired. $\square$

Also, by a result of [24] and [11], the kernel function $K(x,y;p)$ satisfies
the kernel function identity

$\mathcal{D}_{x}(p;u)K(x,y;p)=\mathcal{D}_{y^{-1}}(p;u)K(x,y;p).$ (5.74)

Hence we have

$\begin{split}\mathcal{D}_{x}(p;u)K(x,y;p)w^{\mathrm{sym}}(y;p)&=w^{\mathrm{sym}}(y;p)\mathcal{D}_{y^{-1}}(p;u)K(x,y;p)\\\
&=\mathcal{D}_{y}(p;u)^{\ast}(K(x,y;p)w^{\mathrm{sym}}(y;p)).\end{split}$
(5.75)

We rewrite the symmetric weight function $w^{\mathrm{sym}}(y;p)$ as

$\begin{split}w^{\mathrm{sym}}(y;p)&=\prod_{1\leq i<j\leq
n}\frac{\Gamma(pty_{i}/y_{j};p,q)\Gamma(ty_{j}/y_{i};p,q)}{\Gamma(py_{i}/y_{j};p,q)\Gamma(y_{j}/y_{i};p,q)}\frac{\theta(y_{i}/y_{j};q)}{\theta(ty_{i}/y_{j};q)}\\\
&=\prod_{1\leq i<j\leq
n}\frac{\Gamma(qy_{j}/y_{i};p,q)\Gamma(ty_{j}/y_{i};p,q)}{\Gamma(qy_{j}/ty_{i};p,q)\Gamma(y_{j}/y_{i};p,q)}\cdot\prod_{1\leq
i<j\leq n}\frac{\theta(y_{i}/y_{j};q)}{\theta(ty_{i}/y_{j};q)}\\\
&=\prod_{1\leq i<j\leq
n}\theta(y_{j}/y_{i};p)\frac{\Gamma(ty_{j}/y_{i};p,q)}{\Gamma(qy_{j}/ty_{i};p,q)}\cdot\prod_{1\leq
i<j\leq n}\frac{\theta(y_{i}/y_{j};q)}{\theta(ty_{i}/y_{j};q)}\end{split}$
(5.76)

to obtain

$w^{\mathrm{sym}}(y;p)=w(y;p)g(y),\quad g(y)=\prod_{1\leq i<j\leq
n}\frac{\theta(y_{i}/y_{j};q)}{\theta(ty_{i}/y_{j};q)},$ (5.77)

with the same function $g(y)$ as in the trigonometric case. Then, by the same
argument as in Subsection 3.2, we obtain the kernel function identity

$\mathcal{D}_{x}(p;u)K(x,y;p)w(y;p)=\mathcal{E}_{y,t^{\rho^{\vee}}}(p;u)^{\ast}(K(x,y;p)w(y;p))$
(5.78)

in the elliptic case, where $t^{\rho^{\vee}}=(1,t,\ldots,t^{n-1})$ stands for
the reversal of $t^{\rho}=(t^{n-1},t^{n-2},\ldots,1)$.

###### Theorem 5.7

Suppose that $t^{k}\notin q^{\mathbb{Z}}$ $(k=1,\ldots,n-1)$. Let $\lambda$ be
a partition with $l(\lambda)\leq n$, namely $\lambda\in\mathsf{P}_{+}$,
$\left\langle\epsilon_{n},\lambda\right\rangle\geq 0$, and set
$s=t^{\rho}q^{\lambda}$. Let $\theta$ be a positive constant such that
$\theta\leq\min\left\\{1,1/|t|,|t/q|\right\\}$ and that the Ruijsenaars
function $f(y;s^{\vee};p)$ is holomorphic in the domain

$\mathcal{V}_{\theta}:\quad|y_{2}/y_{1}|<\theta,\ \ \ldots,\ \
|y_{n}/y_{n-1}|<\theta,\ \ |py_{1}/y_{n}|<\theta$ (5.79)

for $|p|<\theta^{n}$, as well as the eigenvalue $\varepsilon(s^{\vee};p;u)$.
Then the integral transform

$\varphi_{\lambda}(x;p)=\int_{C_{r,\sigma}}K(x,y;p)w(y;p)\,y^{\lambda^{\vee}}\\!f(y;s^{\vee};p)d\omega_{n}(y)$
(5.80)

is continued to a symmetric holomorphic function in the domain

$\mathcal{U}_{\theta}:\quad|pq/t|\theta^{-n+1}<|x_{j}/x_{i}|<|t/pq|\theta^{n-1}\qquad(1\leq
i<j\leq n).$ (5.81)

If $|p|<|q|^{n}\theta^{n}$, then $\varphi_{\lambda}(x;p)$ satisfies the joint
eigenfunction equation

$\mathcal{D}_{x}(p;u)\varphi_{\lambda}(x;p)=\varepsilon(s;p;u)\varphi_{\lambda}(x;p)$
(5.82)

in $\mathcal{U}_{|q|\theta}\subset\mathcal{U}_{\theta}$ with the initial
condition $\varphi_{\lambda}(x;0)=b_{\lambda}P_{\lambda}(x)$. Furthermore, as
power series in $p$ we have

$\varphi_{\lambda}(x;p)=b_{\lambda}(p)\mathcal{P}_{\lambda}(x;p)$ (5.83)

with a holomorphic function $b_{\lambda}(p)$ in $|p|<\theta^{n}$.

Proof : The first half of Theorem 5.7 is a consequence of Lemma 5.5. Supposing
that $|p|<|q|^{n}\theta^{n}$, we show that $\varphi_{\lambda}(x;p)$ satisfies
the joint eigenfunction equation (5.82) in the domain
$\mathcal{U}_{|q|\theta}$. Let $\sigma$ be a positive constant such that
$|p|^{\frac{1}{n}}\leq\sigma<|q|\theta$, and suppose that
$x=(x_{1},\ldots,x_{n})\in(\mathbb{C}^{\ast})^{n}$ satisfies

$|p/t|\sigma^{-n+1}<|x_{j}/x_{i}|<|t/p|\sigma^{n-1}\quad(1\leq i<j\leq n).$
(5.84)

Take two indices $k,l\in\left\\{1,\ldots,n\right\\}$ such that
$|x_{l}|\leq|x_{i}|\leq|x_{k}|$ for $i=1,\ldots,n$. Then we have
$|p/t|\sigma^{-n+1}<|x_{l}/x_{k}|\leq 1$. Since
$\sigma^{-n+1}|x_{k}|<|t/p||x_{l}|$, we can take $r>0$ such that
$\sigma^{-n+1}|x_{k}|<r<|t/p||x_{l}|$ so that
$|p/t|r<|x_{l}|\leq|x_{k}|<\sigma^{n-1}r$, namely
$|p/t|r<|x_{i}|<\sigma^{n-1}r$ $(i=1,\ldots,n)$. Then, for $y\in C_{r,\sigma}$
we have $|p/t||y_{j}|<|x_{i}|<|y_{j}|$, namely

$|pq/t|<|qx_{i}/y_{j}|<|x_{i}/y_{j}|<1\qquad(i,j=1,\ldots,n).$ (5.85)

Then we can represent $\mathcal{D}_{x}(p;u)\varphi_{\lambda}(x;p)$ as an
integral over the cycle $C_{r,\sigma}$:

$\begin{split}\mathcal{D}_{x}(p;u)\varphi_{\lambda}(x;p)&=\int_{C_{r,\sigma}}\mathcal{D}_{x}(p;u)K(x,y;p)w(y;p)\,y^{\lambda^{\vee}}\\!f(y;s^{\vee};p)d\omega_{n}(y)\\\
&=\int_{C_{r,\sigma}}(\mathcal{E}_{y,t^{\rho^{\vee}}})^{\ast}\big{(}K(x,y;p)w(y;p)\big{)}\,y^{\lambda^{\vee}}\\!f(y;s^{\vee};p)d\omega_{n}(y).\end{split}$
(5.86)

Recalling that

$\mathcal{E}_{y,t^{\rho^{\vee}}}(p;u)=\sum_{I\subseteq\left\\{1,\ldots,n\right\\}}(-u)^{|I|}t^{\left\langle\epsilon_{I},\rho^{\vee}\right\rangle}\mathcal{B}_{I}(y;p)T_{q,y}^{\epsilon_{I}},$
(5.87)

for each $I\subseteq\left\\{1,\ldots,n\right\\}$, we compute the integral

$\begin{split}&\int_{C_{r,\sigma}}T_{q,y}^{-\epsilon_{I}}\left(\mathcal{B}_{I}(y;p)K(x,y;p)w(y;p)\right)\,y^{\lambda^{\vee}}\\!f(y;s^{\vee};p)d\omega_{n}(y)\\\
&=\int_{C_{r,\sigma}}\mathcal{B}_{I}(q^{-\epsilon_{I}}y;p)K(x,q^{-\epsilon_{I}}y;p)w(q^{-\epsilon_{I}}y;p)\,y^{\lambda^{\vee}}\\!f(y;s^{\vee};p)d\omega_{n}(y)\\\
&=\int_{q^{-\epsilon_{I}}C_{r,\sigma}}\mathcal{B}_{I}(y;p)K(x,y;p)w(y;p)\,(q^{\epsilon_{I}}y)^{\lambda^{\vee}}\\!f(q^{\epsilon_{I}}y,s^{\vee};p)d\omega_{n}(y).\end{split}$
(5.88)

Since the two cycles $q^{-\epsilon_{I}}C_{r,\sigma}$ and $C_{r,\sigma}$ are
homologous in the domain of holomorphy of the integrand, by the Cauchy theorem
we see that this integral equals to

$\begin{split}&\int_{C_{r,\sigma}}K(x,y;p)w(y;p)\,\mathcal{B}_{I}(y;p)(q^{\epsilon_{I}}y)^{\lambda^{\vee}}\\!f(q^{\epsilon_{I}}y,s^{\vee};p)d\omega_{n}(y)\\\
&=\int_{C_{r,\sigma}}K(x,y;p)w(y;p)\,\mathcal{B}_{I}(y;p)T_{q,y}^{\epsilon_{I}}\big{(}y^{\lambda^{\vee}}\\!f(y;s^{\vee};p)\big{)}d\omega_{n}(y).\end{split}$
(5.89)

Note also that the poles $\mathcal{B}_{I}(y;p)$ are cancelled by the factor
$\prod_{1\leq i<j\leq n}\theta(y_{j}/y_{i};p)$ in $w(y;p)$. Hence we have

$\begin{split}\mathcal{D}_{x}(x;p)\varphi_{\lambda}(x;p)&=\int_{C_{r,\sigma}}K(x,y;p)w(y;p)\mathcal{E}_{y,t^{\rho^{\vee}}}(p;u)\big{(}y^{\lambda^{\vee}}\\!f(y;s^{\vee};p)\big{)}d\omega_{n}(y)\\\
&=\int_{C_{r,\sigma}}K(x,y;p)w(y;p)\,(y^{\lambda^{\vee}}\mathcal{E}_{y,t^{\rho^{\vee}}q^{\lambda^{\vee}}}(p;u)\big{(}f(y;s^{\vee};p)\big{)}d\omega_{n}(y)\\\
&=\int_{C_{r,\sigma}}K(x,y;p)w(y;p)\,y^{\lambda^{\vee}}\varepsilon(s^{\vee};p;u)f(y;s^{\vee};p)d\omega_{n}(y)\\\
&=\varepsilon(s^{\vee};p;u)\varphi_{\lambda}(x;p).\end{split}$ (5.90)

This means that

$\mathcal{D}_{x}(p;u)\varphi_{\lambda}(x;p)=\varepsilon(s^{\vee};p;u)\varphi_{\lambda}(x;p),\quad
s=t^{\rho}q^{\lambda}.$ (5.91)

Also, by comparing the integral transform with the one in the trigonometric
case, from Theorem 3.2 we obtain

$\varphi_{\lambda}(x;0)=b_{\lambda}P_{\lambda}(x),\quad\varepsilon(s^{\vee};0;u)=\varepsilon(s;u)=\prod_{i=1}^{n}(1-us_{i}).$
(5.92)

Note also that

$K(x,y;p)=K(x,y)\prod_{i=1}^{n}\prod_{j=1}^{n}\frac{(ptx_{i}/y_{j};p,q)_{\infty}(pqy_{j}/x_{i};p,q)_{\infty}}{(px_{i}/y_{j};p,q)_{\infty}(pqy_{j}/tx_{i};p,q)_{\infty}}.$
(5.93)

From this expression of $K(x,y;p)$, we see that $\varphi_{\lambda}(x;p)$ has
$p$-expansion of the form

$\varphi_{\lambda}(x;p)=\sum_{k=0}^{\infty}p^{k}\varphi_{\lambda,k}(x),\quad\varphi_{\lambda,k}(x)\in(x_{1}\cdots
x_{n})^{-k}\mathbb{C}[[x]]^{\mathfrak{S}_{n}}.$ (5.94)

Since the joint eigenfunction $\varphi_{\lambda}(x)$ has the leading term
$\varphi_{\lambda,0}(x)=b_{\lambda}P_{\lambda}(x)\in\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}$, by the recurrence (4.29) we inductively see
$\varphi_{\lambda,k}(x)\in\mathbb{C}[x^{\pm 1}]^{\mathfrak{S}_{n}}$ for all
$k=0,1,2,\ldots$, which means that $\varphi_{\lambda}(x;p)\in\mathbb{C}[x^{\pm
1}]^{\mathfrak{S}_{n}}[[p]]$. Hence, we conclude that the Laurent expansion of
$\varphi_{\lambda}(x;p)$ around the torus $\mathbb{T}^{n}$ coincides with the
normalized formal solution $\mathcal{P}_{\lambda}(x;p)$ up to multiplication
by a power series in $\mathbb{C}[[p]]$, and that
$\varepsilon(s^{\vee};p;u)=\varepsilon(s;p;u)$. We set

$\varphi_{\lambda}(x;p)=b_{\lambda}(p)\mathcal{P}_{\lambda}(x;p),\quad
b_{\lambda}(p)\in\mathbb{C}[[p]].$ (5.95)

In fact, $b_{\lambda}(p)$ is determined as the constant term of
$x^{-\lambda}\varphi_{\lambda}(x;p)$ with respect to $x$:

$b_{\lambda}(p)=\int_{\mathbb{T}^{n}}x^{-\lambda}\varphi_{\lambda}(x;p)d\omega_{n}(x).$
(5.96)

Since $b_{\lambda}(p)$ is a holomorphic function in $|p|<\theta^{n}$, the
normalized formal solution $\mathcal{P}_{\lambda}(x;p)$ is absolutely
convergent as well in an neighborhood $\mathbb{T}^{n}$ where
$\varphi_{\lambda}(x;p)$ is holomorphic, if $|p|$ is sufficiently small.
$\square$

This implies Theorem 2.3 for the case where $\lambda\in\mathsf{P}_{+}$ is a
partition. In fact, from Theorem 5.7 it follows that
$\varphi_{\lambda}(x;p)=b_{\lambda}(p)\mathcal{P}_{\lambda}(x;p)$ is
holomorphic in the domain

$|p|/\tau<|x_{j}/x_{i}|<\tau/|p|\quad(1\leq i<j\leq n),\quad|p|<\tau$ (5.97)

for any positive $\tau\leq|t/q|\theta^{n-1}$. If we take a sufficiently small
$\tau>0$ such that $b_{\lambda}(p)\neq 0$ for $|p|<\tau$,
$\mathcal{P}_{\lambda}(x;p)$ is holomorphic in the domain (5.97). If
$\lambda\in\mathsf{P}_{+}$ is a general dominant integral weight, there exists
an integer $l\in\mathbb{Z}$ such that $\lambda+(l^{n})$ is a partition. Since

$\mathcal{P}_{\lambda}(x;p)=(x_{1}\cdots
x_{n})^{-l}\mathcal{P}_{\lambda+(l^{n})}(x;p),$ (5.98)

the statement of Theorem 2.3 is reduced to the case of a partition.

###  5.3 Orthogonality relation for $\mathcal{P}_{\lambda}(x;p)$

For each dominant integral weight $\lambda\in\mathsf{P}_{+}$, we use the
notation $\mathcal{P}_{\lambda}(x;p)=\mathcal{P}_{\lambda}(x;p|q,t)$ for the
normalized symmetric joint eigenfunction of the elliptic Ruijsenaars operator
$\mathcal{D}_{x}(p;u)$ with initial condition
$\mathcal{P}_{\lambda}(x;0)=P_{\lambda}(x)$; it is holomorphic in a domain of
the form

$\mathcal{W}_{\tau}=\left\\{{(x;p)\in(\mathbb{C}^{\ast})^{n}\times\mathbb{C}^{\ast}}\big{|}{\
|p|/\tau<|x_{j}/x_{i}|<\tau/|p|\quad(1\leq i<j\leq n),\quad
0<|p|<\tau}\right\\}$ (5.99)

for some $\tau\in(0,1]$. Note that $\mathcal{W}_{\tau}$ is not empty for any
$\tau\in(0,1]$, and $\mathcal{W}_{\tau^{\prime}}\subseteq\mathcal{W}_{\tau}$
if $\tau^{\prime}\leq\tau$. In this subsection, we assume that $|t|<1$ and