Basic Properties of Non-Stationary Ruijsenaars Functions

Basic Properties of Non-Stationary Ruijsenaars
Functions††This paper is a contribution to the Special Issue on Elliptic
Integrable Systems, Special Functions and Quantum Field Theory. The full
collection is available at https://www.emis.de/journals/SIGMA/elliptic-
integrable-systems.html

Edwin LANGMANN †, Masatoshi NOUMI ‡ and Junichi SHIRAISHI §

E. Langmann, M. Noumi and J. Shiraishi

† Physics Department, KTH Royal Institute of Technology, SE-106 91 Stockholm,
Sweden<EMAIL_ADDRESS>
‡ Department 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<EMAIL_ADDRESS>
§ Graduate School of Mathematical Sciences, The University of Tokyo, Komaba,
§ Tokyo 153-8914, Japan<EMAIL_ADDRESS>
Received June 15, 2020, in final form October 08, 2020; Published online
October 21, 2020

For any variable number, a non-stationary Ruijsenaars function was recently
introduced as a natural generalization of an explicitly known asymptotically
free solution of the trigonometric Ruijsenaars model, and it was conjectured
that this non-stationary Ruijsenaars function provides an explicit solution of
the elliptic Ruijsenaars model. We present alternative series representations
of the non-stationary Ruijsenaars functions, and we prove that these series
converge. We also introduce novel difference operators called ${\mathcal{T}}$
which, as we prove in the trigonometric limit and conjecture in the general
case, act diagonally on the non-stationary Ruijsenaars functions.

elliptic integrable systems; elliptic hypergeometric functions; Ruijsenaars
systems

81Q80; 32A17; 33E20; 33E30

##  1 Introduction

The celebrated quantum Calogero–Moser–Sutherland systems [16] have natural
relativistic generalizations discovered by Ruijsenaars [17]. The Ruijsenaars
systems come in four kinds: rational, trigonometric, hyperbolic, and elliptic,
with the latter case being the most general and reducing to the others in
certain limits [17]. While the explicit solution of the trigonometric
Ruijsenaars model is known since a long time: it is given by the celebrated
Macdonald polynomials [13], and a construction of eigenfunctions of the
hyperbolic model was completed recently [10], only partial results about the
explicit solution in the general elliptic case exist [7, 18, 19]. Recently,
one of us (S) conjectured an explicit solution of the elliptic Ruijsenaars
model as a limit of special functions defined by explicit formal power series
and called non-stationary Ruijsenaars functions [21]. In particular, it was
shown in [21] that these functions reduce to the known solutions of the
trigonometric Ruijsenaars model in the trigonometric limit; they have several
remarkable symmetry properties; and they arise in a quantum field theory
related to the elliptic Ruijsenaars system in a way that is a natural
generalization of how the known solutions of the trigonometric Ruijsenaars
model arise in a quantum field theory related to the trigonometric Ruijsenaars
model (this is only a partial list of results in [21]). The validity of this
conjecture was also tested by symbolic computer computations.

In this paper we prove some properties of the non-stationary Ruijsenaars
functions which, we hope, will be useful to find proofs of the conjectures in
[21]. In particular, we give alternative representations of these functions
which are simpler than the original definitions; we prove that the series
defining these functions are absolutely convergent in a suitable domain; and
we present novel difference operators, called ${\mathcal{T}}$, which, we
conjecture, acts diagonally on the non-stationary Ruijsenaars functions (by
this we mean that the latter are eigenfunctions of the former).

Notation: Throughout the paper, the symbols $q$, $t$, $p$, $\kappa$ (complex
parameters) and $N$ (variable number) have special significance. We use the
following standard notation,

$\displaystyle(z;q)_{\infty}\equiv\prod_{n=0}^{\infty}\big{(}1-zq^{n}\big{)}\qquad(|q|<1),$
$\displaystyle(z;q)_{k}\equiv\frac{(z;q)_{\infty}}{(q^{k}z;q)_{\infty}}\qquad(k\in{\mathbb{Z}}),$
$\displaystyle(z;q,p)_{\infty}\equiv\prod_{n,m=0}^{\infty}\big{(}1-q^{n}p^{m}z\big{)}\qquad(|q|<1,\
|p|<1),$ $\displaystyle\theta(z;p)\equiv(z;p)_{\infty}(p/z;p)_{\infty}$

for $z\in{\mathbb{C}}$. Moreover, $T_{q,z}=q^{z\partial_{z}}$, i.e.,

$\displaystyle(T_{q,z}f)(z)=f(qz)$

for functions $f(z)$ of $z\in{\mathbb{C}}$. For $z\in{\mathbb{C}}$,
$\mathrm{Re}(z)$ and $\mathrm{Im}(z)$ are the real- and imaginary parts of
$z$, and $\sin\arg(z)=\mathrm{Im}(z)/|z|$. For $x=(x_{1},\dots,x_{N})$ and
$\lambda=(\lambda_{1},\dots,\lambda_{N})$, $x^{\lambda}$ is short for
$x_{1}^{\lambda_{1}}\cdots x_{N}^{\lambda_{N}}$, $x^{-1}$ is short for
$\big{(}x_{1}^{-1},\dots,x_{N}^{-1}\big{)}$, and $x^{+1}=x$. We denote as
${\mathbb{C}}[[z_{1},\dots,z_{N}]]$ the space of all formal power series
$f(z)=\sum_{\mu\in{\mathbb{Z}}_{\geq 0}^{N}}c_{\mu}z_{1}^{\mu_{1}}\cdots
z_{N}^{\mu}$ in formal variables $z=(z_{1},\dots,z_{N})$ with complex
coefficients $c_{\mu}$.

##  2 Prerequisites

We recall some known facts about the Macdonald polynomials [13] and certain
special functions generalizing the Macdonald polynomials and constructed so as
to solve the trigonometric Ruijsenaars model [15, 20] (Section 2.1). We also
recall the eigenvalue problem defining the elliptic Ruijsenaars model, and the
definition of the non-stationary Ruijsenaars functions (Section 2.2).

###  2.1 Trigonometric Ruijsenaars model

For fixed $N\in{\mathbb{Z}}_{\geq 1}$, the Macdonald polynomials
$P_{\lambda}(x;q,t)=P_{\lambda}\big{(}x;q^{-1},t^{-1}\big{)}$ are symmetric
polynomials in variables $x=(x_{1},\dots,x_{N})\in{\mathbb{C}}^{N}$ depending
on two complex parameters $q$, $t$ and labeled by partitions $\lambda$ of
length less than or equal to $N$, i.e.,
$\lambda=(\lambda_{1},\dots,\lambda_{N})$ with
$\lambda_{i}\in{\mathbb{Z}}_{\geq 0}$ such that
$\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{N}\geq 0$. They can be
defined as common eigenfunctions of the following commuting
Macdonald–Ruijsenaars operators,

$\displaystyle D^{\pm}_{N}(x|q,t)\equiv\sum_{i=1}^{N}\prod_{j\neq
i}^{N}\frac{\big{(}1-t^{\pm
1}x_{i}/x_{j}\big{)}}{(1-x_{i}/x_{j})}T_{q,x_{i}}^{\pm 1}$ (2.1)

with corresponding eigenvalues
$\sum_{i=1}^{N}t^{\pm(N-i)}q^{\pm{\lambda_{i}}}$, together with a convenient
normalization condition [13].

The operators $D^{\pm}_{N}(x|q,t)$ are related by similarity transformations
to the operators defining the trigonometric Ruijsenaars model [17].

As conjectured by one of us (S) [20] and proved by two of us (NS) [15], these
eigenfunctions are naturally generalized to a special function
$f_{N}(x|s|q,t)$ depending on another set of variables,
$s=(s_{1},\dots,s_{N})\in{\mathbb{C}}^{N}$, and determined by the following
requirement, up to normalization: for $\lambda\in{\mathbb{C}}^{N}$, the
function

$\displaystyle x^{\lambda}f_{N}(x|s|q,t),\qquad s_{i}=t^{N-i}q^{\lambda_{i}}$
(2.2)

is a common eigenfunction of $D^{\pm}_{N}(x|q,t)$ with corresponding
eigenvalue $\sum_{j=1}^{N}s_{j}^{\pm 1}$; if $\lambda$ is a partition, then
the function in (2.2) is equal to the Macdonald polynomial
$P_{\lambda}(x;q,t)$ [15]. The function $f_{N}(x|s|q,t)$ is called the
asymptotically free solution of the trigonometric Ruijsenaars model.

One remarkable property of this function is that it has a simple explicit
series representation which converges absolutely in a suitable domain
[15]:111Note that $f_{N}(x|s|q,t)$ here is $p_{N}(x;s|q,t)$ in [15].

$\displaystyle
f_{N}(x|s|q,t)=\sum_{\theta\in\mathsf{M}_{N}}c_{N}(\theta|s|q,t)\prod_{1\leq
i<k\leq N}(x_{k}/x_{i})^{\theta_{ik}}$ (2.3)

with $\mathsf{M}_{N}$ the set of $N\times N$ strictly upper triangular
matrices with nonnegative integer entries:

$\displaystyle\mathsf{M}_{N}\equiv\big{\\{}\theta=(\theta_{ik})_{i,k=1}^{N}\,|\,\theta_{ik}\in{\mathbb{Z}}_{\geq
0}\ (\forall\,i,k),\ \theta_{ik}=0\ (1\leq k\leq i\leq N)\big{\\}},$ (2.4)

and222We write (2.5) in a way that emphasizes the similarity with (3.3) below,
for reasons that will become clear later on. Due to this, we include the empty
factors for $i=N$.

$\displaystyle c_{N}(\theta|s|q,t)=\prod_{i=1}^{N}\prod_{i<j\leq k\leq
N}\frac{\big{(}q^{\sum_{a>k}(\theta_{ia}-\theta_{ja})}ts_{j}/s_{i};q\big{)}_{\theta_{ik}}}{\big{(}q^{\sum_{a>k}(\theta_{ia}-\theta_{ja})}qs_{j}/s_{i};q\big{)}_{\theta_{ik}}}$
$\displaystyle\hphantom{c_{N}(\theta|s|q,t)=}{}\times\prod_{i=1}^{N}\prod_{i\leq
j<k\leq
N}\frac{\big{(}q^{-\theta_{jk}-\sum_{a>k}(\theta_{ja}-\theta_{ia})}qs_{j}/ts_{i};q\big{)}_{\theta_{ik}}}{\big{(}q^{-\theta_{jk}-\sum_{a>k}(\theta_{ja}-\theta_{ia})}s_{j}/s_{i};q\big{)}_{\theta_{ik}}}$
(2.5)

(note that (2.3)–(2.5) is equivalent to (1.10)–(1.11) in [15]).

For later reference, we also define the function333Note that
$\varphi_{N}(x|s|q,t)$ here is $\psi_{N}(x;s|q,t)$ in [15].

$\displaystyle\varphi_{N}(x|s|q,t)\equiv\prod_{1\leq i<j\leq
N}\frac{(qx_{j}/tx_{i};q)_{\infty}}{(qx_{j}/x_{i};q)_{\infty}}f_{N}(x|s|q,t),$
(2.6)

which, as proved in [15], has the following remarkably symmetry properties:

$\displaystyle\varphi_{N}(x|s|q,t)=\varphi_{N}(s|x|q,t)\quad$
$\displaystyle\text{(bispectral duality)},$
$\displaystyle\varphi_{N}(x|s|q,t)=\varphi_{N}(s|x|q,q/t)\quad$
$\displaystyle\text{(Poincar\'{e} duality)}.$ (2.7)

###  2.2 Non-stationary Ruijsenaars functions

The analogue of the operators in (2.1) for the elliptic Ruijsenaars model
depends on a further complex parameter, $p$ such that $|p|<1$:

$\displaystyle D_{N}^{\pm}(x|q,t,p)\equiv\sum_{i=1}^{N}\prod_{j\neq
i}^{N}\frac{\theta\big{(}t^{\pm
1}x_{i}/x_{j};p\big{)}}{\theta(x_{i}/x_{j};p)}T_{q,x_{i}}^{\pm 1}$ (2.8)

with the theta function $\theta(z;p)$ given in the introduction; note that
$D_{N}^{\pm}(x|q,t)=D_{N}^{\pm}(x|q,t,0)$.

The non-stationary Ruijsenaars function
$f^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)$ is a conjectured
eigenfunction of a deformation of the operators in (2.8), depending on a
further complex parameter, $\kappa$, and reducing to the operators in (2.8) in
the limit $\kappa\to 1$ [21].

######  Definition 2.1 (non-stationary Ruijsenaars functions).

For $N\in{\mathbb{Z}}_{\geq 1}$, four parameters $q$, $t$, $p$, $\kappa$, and
two sets of variables $x=(x_{1},\dots,x_{N})$ and $s=(s_{1},\dots,s_{N})$, the
non-stationary Ruijsenaars function is defined as a formal power series in
$(px_{2}/x_{1},\dots,px_{N}/x_{N-1},px_{1}/x_{N})$ as follows,

$\displaystyle
f^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)\equiv\\!\\!\\!\sum_{\lambda^{(1)},\dots,\lambda^{(N)}\in\mathsf{P}}\prod_{i,j=1}^{N}\frac{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}(ts_{j}/s_{i}|q,\kappa)}{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}(s_{j}/s_{i}|q,\kappa)}\prod_{\beta=1}^{N}\prod_{\alpha\geq
1}(px_{\alpha+\beta}/tx_{\alpha+\beta-1})^{\lambda^{(\beta)}_{\alpha}}\\!\\!\\!$
(2.9)

with $x_{\alpha+\ell N}\equiv x_{\alpha}$ for all $\alpha=1,\dots,N$ and
$\ell\in{\mathbb{Z}}_{\geq 1}$, $\mathsf{P}$ the set of all partitions
$\lambda$ of arbitrary length, i.e., $\lambda=(\lambda_{1},\lambda_{2},\dots)$
with $\lambda_{i}\in{\mathbb{Z}}_{\geq 0}$ such that
$\lambda_{1}\geq\lambda_{2}\geq\cdots$ and $\lambda_{i}=0$ for $i\gg 0$, and

$\displaystyle\mathsf{N}^{(k|N)}_{\lambda,\mu}(u|q,\kappa)\equiv\\!\\!\\!\\!\\!\\!\\!\\!\prod_{\genfrac{}{}{0.0pt}{}{b\geq
a\geq 1}{b-a\equiv
k(\mathrm{mod}N)}}\\!\\!\\!\\!\\!\\!\\!\\!\big{(}uq^{-\mu_{a}+\lambda_{b+1}}\kappa^{-a+b};q\big{)}_{\lambda_{b}-\lambda_{b+1}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\prod_{\genfrac{}{}{0.0pt}{}{\beta\geq\alpha\geq
1}{\beta-\alpha\equiv(-k-1)(\mathrm{mod}N)}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\big{(}uq^{\lambda_{\alpha}-\mu_{\beta}}\kappa^{\alpha-\beta-1};q\big{)}_{\mu_{\beta}-\mu_{\beta+1}}\\!\\!\\!$
(2.10)

for $\lambda$, $\mu\in\mathsf{P}$, $k\in{\mathbb{Z}}/N{\mathbb{Z}}$, and
$u\in{\mathbb{C}}$.

As discussed in [21], the expressions in (2.10) are Nekrasov factors [14].
Moreover, by changing $(p,\kappa)\to\big{(}p^{1/N},\kappa^{1/N}\big{)}$ and
scaling variables $x\to
p^{\delta/N}x=\big{(}p^{(N-1)/N}x_{1},p^{(N-2)/2}x_{2},\dots,x_{N}\big{)}$ and
similarly for $s$, one obtains a function,
$f^{\widehat{\mathfrak{gl}}_{N}}\big{(}p^{\delta/N}x,p^{1/N}|\kappa^{\delta/N}s,\kappa^{1/N}|q,q/t\big{)}$,
that converges to the asymptotically free solution of the Ruijsenaars model,
$f_{N}(x|s|q,t)$, in the limit $p\to 0$ [21].

######  Remark 2.2.

To explain the scaling just mentioned, we point out one important technical
point: in Definition 2.1, equations (2.11) and (2.12) below, and equations
(4.10) and (4.11) in Section 4, we use balanced coordinates $x_{B}$, $p_{B}$,
$s_{B}$, $\kappa_{B}$ and $t_{B}$ (written without the subscript $B$ for
simplicity), whereas elsewhere in the paper we use unbalanced coordinates
$x_{U}$, $p_{U}$, $s_{U}$, $\kappa_{U}$ and $t_{U}$ (also written without
subscript $U$) related to the balanced coordinates as follows,

$\displaystyle(x_{B})_{i}=(p_{U})^{(N-i)/N}(x_{U})_{i}\qquad(i=1,\dots,N),\qquad
p_{B}=(p_{U})^{N},$
$\displaystyle(s_{B})_{i}=(\kappa_{U})^{(N-i)/N}(s_{U})_{i}\qquad(i=1,\dots,N),\qquad\kappa_{B}=(\kappa_{U})^{N},\qquad
t_{B}=q/t_{U}.$

Thus, the scaling just described can be understood as a transformation from
balanced to unbalanced coordinates.

The main conjecture in [21] is that eigenfunction of the operator in (2.8) can
be obtained by dividing this rescaled function
$f^{\widehat{\mathfrak{gl}}_{N}}\big{(}p^{\delta/N}x,p^{1/N}|\kappa^{\delta/N}s,\kappa^{1/N}|q,q/t\big{)}$
by a (known) factor
$\alpha\big{(}p^{1/N}|\kappa^{\delta/N}s,\kappa^{1/N}|q,t\big{)}$ and taking
the limit $\kappa\to 1$; see Conjecture 1.14 in [21]. One important open
problem is to find the operator depending on $\kappa$ having these rescaled
non-stationary Ruijsenaars functions as eigenfunctions and reducing to the
Macdonald–Ruijsenaars operator in (2.8) in the limit $\kappa\to 1$.444There
is, however, a recent proposal mentioned in Section 5. At this point, this
operator is only known in limiting cases: the non-relativistic limit $q\to 1$
where the Ruijsenaars systems reduce to the non-stationary elliptic
Calogero–Sutherland system [21], and the limit $t,p\to 0$ with fixed $p/t$
leading to the affine Toda system [21]. We stress that the non-stationary
${\mathcal{T}}$-operators introduced in this paper do not reduce to the
elliptic Macdonald–Ruijsenaars operators in the limit $\kappa\to 1$: the
${\mathcal{T}}$-operators are of a different kind, and they are new even in
the trigonometric limit; only the affine Toda limit of the non-stationary
${\mathcal{T}}$-operator was known before [21].

A natural generalization of the function in (2.6) is

$\displaystyle\varphi^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)\equiv\prod_{1\leq
i<j\leq
N}\frac{\big{(}qp^{j-i}x_{j}/tx_{i};q,p^{N}\big{)}_{\infty}}{\big{(}qp^{j-i}x_{j}/x_{i};q,p^{N}\big{)}_{\infty}}$
$\displaystyle\hphantom{\varphi^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)\equiv}{}\times\prod_{1\leq
i\leq j\leq
N}\frac{\big{(}qp^{N-j+i}x_{i}/tx_{j};q,p^{N}\big{)}_{\infty}}{\big{(}qp^{N-j+i}x_{i}/x_{j};q,p^{N}\big{)}_{\infty}}f^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)$
(2.11)

and, as conjectured in [21], it has the following symmetry properties
generalizing the ones in (2.7).

######  Conjecture 2.3.

The functions in (2.11) satisfy

$\displaystyle\varphi^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)=\varphi^{\widehat{\mathfrak{gl}}_{N}}(s,\kappa|x,p|q,t)\quad$
$\displaystyle\text{$($bispectral duality$)$},$
$\displaystyle\varphi^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)=\varphi^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,q/t)\quad$
$\displaystyle\text{$($Poincar\'{e} duality$)$}.$ (2.12)

##  3 Results on the non-stationary Ruijsenaars function

We give alternative series representations of the non-stationary Ruijsenaars
functions (Section 3.1) and prove convergence of these series in a suitable
domain (Section 3.2).

###  3.1 Alternative series representations

Our first result makes manifest that the non-stationary Ruijsenaars function
in (2.9)–(2.10) is a natural generalization of the asymptotically free
solutions of the trigonometric Ruijsenaars model in (2.3)–(2.5) . For that, we
extend the variables $x=(x_{i})_{i=1}^{N}$ and $s=(s_{i})_{i=1}^{N}$ to
infinitely many variables $\bar{x}=(x_{i})_{i=1}^{\infty}$ and
$\bar{s}=(s_{i})_{i=1}^{\infty}$; as we will see, the pertinent extension is
provided by the parameters $p$ and $\kappa$, respectively — see (3.5).

We first introduce a natural generalization of the function in (2.3)–(2.5) to
infinitely many variables.

######  Definition 3.1.

For $N\in{\mathbb{Z}}_{\geq 1}$, two parameters $q$, $t$, and two sets of
infinitely many variables $\bar{x}=(x_{1},x_{2},\dots)$ and
$\bar{s}=(s_{1},s_{2},\dots)$, let the following define a formal power series
in the infinitely many variables
$(x_{2}/x_{1},x_{3}/x_{2},x_{4}/x_{3},\dots)$,

$\displaystyle
f_{N,\infty}(\bar{x}|\bar{s}|q,t)\equiv\sum_{\theta\in\hat{\mathsf{M}}_{N}}c_{N,\infty}(\theta|\bar{s}|q,t)\prod_{i=1}^{N}\prod_{k>i}(x_{k}/x_{i})^{\theta_{ik}}$
(3.1)

with $\hat{\mathsf{M}}_{N}$ the set of infinite, $N$-periodic, strictly upper
triangular matrices with nonnegative integer entries which are non-zero only
in a finite strip away from the diagonal:

$\displaystyle\hat{\mathsf{M}}_{N}\equiv\big{\\{}\theta=(\theta_{ik})_{i,k=1}^{\infty}\,|\,\theta_{ik}=\theta_{i+N,k+N}\in{\mathbb{Z}}_{\geq
0}\ (i,k\geq 1),\ \theta_{ik}=0\ (k\leq i,\ k\gg i)\big{\\}},$ (3.2)

and

$\displaystyle
c_{N,\infty}(\theta|\bar{s}|q,t)\equiv\prod_{i=1}^{N}\prod_{i<j\leq
k<\infty}\frac{\big{(}q^{\sum_{a>k}(\theta_{ia}-\theta_{ja})}ts_{j}/s_{i};q\big{)}_{\theta_{ik}}}{\big{(}q^{\sum_{a>k}(\theta_{ia}-\theta_{ja})}qs_{j}/s_{i};q\big{)}_{\theta_{ik}}}$
$\displaystyle\hphantom{c_{N,\infty}(\theta|\bar{s}|q,t)\equiv}{}\times\prod_{i=1}^{N}\prod_{i\leq
j<k<\infty}\frac{\big{(}q^{-\theta_{jk}-\sum_{a>k}(\theta_{ja}-\theta_{ia})}qs_{j}/ts_{i};q\big{)}_{\theta_{ik}}}{\big{(}q^{-\theta_{jk}-\sum_{a>k}(\theta_{jb}-\theta_{ia})}s_{j}/s_{i};q\big{)}_{\theta_{ik}}}.$
(3.3)

Note that the product in (3.3) always contains only a finite number of factors
different from $1$. Moreover, by the condition $\theta_{ik}=\theta_{i+N,k+N}$,
a matrix $\theta\in\hat{\mathsf{M}}_{N}$ is fully determined by the matrix
elements $\theta_{ik}$ for $1\leq i\leq N$ and $1\leq k<\infty$. Furthermore,
matrices in $\mathsf{M}_{N}$ can be naturally identified with matrices
$\theta$ in $\hat{\mathsf{M}}_{N}$ by setting $\theta_{ik}=0$ if $i>N$, or
$k>N$, or both.

To state out result we use the $N$-vector
$\delta\equiv(\delta_{1},\dots,\delta_{N})$ with $\delta_{i}=N-i$, and the
notation $p^{\delta/N}x$ and $\kappa^{\delta/N}s$ for the $N$-vectors with
components $\big{(}p^{\delta/N}x\big{)}_{i}=p^{(N-i)/N}x_{i}$ and
$\big{(}\kappa^{\delta/N}s\big{)}_{i}=\kappa^{(N-i)/N}s_{i}$, respectively
($i=1,\dots,N$). As explained in Remark 2.2, this can be understood as a
transformation going from balanced to unbalanced coordinates.

######  Theorem 3.2.

The non-stationary Ruijsenaars function in (2.9)–(2.10) is related to the
function in (3.1)–(3.3) as follows,

$\displaystyle
f^{\widehat{\mathfrak{gl}}_{N}}\big{(}p^{\delta/N}x,p^{1/N}|\kappa^{\delta/N}s,\kappa^{1/N}|q,q/t\big{)}=f_{N,\infty}(\bar{x}|\bar{s}|q,t)$
(3.4)

with the variables $x=(x_{i})_{i=1}^{N}$ and $s=(s_{i})_{i=1}^{N}$ on the
left-hand side extended to variables $\bar{x}=(x_{i})_{i=1}^{\infty}$ and
$\bar{s}=(s_{i})_{i=1}^{\infty}$ on the right-hand side by the
rules555“$x_{i+N}=px_{i}$ $(i\geq 1)$” is short for “$x_{i+kN}=p^{k}x_{i}$
$(i=1,\dots,N,\;k\in{\mathbb{Z}}_{\geq 1})$”.

$\displaystyle x_{i+N}=px_{i},\qquad s_{i+N}=\kappa s_{i}\qquad(i\geq 1).$
(3.5)

(The proof is by straightforward computations given in Appendix A.)

In the following, it is sometimes convenient to use a notation for the
functions $f_{N,\infty}$ that emphasizes that the arguments $\bar{x}$ and
$\bar{s}$ are fixed by $x$, $s$, $p$ and $\kappa$:

######  Definition 3.3.

We write

$\displaystyle f_{N,\infty}(x,p|s,\kappa|q,t)\equiv
f_{N,\infty}(\bar{x}|\bar{s}|q,t)$

if $\bar{x}\,{=}\,(x_{1},x_{2},\dots)$ and $\bar{s}\,{=}\,(s_{1},s_{2},\dots)$
on the right-hand side are determined by $x\,{=}\,(x_{1},\dots,x_{N})$, $p$,
$s=(s_{1},\dots,s_{N})$, and $\kappa$ as in (3.5). Thus

$\displaystyle
f_{N,\infty}(x,p|s,\kappa|q,t)=\sum_{\theta\in\hat{\mathsf{M}}_{N}}c_{N,\infty}(\theta|s,\kappa|q,t)e_{N,\infty}(\theta|x,p)$

with

$\displaystyle c_{N,\infty}(\theta|s,\kappa|q,t)\equiv
c_{N,\infty}(\theta|\bar{s}|q,t),\qquad
e_{N,\infty}(\theta|x,p)\equiv\prod_{i=1}^{N}\prod_{k=i+1}^{\infty}(x_{k}/x_{i})^{\theta_{ik}}$
(3.6)

and the identifications in (3.5) on the right-hand side in (3.6).

Theorem 3.2 makes manifest the following important result in [21]: After
suitably scaling the variables, the non-stationary Ruijsenaars function
reduces the asymptotically free solution of the trigonometric Ruijsenaars
model, $f_{N}(x|s|q,t)$ (2.3)–(2.5), in the limit $p\to 0$; in particular, it
becomes independent of $\kappa$ in this limit:

######  Corollary 3.4.

We have

$\displaystyle\lim_{p\to
0}f^{\widehat{\mathfrak{gl}}_{N}}\big{(}p^{\delta/N}x,p^{1/N}|\kappa^{\delta/N}s,\kappa^{1/N}|q,q/t\big{)}=f_{N}(x|s|q,t).$
(3.7)

###### Proof.

By Theorem 3.2, (3.7) is equivalent to

$\displaystyle\lim_{p\to 0}f_{N,\infty}(x,p|s,\kappa|q,t)=f_{N}(x|s|q,t),$

but this is obvious from definitions: by (3.5), $(x_{k}/x_{i})\to 0$ for $k>N$
as $p\to 0$; therefore, the sum over $\theta\in\hat{\mathsf{M}}_{N}$ on the
right-hand side in (3.1) collapses to a sum over $\theta\in\mathsf{M}_{N}$ in
this limit; obviously, for $\theta\in\mathsf{M}_{N}$, the coefficients
$c_{N,\infty}(\theta|\bar{s}|q,t)$ in (3.3) do not depend on $s_{i>N}$ and are
identical with the coefficients $c_{N}(\theta|s|q,t)$ in (2.5). ∎

We prove Theorem 3.2 by a direct computation in Appendix A. This proof uses an
alternative representation of the function $f_{N,\infty}(\bar{x}|\bar{s}|q,t)$
which is interesting in its own right:

######  Lemma 3.5.

The formal power series in (3.1)–(3.3) can be written as

$\displaystyle
f_{N}(\bar{x}|\bar{s}|q,t)=\sum_{\boldsymbol{\lambda}\in\mathsf{P}^{N}}C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)\prod_{i=1}^{N}\prod_{k\geq
1}(x_{i+k}/x_{i+k-1})^{\lambda^{(i)}_{k}},$ (3.8)

with $\mathsf{P}^{N}$ the set of all $N$-partitions
$\boldsymbol{\lambda}=\big{(}\lambda^{(1)},\lambda^{(2)},\dots,\lambda^{(N)}\big{)}$,
$\lambda^{(i)}$ a partition of arbitrary length for $i=1,\dots,N$, and

$\displaystyle
C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)=\prod_{i=1}^{N}\prod_{i<j\leq
k<\infty}\frac{\big{(}q^{\lambda^{(i)}_{k-i+1}-\lambda^{(j)}_{k-j+1}}ts_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}{\big{(}q^{\lambda^{(i)}_{k-i+1}-\lambda^{(j)}_{k-j+1}}qs_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}$
$\displaystyle\hphantom{C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)=}{}\times\prod_{i=1}^{N}\prod_{i\leq
j<k<\infty}\frac{\big{(}q^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}qs_{j}/ts_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}{\big{(}q^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}s_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}$
(3.9)

setting $\lambda^{(i+N)}_{j}\equiv\lambda^{(i)}_{j}$.

###### Proof.

Straightforward computations, using that
$\theta_{ik}=\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}$ defines a one-to-one
correspondence between multi-partitions
$\boldsymbol{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(N)}\big{)}$ in
$\mathsf{P}^{N}$ and matrices $\theta=(\theta_{ik})_{i,k=1}^{\infty}$ in
$\hat{\mathsf{M}}_{N}$ (the interested reader can find the details in Appendix
A.1). ∎

It is interesting to note that

$\displaystyle\varphi_{N,\infty}(\bar{x}|\bar{s}|q,t)\equiv\prod_{i=1}^{N}\prod_{j>i}\frac{(qx_{j}/tx_{i};q)_{\infty}}{(qx_{j}/x_{i};q)_{\infty}}\,f_{N,\infty}(\bar{x}|\bar{s}|q,t)$

is a natural generalization of the function in (2.6) due to the following
implication of Theorem 3.2.

######  Fact 3.6.

The following holds,

$\displaystyle\varphi^{\widehat{\mathfrak{gl}}_{N}}\big{(}p^{\delta/N}x,p^{1/N}|\kappa^{\delta/N}s,\kappa^{1/N}|q,q/t\big{)}=\varphi_{N,\infty}(\bar{x}|\bar{s}|q,t)$

with the variables $x=(x_{i})_{i=1}^{N}$ and $s=(x_{i})_{i=1}^{N}$ on the
left-hand side extended to variables $\bar{x}=(x_{i})_{i=1}^{\infty}$ and
$\bar{s}=(s_{i})_{i=1}^{\infty}$ on the right-hand side by the rules in (3.5).
Moreover, the conjectures in (2.12) are equivalent to

$\displaystyle\varphi_{N,\infty}(\bar{x}|\bar{s}|q,t)=\varphi_{N,\infty}(\bar{s}|\bar{x}|q,t)\quad$
$\displaystyle\text{(bispectral duality)},$
$\displaystyle\varphi_{N,\infty}(\bar{x}|\bar{s}|q,t)=\varphi_{N,\infty}(\bar{x}|\bar{s}|q,q/t)\quad$
$\displaystyle\text{(Poincar\'{e} duality)},$ (3.10)

under the conditions in (3.5).

###### Proof.

Since
$p^{(j-i)/N}\big{(}p^{\delta/N}x\big{)}_{j}/\big{(}p^{\delta/N}x\big{)}_{i}=x_{j}/x_{i}$
for all $i,j=1,\dots,N$, we only need to show that

$\displaystyle\prod_{i=1}^{N}\prod_{j>i}\frac{(qx_{j}/tx_{i};q)_{\infty}}{(qx_{j}/x_{i};q)_{\infty}}=\prod_{1\leq
i<j\leq
N}\frac{(qx_{j}/tx_{i};q,p)_{\infty}}{(qx_{j}/x_{i};q,p)_{\infty}}\prod_{1\leq
i\leq j\leq
N}\frac{(qpx_{i}/tx_{j};q,p)_{\infty}}{(qpx_{i}/x_{j};q,p)_{\infty}}.$

This is proved in Appendix D, Lemma D.1. ∎

.

###  3.2 Convergence

We prove that the non-stationary Ruijsenaars functions
$f_{N}\\!(x,p|s,\kappa|q,t)$ in Definitions 3.1 and 3.3 are absolutely
convergent in a certain domain of variables and parameters.

######  Theorem 3.7.

For fixed $N\in{\mathbb{Z}}_{\geq 1}$, assume that the variables
$s=(s_{1},\dots,s_{N})\in{\mathbb{C}}^{N}$ and the parameters $q$ and $\kappa$
satisfy the following conditions,

  * $(i)$

for some $\sigma>0$,

$\displaystyle{}|\sin\arg(s_{i}/s_{j})|>\sigma\qquad(1\leq i<j\leq N),$

  * $(ii)$

$q$ and $\kappa$ both are real, and either $|q|<1$ and $|\kappa|>1$, or
$|q|>1$ and $|\kappa|<1$.

Then, there exists a constant $\rho>0$ such that the formal power series

$\displaystyle
f_{N,\infty}(x,p;s,\kappa|q,t)\in{\mathbb{C}}[[x_{2}/x_{1},\dots,x_{N}/x_{N-1},px_{1}/x_{N}]]$

in Definitions 3.1 and 3.3 is absolutely convergent in the domain

$\displaystyle|p|<\rho^{N},\qquad|x_{2}/x_{1}|<\rho,\qquad\dots,\qquad|x_{N}/x_{N-1}|<\rho,\qquad|px_{1}/x_{N}|<\rho.$
(3.11)

######  Remark 3.8.

In our proof, we actually show convergence for any $\rho<1/C_{1}C_{2}$ where

$\displaystyle
C_{1}=1+|1-t/q|\max\left(\frac{1}{\sigma},\frac{|\kappa|}{|1-|\kappa||}\right)\\!,$
$\displaystyle
C_{2}=1+|1-q/t|\max\left(\frac{1}{\sigma},\frac{1}{|1-|q||}\right)\\!.$ (3.12)

######  Remark 3.9.

We believe that it is possible to refine this convergence result. In
particular, we believe that there are regions of convergence where
$s_{i}/s_{j}$, $1\leq i<j\leq N$, are real and $q$ and $\kappa$ have non-
trivial imaginary parts.

###### Proof of Theorem 3.7.

Our strategy of proof is to show that our assumptions imply simple upper
bounds on the terms appearing in the series in (3.8)–(3.9):

$\displaystyle\Bigg{|}\prod_{i=1}^{N}\prod_{k\geq
1}(x_{i+k}/x_{i+k-1})^{\lambda^{(i)}_{k}}\Bigg{|}\leq\rho^{|\boldsymbol{\lambda}|},\qquad\left|C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)\right|\leq
C_{1}^{|\boldsymbol{\lambda}|}C_{2}^{|\boldsymbol{\lambda}|}$ (3.13)

with $|\boldsymbol{\lambda}|\equiv\sum_{i=1}^{N}\sum_{k\geq
1}\lambda^{(i)}_{k}$ and $\alpha=\rho C_{1}C_{2}<1$. With that, absolute
convergence follows from the comparison test: the series in (3.8)–(3.9) is of
the form
$\sum_{\boldsymbol{\lambda}\in\mathsf{P}^{N}}a_{\boldsymbol{\lambda}}$ with
$|a_{\boldsymbol{\lambda}}|\leq\alpha^{|\boldsymbol{\lambda}|}$ for all
$\boldsymbol{\lambda}\in\mathsf{P}^{N}$, and the series
$\sum_{\boldsymbol{\lambda}\in\mathsf{P}^{N}}\alpha^{|\boldsymbol{\lambda}|}$
converges absolutely for $|\alpha|<1$.

The first estimate in (3.13) is a simple consequence of the conditions in
(3.11): since $x_{i+N}=px_{i}$ for all $i\geq 1$, these conditions are
equivalent to

$\displaystyle|x_{i+1}/x_{i}|<\rho\qquad(i\geq 1),$

which clearly implies the result.

The proof of the second estimate in (3.13) is more involved and, for this
reason, we supplement our somewhat descriptive arguments in the main text
below by a detailed argument in Appendix B.

We observe that $C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)$ in (3.9) is a
product of fractions $\big{(}1-q^{l}au\big{)}/\big{(}1-q^{l}u\big{)}$ with
$a=t/q$ in the first group of products and $a=q/t$ in the second group,
$l\in{\mathbb{Z}}$, and $u=s_{j}/s_{i}$ for $i=1,\dots,N$ and $j\geq i$;
moreover, $s_{j+\ell N}=\kappa^{\ell}s_{j}$ for $\ell\in{\mathbb{Z}}_{\geq
1}$. Such a fraction can be estimated in a simple way:

$\displaystyle\left|\frac{1-q^{l}au}{1-q^{l}u}\right|=\left|1+(1-a)\frac{q^{l}u}{1-q^{l}u}\right|\leq
1+|1-a|\left|\frac{q^{l}u}{1-q^{l}u}\right|\\!.$

If $j-i$ is not an integer multiple of $N$, we can estimate this further using

$\displaystyle\left|\frac{z}{1-z}\right|\leq\frac{1}{|\sin\arg(z)|}\qquad(z\in{\mathbb{C}}\setminus\\{{\mathbb{R}}\\})$
(3.14)

(to see that the latter inequality holds, write $z=|z|{\rm e}^{{\rm
i}\varphi}$ and note that (3.14) is equivalent to

$\displaystyle|z|^{2}\sin^{2}\varphi\leq
1+|z|^{2}-2|z|\cos\varphi\Leftrightarrow 0\leq(1-|z|\cos\varphi)^{2},$

which is obvious). Since we assume that $q$ and $\kappa$ both are real,

$\displaystyle\big{|}\sin\arg\big{(}q^{l}\kappa^{\ell}s_{j}/s_{i}\big{)}\big{|}=|\sin\arg(s_{j}/s_{i})|\geq\sigma>0\qquad(j-i\neq
N{\mathbb{Z}}_{\geq 0})$

for all integers $l,\ell$, we get a simple universal bound for these
fractions:

$\displaystyle\left|\frac{1-q^{l}as_{j}/s_{i}}{1-q^{l}s_{j}/s_{i}}\right|\leq
1+|1-a|\frac{1}{\sigma}\qquad(j-i\notin N{\mathbb{Z}}_{\geq 0})$

for all integers $l$. However, this bound does not work for $j=i+\ell N$ with
$\ell\in{\mathbb{Z}}_{\geq 0}$ since, in these cases,
$q^{l}u=q^{l}s_{j}/s_{i}=q^{l}\kappa^{\ell}$ is real. However, one can check
that, in all these latter cases, either $l\leq 0$ and $\ell>0$, or $l<0$ and
$\ell\geq 0$, and thus, by our assumptions, $z\equiv
q^{l}u=q^{l}\kappa^{\ell}$ always satisfies either
$|z|\geq\min\big{(}|q|^{-1},|\kappa|\big{)}>1$ (if $|q|<1$ and $|\kappa|>1$)
or $|z|\leq\max\big{(}|q|,|\kappa|^{-1}\big{)}<1$ (if $|q|>1$ and
$|\kappa|<1$); we therefore can use the inequality

$\displaystyle\left|\frac{z}{1-z}\right|\leq\frac{|z|}{|1-|z||}\qquad(|z|\neq
1)$

to get simple universal bounds for the cases $j=i+\ell N$ with
$\ell\in{\mathbb{Z}}_{\geq 0}$ as well (we spell our the details of this
argument in Appendix B.2.2). We thus get estimates

$\displaystyle\left|\frac{1-q^{l}au}{1-q^{l}u}\right|\leq C_{1,2}$

with different upper bounds, $C_{1}$ and $C_{2}$, for all fractions in the
first and second groups of products on the right-hand side in (3.9),
respectively. The arguments above allow to compute the constants $C_{1}$ and
$C_{2}$ and give the results in (3.12); the interested reader can find the
details of this computation in Appendix B.

Inserting these bounds into (3.9) we obtain

$\displaystyle|C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)|\leq\prod_{i=1}^{N}\Bigg{(}\prod_{i<j\leq
k<\infty}C_{1}^{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}\Bigg{)}\Bigg{(}\prod_{i\leq
j<k<\infty}C_{2}^{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}\Bigg{)}$
$\displaystyle\hphantom{|C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)|}{}=\prod_{i=1}^{N}\Bigg{(}\prod_{i<j<\infty}C_{1}^{\lambda^{(i)}_{j-i}}\Bigg{)}\Bigg{(}\prod_{i\leq
j<\infty}C_{2}^{\lambda^{(i)}_{j+1-i}}\Bigg{)}$
$\displaystyle\hphantom{|C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)|}{}=\prod_{i=1}^{N}\Bigg{(}\prod_{k\geq
1}C_{1}^{\lambda^{(i)}_{k}}\Bigg{)}\Bigg{(}\prod_{k\geq
1}C_{2}^{\lambda^{(i)}_{k}}\Bigg{)}=C_{1}^{|\boldsymbol{\lambda}|}C_{2}^{|\boldsymbol{\lambda}|},$
(3.15)

computing telescoping products in the second step and using
$\sum_{i=1}^{N}\sum_{k\geq 1}\lambda^{(i)}_{k}=|\boldsymbol{\lambda}|$ in the
last step. This proves the second estimate in (3.13).

To conclude, we prove that the series
$\sum_{\boldsymbol{\lambda}\in\mathsf{P}^{N}}\alpha^{|\boldsymbol{\lambda}|}$
for $|\alpha|<1$ is absolutely convergent by the following computation,

$\displaystyle\sum_{\boldsymbol{\lambda}\in\mathsf{P}^{N}}\alpha^{|\boldsymbol{\lambda}|}=\sum_{\lambda^{(1)},\dots,\lambda^{(N)}\in\mathsf{P}}\prod_{i=1}^{N}\alpha^{|\lambda^{(i)}|}=\prod_{i=1}^{N}\sum_{\lambda^{(i)}\in\mathsf{P}}\alpha^{|\lambda^{(i)}|}=\left(\sum_{\lambda\in\mathsf{P}}\alpha^{|\lambda|}\right)^{N}=\frac{1}{(\alpha;\alpha)_{\infty}^{N}},$

using the definition $|\lambda|\equiv\sum_{k\geq 1}\lambda_{k}$ for partitions
$\lambda$; for clarity, and for the convenience of the reader, we give in
Appendix D the well-known identity used in the last step, together with its
elementary proof making absolute convergence manifest; see (D.1)–(D.2). ∎

##  4 ${\mathcal{T}}$-operators

For fixed $N\in{\mathbb{Z}}_{\geq 1}$, we define an operator ${\mathcal{T}}$
which acts diagonally on the asymptotically free solution of the trigonometric
Ruijsenaars model (Section 4.1). We also present a natural non-stationary
generalization of this operator which, as we conjecture, acts diagonally on
the corresponding non-stationary Ruijsenaars function (Section 4.2).

###  4.1 Trigonometric case

We find it convenient to work with formal power series.

######  Definition 4.1.

For

$\displaystyle\Delta\equiv\sum_{i=1}^{N}\left(x_{i}\partial_{x_{i}}+(N-i)\beta\right)^{2}$
(4.1)

with $\beta=\log(t)/\log(q)$, let

$\displaystyle{\mathcal{T}}_{N}(x|q,t)\equiv\sum_{\theta\in\mathsf{M}_{N}}\prod_{1\leq
i<j\leq
N}(x_{j}/x_{i})^{\theta_{ij}}q^{\frac{1}{2}\Delta}c_{N}(\theta|x|q,t)\prod_{1\leq
i<j\leq N}\frac{(x_{j}/x_{i};q)_{\infty}}{(tx_{j}/x_{i};q)_{\infty}}$ (4.2)

on $x^{\lambda}{\mathbb{C}}[[x_{2}/x_{1},\dots,x_{N}/x_{N-1}]]$ for
$\lambda\in{\mathbb{C}}^{N}$, with $\mathsf{M}_{N}$ in (2.4) and
$c_{N}(\theta|s|q,t)$ in (2.5).

Clearly, the operator ${\mathcal{T}}_{N}(x|q,t)$ is complicated: it has the
same complexity as the function $f_{N}(x|s|q,t)$; cf. (2.3). Still, it is
interesting since, different from the elliptic Macdonald–Ruijsenaars operators
in (2.8), we know its natural generalization to the non-stationary case; see
Section 4.2.

The following is our main result in this section.

######  Proposition 4.2.

The ${\mathcal{T}}$-operator in (4.2) is well-defined, it commutes with the
trigonometric Macdonald–Ruijsenaars operators in (2.1):

$\displaystyle\big{[}{\mathcal{T}}_{N}(x|q,t),D_{N}^{\pm}(x|q,t)\big{]}=0$
(4.3)

on $x^{\lambda}{\mathbb{C}}[[x_{2}/x_{1},\dots,x_{N}/x_{N-1}]]$ for all
$\lambda\in{\mathbb{C}}^{N}$, and it acts diagonally on the asymptotically
free solutions of the trigonometric Ruijsenaars model in (2.3)–(2.5):

$\displaystyle{\mathcal{T}}_{N}(x|q,t)x^{\lambda}f_{N}(x|s|q,t)=\varepsilon_{N}(s|q)x^{\lambda}f_{N}(x|s|q,t),\qquad
s_{i}=t^{N-i}q^{\lambda_{i}},$ (4.4)
$\displaystyle\varepsilon_{N}(s|q)=q^{\frac{1}{2}\sum_{i=1}^{N}[\log(s_{i})/\log(q)]^{2}}$
(4.5)

$($note that $\log(s_{i})/\log(q)=\lambda_{i}+\beta(N-i))$.

(A proof based on results in the rest of this section can be found in Appendix
C.)

Our proof of Proposition 4.2 is based on the following convenient
representation of the ${\mathcal{T}}$-operator.

######  Lemma 4.3.

For $f(x)\in{\mathbb{C}}[[x_{2}/x_{1},\dots,x_{N}/x_{N-1}]]$ and
$\lambda\in{\mathbb{C}}^{N}$,

$\displaystyle{\mathcal{T}}_{N}(x|q,t)x^{\lambda}f(x)=\varepsilon(\lambda)x^{\lambda}\Bigg{[}\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\chi_{N}(x|y|q,t)\\!\\!\prod_{1\leq
i<j\leq N}\\!\\!(1-y_{j}/y_{i})f(y)\Bigg{]}_{1,y}$ (4.6)

with

$\displaystyle\varepsilon(\lambda)=q^{\frac{1}{2}\sum_{i^{1}}^{N}(\lambda_{i}+(N-i)\beta)^{2}},$

$\vartheta_{3}(z|q)\equiv\sum_{n\in{\mathbb{Z}}}q^{\frac{1}{2}n^{2}}z^{n}$ the
third Jacobi theta function,

$\displaystyle\chi_{N}(x|y|q,t)\equiv f_{N}(x|y|q,t)\prod_{1\leq i<j\leq
N}\frac{(qy_{j}/y_{i};q)_{\infty}}{(ty_{j}/y_{i};q)_{\infty}},$ (4.7)

and $[\cdots]_{1,y}$ is the constant term in $y$, i.e., for formal Laurent
series $g(y)=\sum_{\mu\in{\mathbb{Z}}^{N}}g_{\mu}y^{\mu}$ as in (4.6),
$[g(y)]_{1,y}=g_{0}$.

######  Remark 4.4.

We use $[\cdots]_{1,y}$ only for
$g(y)\in\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q){\mathbb{C}}[[y_{2}/y_{1},\dots,y_{N}/y_{N-1}]]$,
and our definition of $[\cdots]_{1,y}$ is non-ambiguous for these.

###### Proof of Lemma 4.3.

We use that ${\mathbb{C}}[[x_{2}/x_{1},\dots,x_{N}/x_{N-1}]]$ is spanned by (a
subset of) monomials $x^{\mu}$ with $\mu\in{\mathbb{Z}}^{N}$. For fixed
$\lambda\in{\mathbb{C}}^{N}$, we compute the action of $q^{\frac{1}{2}\Delta}$
on $x^{\lambda}x^{\mu}$, $\mu\in{\mathbb{Z}}^{N}$:

$\displaystyle
q^{\frac{1}{2}\Delta}x^{\lambda}x^{\mu}=q^{\frac{1}{2}\sum_{i=1}^{N}(\lambda_{i}+\mu_{i}+(N-i)\beta)^{2}}x^{\lambda+\mu}=\varepsilon(\lambda)x^{\lambda}\prod_{i=1}^{N}(x_{i}s_{i})^{\mu_{i}}q^{\frac{1}{2}\sum_{i=1}^{N}\mu_{i}^{2}}$

equal to

$\displaystyle\varepsilon(\lambda)x^{\lambda}\left[\,\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)y^{\mu}\right]_{1,y}\\!,$

and thus

$\displaystyle
q^{\frac{1}{2}\Delta}x^{\lambda}f(x)=\varepsilon(\lambda)x^{\lambda}\left[\,\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)f(y)\right]_{1,y}$

for all $f(x)\in{\mathbb{C}}[[x_{2}/x_{1},\dots,x_{N}/x_{N-1}]]$. This and the
definition in (4.2) give

$\displaystyle{\mathcal{T}}_{N}(x|q,t)x^{\lambda}f(x)=\varepsilon(\lambda)x^{\lambda}\Biggl{[}\,\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)$
$\displaystyle\hphantom{{\mathcal{T}}_{N}(x|q,t)x^{\lambda}f(x)=}{}\times\sum_{\theta\in\mathsf{M}_{N}}\prod_{1\leq
i<j\leq N}(x_{j}/x_{i})^{\theta_{ij}}c_{N}(\theta|y|q,t)\prod_{1\leq i<j\leq
N}\frac{(y_{j}/y_{i};q)_{\infty}}{(ty_{j}/y_{i};q)_{\infty}}f(y)\Biggr{]}_{1,y},$

and using (2.3) and the definition in (4.7) we obtained (4.6). ∎

We note that $\chi_{N}(x|y|q,t)=\chi_{N}(y|x|q,q/t)$ (this is proved in
Appendix C, Lemma C.1); inserting this in (4.6) and backtracking, one obtains
the following alternative representation of the ${\mathcal{T}}$-operator:

$\displaystyle{\mathcal{T}}_{N}(x|q,t)\equiv\frac{(qx_{j}/x_{i};q)_{\infty}}{(qx_{j}/tx_{i};q)_{\infty}}\sum_{\theta\in\mathsf{M}_{N}}c_{N}(\theta|x|q,q/t)q^{\frac{1}{2}\Delta}\\!\\!\\!\prod_{1\leq
i<j\leq N}\\!\\!\\!(x_{j}/x_{i})^{\theta_{ij}}\\!\\!\\!\prod_{1\leq i<j\leq
N}\\!\\!\\!(1-x_{j}/x_{i}).$ (4.8)

###  4.2 Non-stationary case

We present a non-stationary generalization of the ${\mathcal{T}}$-operator.

######  Definition 4.5.

For $\Delta$ in (4.1) with $\beta=\log(t)/\log(q)$, let

$\displaystyle{\mathcal{T}}_{N,\infty}(x,p|q,t,\kappa)=\sum_{\theta\in\hat{\mathsf{M}}_{N}}\prod_{i=1}^{N}\prod_{j>i}(x_{j}/x_{i})^{\theta_{ij}}q^{\frac{1}{2}\Delta}\,T_{\kappa,p}c_{N,\infty}(\theta|\bar{x}|q,t)\prod_{i=1}^{N}\prod_{j>i}\frac{(x_{j}/x_{i};q)_{\infty}}{(tx_{j}/x_{i};q)_{\infty}}$
(4.9)

with $x_{i+N}=px_{i}$ for $i\geq 1$ on
$x^{\lambda}{\mathbb{C}}[[x_{2}/x_{1},x_{3}/x_{2},\dots,x_{N}/x_{N-1},px_{1}/x_{N}]]$
for $\lambda\in{\mathbb{C}}^{N}$, with $\hat{\mathsf{M}}_{N}$ in (3.2) and
$c_{N,\infty}(\theta|\bar{x}|q,t)$ in (3.3).

######  Remark 4.6.

To make the $p$-dependence of this operator manifest, one can write it as

$\displaystyle{\mathcal{T}}_{N,\infty}(x,p|q,t,\kappa)=\sum_{\theta\in\hat{\mathsf{M}}_{N}}e_{N,\infty}(x,p)q^{\frac{1}{2}\Delta}\,T_{\kappa,p}c_{N,\infty}(\theta|x,p|q,t)$
$\displaystyle\hphantom{{\mathcal{T}}_{N,\infty}(x,p|q,t,\kappa)=}{}\times\prod_{1\leq
i<j\leq
N}\frac{(x_{j}/x_{i};q,p)_{\infty}}{(tx_{j}/x_{i};q,p)_{\infty}}\prod_{1\leq
i\leq j\leq
N}\frac{(px_{i}/x_{j};q,p)_{\infty}}{(ptx_{i}/x_{j};q,p)_{\infty}}$

using the definitions in (3.6) and Lemma D.1 in Appendix D.

By comparing with (3.1)–(3.3), it is clear that the operator in (4.9) is a
natural non-stationary generalization of the trigonometric
${\mathcal{T}}$-operators in (4.2); however, there is one important new
feature: the shift operator $T_{\kappa,p}$ acting on $p$.

We propose the following generalization to Proposition 4.2; this conjecture is
a complement to the ones in [21].

######  Conjecture 4.7.

The non-stationary ${\mathcal{T}}$-operator in (4.9) has a well-defined
diagonal action on the non-stationary Ruijsenaars function in Definitions 2.1
and 3.3:

$\displaystyle{\mathcal{T}}_{N,\infty}(x,p|q,t,\kappa)x^{\lambda}f_{N,\infty}(x,p|s,\kappa|q,t)=\varepsilon_{N}(s|q)x^{\lambda}f_{N,\infty}(x,p|s,\kappa|q,t),\qquad
s_{i}=t^{N-i}q^{\lambda_{i}}$

with $\varepsilon_{N}(s|q)$ given in (4.5).

In the rest of this section, we present two generalizations of results about
the trigonometric ${\mathcal{T}}$-operators: (i) the constant-term
representation of the ${\mathcal{T}}$-operator in Lemma 4.3(ii) the
alternative representation in (4.8) obtained with the duality in (2.12). We
also rephrase Conjecture 4.7 in terms of the non-stationary Ruijsenaars
functions as defined in [21].

One can adapt the proof Lemma 4.3 to obtain the following constant-term
representation of the ${\mathcal{T}}$-operator in (4.9):

######  Lemma 4.8.

For
$f(x,p)\in{\mathbb{C}}[[x_{2}/x_{1},px_{3}/x_{2},\dots,x_{N}/x_{N-1},px_{1}/x_{N}]]$
and $\lambda\in{\mathbb{C}}^{N}$,

$\displaystyle{\mathcal{T}}_{N,\infty}(x,p|q,t,\kappa)x^{\lambda}f(x,p)$
$\displaystyle\qquad{}=\varepsilon(\lambda)x^{\lambda}\Biggl{[}\,\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\frac{1}{1-\kappa
p/u}\chi_{N,\infty}(x,p|y,u|q,t)\prod_{i=1}^{N}\prod_{j>i}(1-y_{j}/y_{i})f(y,u)\Biggr{]}_{1,y,u}$

with

$\displaystyle\chi_{N,\infty}(x,p|y,u|q,t)=f_{N,\infty}(x,p|y,u|q,t)\prod_{i=1}^{N}\prod_{j>i}\frac{(qy_{j}/y_{i};q)_{\infty}}{(ty_{j}/y_{i};q)_{\infty}}$

setting $y_{i+N}=uy_{i}$ for all $i\geq 1$, $f_{N,\infty}(x,p|y,u|q,t)$ in
(3.1)–(3.3), and $[\cdots]_{1;y,u}$, short for $[[\cdots]_{1,y}]_{1,u}$, the
constant term in $y\in{\mathbb{C}}^{N}$ and $u\in{\mathbb{C}}$.

###### Proof.

This is proved by a straightforward generalization of the arguments given in
the proof of Lemma 4.3; the only new ingredient is

$\displaystyle T_{\kappa,p}p^{n}=(\kappa p)^{n}=\left[\frac{1}{1-\kappa
p/u}u^{n}\right]_{1,u}\qquad(n\in{\mathbb{Z}}_{\geq 0}),$

and therefore

$\displaystyle
q^{\frac{1}{2}\Delta}T_{\kappa,p}x^{\lambda}f(x,p)=\varepsilon(\lambda)x^{\lambda}\left[\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\frac{1}{1-\kappa
p/u}f(y,u)\right]_{1,y,u}$

for all
$f(x,p)\in{\mathbb{C}}[[x_{2}/x_{1},px_{3}/x_{2},\dots,x_{N}/x_{N-1},px_{1}/x_{N}]]$.
∎

Moreover, by an argument similar to the one for the trigonometric
${\mathcal{T}}$-operator above, the conjectured duality in (3.10) implies

$\displaystyle\chi_{N,\infty}(x,p|y,u|q,t)=\chi_{N,\infty}(y,u|x,p|q,q/t)$

and the following alternative representation of this ${\mathcal{T}}$-operator

$\displaystyle{\mathcal{T}}_{N,\infty}(x,p|q,t,\kappa)=\prod_{i=1}^{N}\prod_{j>i}\frac{(qx_{j}/x_{i};q)_{\infty}}{(qx_{j}/tx_{i};q)_{\infty}}\sum_{\theta\in\hat{\mathsf{M}}_{N}}c_{N,\infty}(\theta|\bar{x}|q,t)q^{\frac{1}{2}\Delta}T_{\kappa,p}$
$\displaystyle\hphantom{{\mathcal{T}}_{N,\infty}(x,p|q,t,\kappa)=}{}\times\prod_{i=1}^{N}\prod_{j>i}(x_{j}/x_{i})^{\theta_{ij}}\prod_{i=1}^{N}\prod_{j>i}(1-x_{j}/x_{i})$

with $x_{i+N}=px_{i}$ for all $i\geq 1$.

To conclude, we rephrase Conjecture 4.7 using balanced coordinates.

######  Definition 4.9.

For $\Delta$ as in (4.1) with $\beta=\log(q/t)/\log(q)$, let

$\displaystyle{\mathcal{T}}^{\widehat{\mathfrak{gl}}_{N}}(x,p|q,t,\kappa)$
$\displaystyle\qquad{}\equiv\sum_{\lambda^{(1)},\dots,\lambda^{(N)}\in\mathsf{P}}\prod_{\beta=1}^{N}\prod_{\alpha\geq
1}(px_{\alpha+\beta}/tx_{\alpha+\beta-1})^{\lambda^{(\beta)}_{\alpha}}q^{\frac{1}{2}\Delta}T_{\kappa,p}\prod_{i,j=1}^{N}\frac{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}(tx_{j}/x_{i}|q,p)}{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}(x_{j}/x_{i}|q,p)}$
$\displaystyle\qquad\phantom{\equiv}{}\times\prod_{1\leq i<j\leq
N}\frac{\big{(}p^{j-i}x_{j}/x_{i};q,p^{N}\big{)}_{\infty}}{\big{(}p^{j-i}qx_{j}/tx_{i};q,p^{N}\big{)}_{\infty}}\prod_{1\leq
i\leq j\leq
N}\frac{\big{(}p^{N-j+i}x_{i}/x_{j};q,p^{N}\big{)}_{\infty}}{\big{(}p^{N-j+i}qx_{i}/tx_{j};q,p^{N}\big{)}_{\infty}}$
(4.10)

with $x_{i+N}=x_{i}$ for $i\geq 1$ on
$x^{\lambda}{\mathbb{C}}[[px_{2}/x_{1},px_{3}/x_{2},\dots,px_{N}/x_{N-1},px_{1}/x_{N}]]$
for $\lambda\in{\mathbb{C}}^{N}$, with $\mathsf{P}$ the set of all partitions
and $\mathsf{N}^{(k|N)}_{\lambda,\mu}(u|q,\kappa)$ the Nekrasov factors given
in (2.10).

######  Fact 4.10.

Conjecture 4.7 is equivalent to the following diagonal action of the non-
stationary ${\mathcal{T}}$-operator in (4.10) on the non-stationary
Ruijsenaars function in (2.9)–(2.10),

$\displaystyle{\mathcal{T}}^{\widehat{\mathfrak{gl}}_{N}}(x,p|q,t)x^{\lambda}f^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)\,{=}\,\varepsilon_{N}(s|q)x^{\lambda}f^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t),\quad
s_{i}\,{=}\,(q/t)^{N-i}q^{\lambda_{i}}.$ (4.11)

###### Proof.

This is implied by Theorem 3.2, using that the shift operator $T_{\kappa,p}$
commutes with the following operator, $\Phi$, switching from unbalanced to
balanced coordinates:

$\displaystyle(\Phi f)(x,p|s,\kappa|q,t)\equiv
f\big{(}p^{-\delta}x,p^{N}|\kappa^{-\delta}s,\kappa^{N}|q,t/q\big{)},$

and noting that Theorem 3.2 implies $(\Phi
f_{N,\infty})(x,p|s,\kappa|q,t)=f^{\widehat{\mathfrak{gl}}_{N}}(x,p|s,\kappa|q,t)$.
∎

##  5 Final remarks

The main conjecture in [21, Conjecture 1.14] can be tested systematically
using a perturbative solution of the elliptic Ruijsenaars model that
generalizes the perturbative solution of the elliptic Calogero–Sutherland
(eCS) model in [12]. We plan to present this elsewhere.

As already mentioned, one important outstanding problem is to find
$\kappa$-deformations of the elliptic Ruijsenaars operators in (2.8) that have
the non-stationary Ruijsenaars functions as eigenfunctions. As conjectured in
[21], the limit $q\to 1$ of this hypothetical non-stationary Ruijsenaars model
is a known non-stationary eCS model depending on parameters $\beta$, $p$ and
$\kappa$ related to the non-stationary Ruijsenaars parameters as follows,
$t=q^{\beta}$ and $\kappa=q^{-\varkappa}$.666The elliptic deformation
parameter $p$ is the same in both cases. Recently, a rigorous construction of
integral representations of eigenfunctions of the non-stationary eCS model for
$\varkappa=\beta$ was presented [2]. We hope that, by combining the latter
results with recent results on the non-stationary Ruijsenaars functions for
the corresponding special value of $\kappa$ [9], it will be possible to prove
the main conjecture in [21] in the non-stationary eCS limit $q\to 1$ and for
$\varkappa=\beta$. Another possible strategy to prove the conjecture in [21]
for $q\to 1$ and general $\varkappa$-values is to try to generalize the
perturbative solution of the non-stationary Lamé equation in [1] (note that
the latter equation reduces to the non-stationary eCS model for $N=2$ in
special cases).

The elliptic Ruijsenaars model is invariant under the exchange
$p\leftrightarrow q$ [17].777We thank S. Ruijsenaars for pointing this out at
the NORDITA workshop “Elliptic integrable systems, special functions and
quantum field theory” in June 2019. The non-stationary Ruijsenaars functions
do not have this property manifest; we plan to report elsewhere on how this
duality is recovered from the non-stationary Ruijsenaars function.

It was suggested more than 20 years ago that the elliptic Ruijsenaars model
has a double-elliptic generalization with remarkable duality properties [4,
8], and recently an explicit formula for an operator defining such a model was
conjectured [11]. It would be interesting to obtain a better understanding of
the relation between the non-stationary Ruijsenaars functions and this double
elliptic system recently proposed in [3].

Since the non-stationary ${\mathcal{T}}$-operator proposed in this paper
contains a factor $q^{\frac{1}{2}\Delta}T_{\kappa,p}$, its eigenvalue equation
can be regarded as a $q$-deformed heat equation. We mention the work of Felder
and Varchenko on the $q$-deformed KZB heat equation [5, 6] which seems
related; it would be interesting to understand this relation in detail.

##  Appendix A Alternative series representation

We prove Theorem 3.2. We start with details complementing the concise proof of
Lemma 3.5 in the main text (Appendix A.1). The main part of the proof is in
Appendix A.2.

###  A.1 Details on Lemma 3.5

One can check that the following two formulas provide a correspondence between
multi-partitions
$\boldsymbol{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(N)}\big{)}$ in
$\mathsf{P}^{N}$ and matrices $\theta=(\theta_{ik})_{i,k=1}^{\infty}$ in
$\hat{\mathsf{M}}_{N}$ that is one-to-one:

$\displaystyle\theta_{ik}=\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}$

and

$\displaystyle\lambda^{(i)}_{k-i}=\sum_{a\geq k}\theta_{ia}$ (A.1)

setting $\lambda^{(i+N)}=\lambda^{(i)}$ for all $i\geq 1$. With this
identification, $C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)$ in (3.9) is
clearly equal to $c_{N,\infty}(\theta|\bar{s}|q,t)$ in (3.3), and

$\displaystyle\begin{split}&\prod_{i=1}^{N}\prod_{k=i+1}^{\infty}(x_{k}/x_{i})^{\theta_{ik}}=\prod_{i=1}^{N}\prod_{k=i+1}^{\infty}[(x_{k}/x_{k-1})(x_{k-1}/x_{k-2})\cdots(x_{i+1}/x_{i})]^{\theta_{ik}}\\\
&\hphantom{\prod_{i=1}^{N}\prod_{k=i+1}^{\infty}(x_{k}/x_{i})^{\theta_{ik}}}{}=\prod_{i=1}^{N}\prod_{i<j\leq
k<\infty}(x_{j}/x_{j-1})^{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}=\prod_{i=1}^{N}\prod_{j>i}(x_{j}/x_{j-1})^{\lambda^{(i)}_{j-i}}\\\
&\hphantom{\prod_{i=1}^{N}\prod_{k=i+1}^{\infty}(x_{k}/x_{i})^{\theta_{ik}}}{}=\prod_{i=1}^{N}\prod_{k\geq
1}(x_{i+k}/x_{i+k-1})^{\lambda^{(i)}_{k}},\end{split}$

inserting a telescoping product in the second step, using (A.1) in the third,
and computing a telescoping product in the fourth. This proves the result.

###  A.2 Proof of Theorem 3.2

We show by direct computations that the function on the left-hand side in
(3.7) is equal to the function $f_{N,\infty}(\bar{x}|\bar{s}|q,t)$ in
(3.8)–(3.9) with $x_{i+N}=px_{i}$ and $s_{i+N}=\kappa s_{i}$, for all $i\geq
1$. This, together with Lemma 3.5, proves the result.

We compute the function on the left-hand side in (3.7) using (2.9)–(2.10):

$\displaystyle
f^{\widehat{\mathfrak{gl}}_{N}}\big{(}p^{\delta/N}x,p^{1/N}|\kappa^{\delta/N}s,\kappa^{1/N}|q,q/t\big{)}$
$\displaystyle\qquad{}=\sum_{\lambda^{(1)},\dots,\lambda^{(N)}\in\mathsf{P}}\prod_{i,j=1}^{N}\frac{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}(q/t)\big{(}\kappa^{\delta/N}s\big{)}_{j}/(\kappa^{\delta/N}s)_{i}|q,\kappa^{1/N}\big{)}}{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}\big{(}\kappa^{\delta/N}s\big{)}_{j}/\big{(}\kappa^{\delta/N}s\big{)}_{i}|q,\kappa\big{)}}$
$\displaystyle\phantom{\qquad=}{}\times\prod_{\beta=1}^{N}\prod_{\alpha\geq
1}\big{(}p^{1/N}\big{(}p^{\delta/N}\big{)}_{\alpha+\beta}t/q\big{(}p^{\delta/N}\big{)}_{\alpha+\beta-1}\big{)}^{\lambda^{(\beta)}_{\alpha}}.$

By Definition 2.1 of the non-stationary Ruijsenaars functions, the variables
$\big{(}p^{\delta/N}x\big{)}_{i}$ above are extended from $i=1,\dots,N$ to all
$i\geq 1$ by the rule
$\big{(}p^{\delta/N}x\big{)}_{i+N}=\big{(}p^{\delta/N}x\big{)}_{i}$, whereas
$x_{i+N}=px_{i}$ for all $i\geq 1$ implies
$\big{(}p^{\delta/N}x\big{)}_{i+kN}=p^{(N-i)/N}x_{i}=p^{[N-(i+kN)]/N}x_{i+kN}$
for all $i=1,\dots,N$ and $k\in{\mathbb{Z}}_{\geq 1}$, and thus

$\displaystyle\big{(}p^{\delta/N}x\big{)}_{i}=p^{(N-i)/N}x_{i}\qquad(i\geq
1).$

Therefore,

$\displaystyle\prod_{\beta=1}^{N}\prod_{\alpha\geq
1}\big{(}p^{1/N}\big{(}p^{\delta/N}x\big{)}_{\alpha+\beta}t/q\big{(}p^{\delta/N}x\big{)}_{\alpha+\beta-1}\big{)}^{\lambda^{(\beta)}_{\alpha}}$
$\displaystyle\qquad{}=\prod_{\beta=1}^{N}\prod_{\alpha\geq
1}\big{(}p^{1/N}p^{(N-\alpha-\beta)/N}x_{\alpha+\beta}t/qp^{(N-\alpha-\beta+1)/N}x_{\alpha+\beta-1}\big{)}^{\lambda^{(\beta)}_{\alpha}}$
$\displaystyle\qquad{}=(t/q)^{|\boldsymbol{\lambda}|}\prod_{\beta=1}^{N}\prod_{\alpha\geq
1}(x_{\alpha+\beta}/x_{\alpha+\beta-1})^{\lambda^{(\beta)}_{\alpha}}$

using the abbreviation
$|\boldsymbol{\lambda}|\equiv\sum_{\beta=1}^{N}\sum_{\alpha\geq
1}\lambda^{(\beta)}_{\alpha}$. Renaming indices $(\alpha,\beta)\to(k,i)$, we
thus can write the function on the left-hand side in (3.7) as

$\displaystyle
f^{\widehat{\mathfrak{gl}}_{N}}\big{(}p^{\delta/N}x,p^{1/N}|\kappa^{\delta/N}s,\kappa^{1/N}|q,q/t\big{)}=\sum_{\boldsymbol{\lambda}\in\mathsf{P}^{N}}\tilde{C}_{N}(\boldsymbol{\lambda};s|q,t,\kappa)\prod_{i=1}^{N}\prod_{k\geq
1}(x_{i+k}/x_{i+k-1})^{\lambda^{(i)}_{k}}$

with $x_{i+N}=px_{i}$ for all $i\geq 1$ and

$\displaystyle\tilde{C}_{N}(\boldsymbol{\lambda};s|q,t,\kappa)=(t/q)^{|\boldsymbol{\lambda}|}\prod_{i,j=1}^{N}\frac{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}(q/t)\big{(}\kappa^{\delta/N}s\big{)}_{j}/\big{(}\kappa^{\delta/N}s\big{)}_{i}|q,\kappa^{1/N}\big{)}}{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}\big{(}\kappa^{\delta/N}s\big{)}_{j}/\big{(}\kappa^{\delta/N}s\big{)}_{i}|q,\kappa^{1/N}\big{)}}.$
(A.2)

To complete the proof, we have to show that
$\tilde{C}_{N}(\boldsymbol{\lambda};s|q,t,\kappa)$ in (A.2) is equal to
$C_{N,\infty}(\boldsymbol{\lambda};\bar{s}|q,t)$ in (3.9) for $s_{i+N}=\kappa
s_{i}$ ($i\geq 1$). For that, we compute the Nekrasov factors in (2.10),
partially specializing to the variables we need:

$\displaystyle\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}u|q,\kappa^{1/N}\big{)}=\prod_{\genfrac{}{}{0.0pt}{}{b\geq
a\geq
1}{b-a\equiv(j-i)(\mathrm{mod}N)}}\big{(}uq^{-\lambda^{(j)}_{a}+\lambda^{(i)}_{b+1}}\kappa^{(-a+b)/N};q\big{)}_{\lambda^{(i)}_{b}-\lambda^{(i)}_{b+1}}$
$\displaystyle\hphantom{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}u|q,\kappa^{1/N}\big{)}=}{}\times\prod_{\genfrac{}{}{0.0pt}{}{\beta\geq\alpha\geq
1}{\beta-\alpha\equiv(i-j-1)(\mathrm{mod}N)}}\big{(}uq^{\lambda^{(i)}_{\alpha}-\lambda^{(j)}_{\beta}}\kappa^{(\alpha-\beta-1)/N};q\big{)}_{\lambda^{(j)}_{\beta}-\lambda^{(j)}_{\beta+1}}.$

We note that the constraints on $b$ in the first product is solved by
$b=a+j+\ell N-i$ with $\ell$ an arbitrary integer $\geq\chi(i>j)$, using the
definition $\chi(i>j)=1$ for $j<i$ and $0$ otherwise; similarly, the
constraints on $\beta$ in the second product is solved by
$\beta=\alpha+i+\ell^{\prime}N-j-1$ with arbitrary integer
$\ell^{\prime}\geq\chi(j\geq i)$. We thus can write these Nekrasov factors as

$\displaystyle\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}u|q,\kappa^{1/N}\big{)}=\prod_{a\geq
1}\prod_{\ell\geq\chi(i>j)}\\!\\!\\!\\!\big{(}uq^{-\lambda^{(j)}_{a}+\lambda^{(i)}_{a+j+\ell
N-i+1}}\kappa^{(j+\ell N-i)/N};q\big{)}_{\lambda^{(i)}_{a+j+\ell
N-i}-\lambda^{(i)}_{a+j+\ell N-i+1}}$
$\displaystyle\hphantom{\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}u|q,\kappa^{1/N}\big{)}}{}\times\prod_{\alpha\geq
1}\prod_{\ell^{\prime}\geq\chi(j\geq
i)}\\!\\!\\!\\!\\!\\!\big{(}uq^{\lambda^{(i)}_{\alpha}-\lambda^{(j)}_{\alpha+i+\ell^{\prime}N-j-1}}\kappa^{(j-i-\ell^{\prime}N)/N};q\big{)}_{\lambda^{(j)}_{\alpha+i+\ell^{\prime}N-j-1}-\lambda^{(j)}_{\alpha+i+\ell^{\prime}N-j}}\\!.$

We now specialize further to the arguments of interest to us:

$\displaystyle
u=c\big{(}\kappa^{\delta/N}s\big{)}_{j}/\big{(}\kappa^{\delta/N}s\big{)}_{i}=c\kappa^{(N-j)/N}s_{j}/\kappa^{(N-i)/N}s_{i}=c\kappa^{(i-j)/N}s_{j}/s_{i},\qquad
c\in\\{1,q/t\\}.$

For these arguments, the manifest $\kappa$-dependence disappears:

$\displaystyle\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}c\big{(}\kappa^{\delta/N}s\big{)}_{j}/\big{(}\kappa^{\delta/N}s\big{)}_{i}|q,\kappa^{1/N}\big{)}$
$\displaystyle\qquad=\prod_{a\geq
1}\prod_{\ell\geq\chi(i>j)}\big{(}cq^{-\lambda^{(j)}_{a}+\lambda^{(i)}_{a+j+\ell
N-i+1}}s_{j+\ell N}/s_{i};q\big{)}_{\lambda^{(i)}_{a+j+\ell
N-i}-\lambda^{(i)}_{a+j+\ell N-i+1}}$
$\displaystyle\qquad\phantom{=}{}\times\prod_{\alpha\geq
1}\prod_{\ell^{\prime}\geq\chi(j\geq
i)}\big{(}cq^{\lambda^{(i)}_{\alpha}-\lambda^{(j)}_{\alpha+i+\ell^{\prime}N-j-1}}s_{j}/s_{i+\ell^{\prime}N};q\big{)}_{\lambda^{(j)}_{\alpha+i+\ell^{\prime}N-j-1}-\lambda^{(j)}_{\alpha+i+\ell^{\prime}N-j}}$

using $\kappa^{\ell}s_{j}/s_{i}=s_{j+\ell N}/s_{i}$ and
$\kappa^{-\ell^{\prime}}s_{j}/s_{i}=s_{j}/s_{i+\ell^{\prime}N}$ implied by
$s_{i+N}=\kappa s_{i}$ for $i\geq 1$. We now take the product of these
Nekrasov factors over $i,j=1,\dots,N$, change variables $j+\ell N\to j$ in the
first group of products and $i+\ell^{\prime}N\to i$ in the second group, and
obtain

$\displaystyle\prod_{i,j=1}^{N}\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}c\big{(}\kappa^{\delta/N}s\big{)}_{j}/\big{(}\kappa^{\delta/N}s\big{)}_{i}|q,\kappa^{1/N}\big{)}$
$\displaystyle\qquad{}=\prod_{i=1}^{N}\prod_{j\geq i}\prod_{a\geq
1}\big{(}cq^{-\lambda^{(j)}_{a}+\lambda^{(i)}_{a+j-i+1}}s_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{a+j-i}-\lambda^{(i)}_{a+j-i+1}}$
$\displaystyle\qquad\phantom{=}{}\times\prod_{j=1}^{N}\prod_{i>j}\prod_{\alpha\geq
1}\big{(}cq^{\lambda^{(i)}_{\alpha}-\lambda^{(j)}_{\alpha+i-j-1}}s_{j}/s_{i};q\big{)}_{\lambda^{(j)}_{\alpha+i-j-1}-\lambda^{(j)}_{\alpha+i-j}}$
$\displaystyle\qquad{}=\prod_{i=1}^{N}\prod_{j\geq
i}\prod_{k>j}\big{(}cq^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}s_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}$
$\displaystyle\qquad\phantom{=}{}\times\prod_{j=1}^{N}\prod_{i>j}\prod_{k\geq
i}\big{(}cq^{\lambda^{(i)}_{k-i+1}-\lambda^{(j)}_{k-j}}s_{j}/s_{i};q\big{)}_{\lambda^{(j)}_{k-j}-\lambda^{(j)}_{k-j+1}},$

where we changed variables $a\to k=a+j$ and $\alpha\to k=\alpha+i-1$ in the
last step. We find it convenient to write this result as

$\displaystyle\prod_{i,j=1}^{N}\\!\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}\big{(}c\big{(}\kappa^{\delta/N}s\big{)}_{j}/\big{(}\kappa^{\delta/N}s\big{)}_{i}|q,\kappa^{1/N}\big{)}=\prod_{i=1}^{N}\prod_{i\leq
j<k<\infty}\\!\\!\\!\\!\\!\big{(}cq^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}s_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}$
$\displaystyle\hphantom{\prod_{i,j=1}^{N}\mathsf{N}^{(j-i|N)}_{\lambda^{(i)},\lambda^{(j)}}(c(\kappa^{\delta/N}s)_{j}/(\kappa^{\delta/N}s)_{i}|q,\kappa^{1/N})=}{}\times\prod_{i=1}^{N}\prod_{i<j\leq
k<\infty}\\!\\!\\!\\!\\!\big{(}cq^{\lambda^{(j)}_{k-j+1}-\lambda^{(i)}_{k-i}}s_{i}/s_{j};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}$

swapping variable names $i\leftrightarrow j$ in the second group of products.
We insert this into (A.2) to obtain

$\displaystyle\tilde{C}_{N}(\boldsymbol{\lambda};s|q,t,\kappa)=(t/q)^{|\boldsymbol{\lambda}|}\prod_{i=1}^{N}\prod_{i\leq
j<k<\infty}\frac{\big{(}q^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}qs_{j}/ts_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}{\big{(}q^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}s_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}$
$\displaystyle\hphantom{\tilde{C}_{N}(\boldsymbol{\lambda};s|q,t,\kappa)=}{}\times\prod_{i=1}^{N}\prod_{i<j\leq
k<\infty}\frac{\big{(}q^{\lambda^{(j)}_{k-j+1}-\lambda^{(i)}_{k-i}}qs_{i}/ts_{j};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}{\big{(}q^{\lambda^{(j)}_{k-j+1}-\lambda^{(i)}_{k-i}}s_{i}/s_{j};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}.$

To proceed, we use the well-known identity

$\displaystyle\frac{(q^{-m}/a;q)_{m}}{(q^{-m}/b;q)_{m}}=(b/a)^{m}\frac{(qa;q)_{m}}{(qb;q)_{m}}\qquad(a,b\in{\mathbb{C}},\
m\in{\mathbb{Z}}_{\geq 0}).$

Applying this to the factors in the second group of products for
$m=\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}$,
$b=q^{\lambda^{(i)}_{k-i+1}-\lambda^{(j)}_{k-j+1}}s_{j}/s_{i}$, $a=tb/q$
yields

$\displaystyle\tilde{C}_{N}(\boldsymbol{\lambda};s|q,t,\kappa)=(t/q)^{|\boldsymbol{\lambda}|}\prod_{i=1}^{N}\prod_{i\leq
j<k<\infty}\frac{\big{(}q^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}qs_{j}/ts_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}{\big{(}q^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}s_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}$
$\displaystyle\hphantom{\tilde{C}_{N}(\boldsymbol{\lambda};s|q,t,\kappa)=}{}\times\prod_{i=1}^{N}\prod_{i<j\leq
k<\infty}(q/t)^{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}\frac{\big{(}q^{\lambda^{(i)}_{k-i+1}-\lambda^{(j)}_{k-j+1}}ts_{j}/s_{i}\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}{\big{(}q^{\lambda^{(i)}_{k-i+1}-\lambda^{(j)}_{k-j+1}}qs_{j}/s_{i}\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}.$

To complete the proof that $\tilde{C}_{N}(\boldsymbol{\lambda};s|q,t,\kappa)$
in (A.2) is identical with $C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)$ in
(3.9), we swap the order of the two groups of products and compute the overall
power of $(q/t)$:

$\displaystyle\prod_{i=1}^{N}\prod_{i<j\leq
k<\infty}(q/t)^{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}=\prod_{i=1}^{N}\prod_{j=i+1}^{\infty}\prod_{k=j}^{\infty}(q/t)^{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}$
$\displaystyle\hphantom{\prod_{i=1}^{N}\prod_{i<j\leq
k<\infty}(q/t)^{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}{}=\prod_{i=1}^{N}\prod_{j=i+1}^{\infty}(q/t)^{\lambda^{(i)}_{j-i}}=\prod_{i=1}^{N}\prod_{k\geq
1}(q/t)^{\lambda^{(i)}_{k}}=(q/t)^{|\boldsymbol{\lambda}|},$

cancelling the factor $(t/q)^{|\boldsymbol{\lambda}|}$. This proves the
identity in (3.4) with $f_{N,\infty}(\bar{x}|\bar{s}|q,t)$ in (3.8)–(3.9) and
$x_{i+N}=px_{i}$, $s_{i+N}=\kappa s_{i}$ ($i\geq 1$). This, together with
Lemma 3.5, implies the result.

##  Appendix B Estimates

We give a complementary proof of the second estimate in (3.13), to compute the
upper bounds $C_{1,2}$ in Theorem 3.7.

###  B.1 Complementary proof of the second estimate in (3.13)

We prove that, under the assumptions in Theorem 3.7, the following estimates
hold true for the fractions appearing in the formula (3.9) for
$C_{N,\infty}(\boldsymbol{\lambda}|\bar{s}|q,t)$,

$\displaystyle\left|\frac{\big{(}q^{\lambda^{(i)}_{k-i+1}-\lambda^{(j)}_{k-j+1}}ts_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}{\big{(}q^{\lambda^{(i)}_{k-i+1}-\lambda^{(j)}_{k-j+1}}qs_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}\right|\leq
C_{1}^{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}\ \ (1\leq i\leq N,\ i<j\leq
k<\infty),\\!\\!\\!$ (B.1)
$\displaystyle\left|\frac{\big{(}q^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}qs_{j}/ts_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}{\big{(}q^{-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1}}s_{j}/s_{i};q\big{)}_{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}}\right|\leq
C_{2}^{\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}}\ \ (1\leq i\leq N,\ i\leq
j<k<\infty)\\!\\!$ (B.2)

for all $N$-partitions
$\boldsymbol{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(N)}\big{)}\in\mathsf{P}^{N}$,
with $C_{1}$ and $C_{2}$ in (3.12). This and (3.9) imply the estimate in
(3.15) which, by the computation in (3.15), is equivalent to the second
estimate in (3.13).

We observe all estimates in (B.1)–(B.2) are of the form

$\displaystyle\left|\frac{\big{(}q^{l}as_{i}/s_{j};q\big{)}_{\theta}}{\big{(}q^{l}s_{i}/s_{j};q\big{)}_{\theta}}\right|\leq
C^{\theta},$

where

$\displaystyle l=\lambda^{(i)}_{k-i+1}-\lambda^{(j)}_{k-j+1}+1,\qquad
a=t/q,\qquad\theta=\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1},\qquad C=C_{1}$
(B.3)

in (B.1) and

$\displaystyle l=-\lambda^{(j)}_{k-j}+\lambda^{(i)}_{k-i+1},\qquad
a=q/t,\qquad\theta=\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1},\qquad C=C_{2}$
(B.4)

in (B.2). We prove (B.1)–(B.2) using three different kinds of estimates:

######  Lemma B.1.

Let $\theta\in{\mathbb{Z}}_{\geq 0}$, $a\in{\mathbb{C}}$,
$q,\kappa\in{\mathbb{R}}$ with either $|q|<1$ and $|\kappa|>1$ or $|q|>1$ and
$|\kappa|<1$. Then the following estimates hold true,

  * $(a)$

for all $l\in{\mathbb{Z}}$ and $u\in{\mathbb{C}}\setminus\\{{\mathbb{R}}\\}$:

$\displaystyle\left|\frac{\big{(}q^{l}au;q\big{)}_{\theta}}{\big{(}q^{l}u;q\big{)}_{\theta}}\right|\leq\left(1+\frac{|1-a|}{|\sin\arg(u)|}\right)^{\theta}\\!,$
(B.5)

  * $(b)$

for all $m\in{\mathbb{Z}}_{\geq 0}$, $\ell\in{\mathbb{Z}}_{\geq 1}$:

$\displaystyle\left|\frac{\big{(}q^{-\theta-m+1}a\kappa^{\ell}\big{)}_{\theta}}{\big{(}q^{-\theta-m+1}\kappa^{\ell}\big{)}_{\theta}}\right|\leq\left(1+|1-a|\frac{|\kappa|}{|1-|\kappa||}\right)^{\theta}\\!,$
(B.6)

  * $(c)$

for all $m\in{\mathbb{Z}}_{\geq 0}$, $\ell\in{\mathbb{Z}}_{\geq 0}$:

$\displaystyle\left|\frac{\big{(}q^{-\theta-m}a\kappa^{\ell}\big{)}_{\theta}}{\big{(}q^{-\theta-m}\kappa^{\ell}\big{)}_{\theta}}\right|\leq\left(1+|1-a|\frac{1}{|1-|q||}\right)^{\theta}\\!.$
(B.7)

(The proof is given in Appendix B.2.)

Case A: For $j-i\notin N{\mathbb{Z}}_{\geq 0}$, we can use the estimate in
(B.5): Since $s_{j+N}=\kappa s_{j}$ and $\kappa$ is real, we have
$|\sin\arg(s_{j}/s_{i})|=|\sin\arg(s_{j+N}/s_{i})|=|\sin\arg(s_{i}/s_{j})|$
for all $j\geq i$; since $|\sin\arg(s_{j}/s_{i})|\geq\sigma$ for all $1\leq
i<j\leq N$ by assumption, $|\sin\arg(s_{j}/s_{i})|\geq\sigma$ for all $1\leq
i\leq N$ and $j\geq i$ such that $j-i\neq N{\mathbb{Z}}_{\geq 0}$, and we get

$\displaystyle\left|\frac{(q^{l}as_{i}/s_{j};q)_{\theta}}{(q^{l}s_{i}/s_{j};q)_{\theta}}\right|\leq\left(1+\frac{|1-a|}{\sigma}\right)^{\theta}\qquad(j-i\notin
N{\mathbb{Z}}_{\geq 0})$

for all cases in (B.3)–(B.4). This proves that the estimates in (B.1)–(B.2)
for all

$\displaystyle C_{1}\geq 1+\frac{|1-t/q|}{\sigma},\qquad C_{2}\geq
1+\frac{|1-q/t|}{\sigma}$ (B.8)

and for all cases $j-i\notin N{\mathbb{Z}}_{\geq 0}$.

We consider the remaining cases for (B.1) and (B.2) below in Cases B and C,
respectively.

Case B: For $j-i\in N{\mathbb{Z}}_{\geq 1}$, we have $s_{j}/s_{i}=s_{i+\ell
N}/s_{i}=\kappa^{\ell}$ for some $\ell\in{\mathbb{Z}}_{\geq 1}$, and we can
use the estimate in (B.6):

$\displaystyle\left|\frac{\big{(}q^{-\theta-m+1}as_{i}/s_{j};q\big{)}_{\theta}}{\big{(}q^{-\theta-m+1}s_{i}/s_{j};q\big{)}_{\theta}}\right|\leq\left(1+|1-a|\frac{|\kappa|}{||\kappa|-1|}\right)^{\theta}\qquad(j-i\in
N{\mathbb{Z}}_{\geq 1},\ m\in{\mathbb{Z}}_{\geq 0}).$

We check that all cases in (B.1) for $j-i\in N{\mathbb{Z}}_{\geq 1}$ are
covered by this: all $l$ in (B.3) for $j=i+\ell N$ can be written as (recall
that $\lambda^{(i+\ell N)}_{k}=\lambda^{(i)}_{k}$)

$\displaystyle
l=-\big{[}\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}\big{]}-\big{[}\lambda^{(i)}_{k-i-\ell
N+1}-\lambda^{(i)}_{k-i}\big{]}+1=-\theta-m+1$

with $m=\lambda^{(i)}_{k-i-\ell N+1}-\lambda^{(i)}_{k-i}\geq 0$ since
$\ell\geq 1$ and
$\lambda^{(i)}=\big{(}\lambda_{1}^{(i)},\lambda_{2}^{(i)},\dots\big{)}$ is a
partition. This proves that (B.1) holds true if

$\displaystyle C_{1}\geq 1+|1-t/q|\frac{|\kappa|}{|1-|\kappa||}$ (B.9)

for all cases $j-i\in N{\mathbb{Z}}_{\geq 1}$.

Case C: For $j-i\in N{\mathbb{Z}}_{\geq 0}$, we have $s_{j}/s_{i}=s_{i+\ell
N}/s_{i}=\kappa^{\ell}$ for some $\ell\in{\mathbb{Z}}_{\geq 0}$, and we can
use the estimate in (B.7):

$\displaystyle\left|\frac{\big{(}q^{-\theta-m}as_{i}/s_{j};q\big{)}_{\theta}}{\big{(}q^{-\theta-m}s_{i}/s_{j};q\big{)}_{\theta}}\right|\leq\left(1+|1-a|\frac{1}{|1-|q||}\right)^{\theta}\qquad(j-i\in
N{\mathbb{Z}}_{\geq 0},\ m\in{\mathbb{Z}}_{\geq 0}).$

We check that all cases in (B.2) for $j-i\in N{\mathbb{Z}}_{\geq 0}$ are
covered by this: all $l$ in can be written as

$\displaystyle
l=-\big{[}\lambda^{(i)}_{k-i}-\lambda^{(i)}_{k-i+1}\big{]}-\big{[}\lambda^{(i)}_{k-i-\ell
N}-\lambda^{(i)}_{k-i}\big{]}=-\theta-m$

with $m=\lambda^{(i)}_{k-i-\ell N}-\lambda^{(i)}_{k-i}\geq 0$. This proves
that (B.2) holds true if

$\displaystyle C_{2}\geq 1+|1-q/t|\frac{1}{|1-|q||}$ (B.10)

for all cases $j-i\in N{\mathbb{Z}}_{\geq 0}$.

We proved that (B.1)–(B.2) holds true for all cases provided the conditions in
(B.8), (B.9) and (B.10) all hold true; this is the case if we choose $C_{1}$
and $C_{2}$ as in (3.12).

###  B.2 Proof of Lemma B.1

####  B.2.1 Proof of the estimate in (B.5)

We have

$\displaystyle\text{LHS}=\left|\prod_{n=0}^{\theta-1}\frac{1-q^{l+n}au}{1-q^{l+n}u}\right|=\prod_{n=0}^{\theta-1}\left|1+(1-a)\frac{q^{l+n}u}{1-q^{l+n}u}\right|$
$\displaystyle\hphantom{\text{LHS}}{}\leq\prod_{n=0}^{\theta-1}\left(1+|1-a|\left|\frac{q^{l+n}u}{1-q^{l+n}u}\right|\right)\leq\prod_{n=0}^{\theta-1}\left(1+\frac{|1-a|}{|\sin\arg(u)|}\right)=\text{RHS},$

using the estimate in (3.14) and
$\big{|}\sin\arg\big{(}q^{l+n}u\big{)}\big{|}=|\sin\arg(u)|$ since $q$ is
real.

####  B.2.2 Proof of the estimate in (B.6)

We have

$\displaystyle\text{LHS}=\left|\prod_{n=0}^{\theta-1}\frac{1-q^{n-\theta-m+1}a\kappa^{\ell}}{1-q^{n-\theta-m+1}\kappa^{\ell}}\right|=\prod_{n=0}^{\theta-1}\left|1+(1-a)\frac{q^{-n-m}\kappa^{\ell}}{1-q^{-n-m}\kappa^{\ell}}\right|$
$\displaystyle\hphantom{\text{LHS}}{}\leq\prod_{n=0}^{\theta-1}\left(1+|1-a|\left|\frac{q^{-n-m}\kappa^{\ell}}{1-q^{-n-m}\kappa^{\ell}}\right|\right)\leq\prod_{n=0}^{\theta-1}\left(1+|1-a|\frac{|\kappa|}{|1-|\kappa||}\right)=\text{RHS},$

using

$\displaystyle\left|\frac{q^{-l}\kappa^{\ell}}{1-q^{-l}\kappa^{\ell}}\right|\leq\frac{|\kappa|}{|1-|\kappa||}\qquad(l\geq
0,\ell\geq 1);$

the latter follows for the case $|\kappa|<1$ and $|q|>1$ from the following
inequality: $x/(1-x)<y/(1-x)$ for $0\leq x<y<1$, and for the case $|\kappa|>1$
and $|q|<1$:

$\displaystyle\left|\frac{q^{-l}\kappa^{\ell}}{1-q^{-l}\kappa^{\ell}}\right|=\left|\frac{1}{1-q^{l}\kappa^{-\ell}}\right|\leq\frac{1}{1-|1/\kappa|}=\frac{|\kappa|}{|1-|\kappa||}$

since $1/(1-x)<1/(1-y)$ for $0\leq x<y<1$.

####  B.2.3 Proof of the estimate in (B.7)

We have

$\displaystyle\text{LHS}=\left|\prod_{n=0}^{\theta-1}\frac{1-q^{n-\theta-m}a\kappa^{\ell}}{1-q^{n-\theta-m+1}\kappa^{\ell}}\right|=\prod_{n=1}^{\theta}\left|1+(1-a)\frac{q^{-n-m}\kappa^{\ell}}{1-q^{-n-m}\kappa^{\ell}}\right|$
$\displaystyle\hphantom{\text{LHS}}{}\leq\prod_{n=1}^{\theta}\left(1+|1-a|\left|\frac{q^{-n-m}\kappa^{\ell}}{1-q^{-n-m}\kappa^{\ell}}\right|\right)\leq\prod_{n=1}^{\theta}\left(1+|1-a|\frac{1}{|1-|q||}\right)=\text{RHS},$

using

$\displaystyle\left|\frac{q^{-l}\kappa^{\ell}}{1-q^{-l}\kappa^{\ell}}\right|\leq\frac{|q^{-1}|}{||q^{-1}|-1|}=\frac{1}{|1-|q||}\qquad(l\geq
1,\ \ell\geq 0),$

as in the proof of (B.6).

##  Appendix C Proof of Proposition 4.2

We prove Proposition 4.2 using Lemma 4.3 in the main text.

###  C.1 Proof of commutativity

We prove (4.3). We note that the action of the Macdonald–Ruijsenaars operators
in (2.1) on functions $x^{\lambda}f(x)$ can be written as [15]

$\displaystyle
D_{N}^{\pm}(x|q,t)x^{\lambda}f(x)=x^{\lambda}E^{\pm}_{N}(x|s|q,t)f(x),\qquad
s_{i}=t^{N-i}q^{\lambda_{i}}$ (C.1)

with the modified Macdonald–Ruijsenaars operators

$\displaystyle
E^{\pm}_{N}(x|s|q,t)=\sum_{i=1}^{N}A_{N,i}\big{(}x|t^{\pm}\big{)}s_{i}^{\pm
1}T_{q,x_{i}}^{\pm 1},$ (C.2) $\displaystyle A_{N,i}(x|t^{\pm
1})=\prod_{j=1}^{i-1}\frac{1-t^{\pm
1}x_{i}/x_{j}}{1-x_{i}/x_{j}}\prod_{k=i+1}^{N}\frac{1-t^{\mp
1}x_{k}/x_{i}}{1-x_{k}/x_{i}}$ (C.3)

(this can be proved by simple computations which we skip).

We also need properties of the function $\chi_{N}(x|y|q,t)$ in (4.7) which we
summarize as follows.

######  Lemma C.1.

The function $\chi_{N}(x|y|q,t)$ satisfies the following duality relation,

$\displaystyle\chi_{N}(x|y|q,t)=\chi_{N}(y|x|q,t/q).$ (C.4)

Moreover,

$\displaystyle\begin{split}&E^{\pm}_{N}(x|y|q,t)\chi_{N}(x|y|q,t)=e_{1}\big{(}y^{\pm
1}\big{)}\chi_{N}(x|y|q,t),\\\\[2.15277pt]
&E^{\pm}_{N}(y|x|q,q/t)\chi_{N}(x|y|q,t)=e_{1}\big{(}x^{\pm
1}\big{)}\chi_{N}(x|y|q,t)\end{split}$ (C.5)

with

$\displaystyle e_{1}\big{(}x^{\pm 1}\big{)}=x_{1}^{\pm 1}+\cdots+x_{N}^{\pm
1}.$ (C.6)

###### Proof.

The definitions in (2.6) and (4.7) imply

$\displaystyle\chi_{N}(x|y|q,t)=\prod_{1\leq i<j\leq
N}\frac{(qx_{j}/x_{i};q)_{\infty}}{(qx_{j}/tx_{i};q)_{\infty}}\frac{(qy_{j}/y_{i};q)_{\infty}}{(ty_{j}/y_{i};q)_{\infty}}\,\varphi_{N}(x|y|q,t).$

The product on the right-hand side is manifestly invariant under the
transformation $(x,y,t)\mapsto(y,x,q/t)$; the function $\varphi_{N}(x|y|q,t)$
has this invariance by (2.7). This proves (C.4).

The first identity in (LABEL:cDpsi) is implied by
$E^{\pm}_{N}(x|s|q,t)f_{N}(x|s|q,t)=e_{1}\big{(}s^{\pm
1}\big{)}f_{N}(x|s|q,t)$ proved in [15]; the second follows from the first and
the duality in (C.4). ∎

Equation (C.1) and Lemma 4.3 imply that the result we want to prove:

$\displaystyle
D_{N}^{\pm}(x|q,t){\mathcal{T}}_{N}(x|q,t)x^{\lambda}f(x)={\mathcal{T}}_{N}(x|q,t)D_{N}^{\pm}(x|q,t)x^{\lambda}f(x),$

is equivalent to

$\displaystyle\Bigg{[}E^{\pm}_{N}(x|s|q,t)\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\chi_{N}(x|y|q,t)\prod_{1\leq
i<j\leq N}(1-y_{j}/y_{i})f(y)\Bigg{]}_{1,y}$
$\displaystyle\qquad{}=\Bigg{[}\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\chi_{N}(x|y|q,t)\prod_{1\leq
i<j\leq
N}(1-y_{j}/y_{i})\big{(}E^{\pm}_{N}(y|s|q,t)f(y)\big{)}\Bigg{]}_{1,y}\\!.$

The latter is obviously implied by the following two identities: first,

$\displaystyle
E^{\pm}_{N}(x|s|q,t)\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\chi_{N}(x|y|q,t)$
$\displaystyle\qquad{}=E^{\mp}_{N}\big{(}y\big{|}s^{-1}\big{|}q,q/t\big{)}\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\chi_{N}(x|y|q,t),$
(C.7)

and second,

$\displaystyle\Bigg{[}\Bigg{(}E^{\mp}_{N}\big{(}y\big{|}s^{-1}\big{|}q,q/t\big{)}\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\chi_{N}(x|y|q,t)\Bigg{)}\prod_{1\leq
i<j\leq N}(1-y_{j}/y_{i})f(y)\Bigg{]}_{1,y}$
$\displaystyle\qquad{}=\Bigg{[}\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\chi_{N}(x|y|q,t)\prod_{1\leq
i<j\leq
N}(1-y_{j}/y_{i})\big{(}E^{\pm}_{N}(y|s|q,t)f(y)\big{)}\Bigg{]}_{1,y}\\!.$
(C.8)

We first prove (C.7) in three steps, using the shorthand notation in (C.6). We
start with

$\displaystyle
E^{\pm}_{N}(x|s|q,t)\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)=\frac{q^{-1/2}}{1-q^{-1}}\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\big{[}e_{1}\big{(}x^{\mp
1}\big{)},E^{\pm}_{N}(x|y|q,t)\big{]}$ (C.9)

proved by the following computation (we insert definitions and change the
summation variable $n_{i}\pm 1\to n_{i}$ in the third equality),

$\displaystyle\text{LHS}=\sum_{i=1}^{N}A_{N,i}\big{(}x|t^{\pm 1}\big{)}s^{\pm
1}_{i}T_{q,x_{i}}^{\pm
1}\sum_{n\in{\mathbb{Z}}^{N}}\prod_{j=1}^{N}\left(\frac{s_{j}x_{j}}{y_{j}}\right)^{n_{j}}q^{\frac{1}{2}n_{j}^{2}}$
$\displaystyle\hphantom{\text{LHS}}{}=\sum_{n\in{\mathbb{Z}}^{N}}\sum_{i=1}^{N}A_{N,i}\big{(}x|t^{\pm
1}\big{)}\left(\frac{y_{i}}{x_{i}}\right)^{\pm
1}\left(\frac{s_{i}x_{i}}{y_{i}}\right)^{n_{i}\pm 1}q^{\frac{1}{2}n_{i}^{2}\pm
n_{i}}\prod_{\genfrac{}{}{0.0pt}{}{j=1}{j\neq
i}}^{N}\left(\frac{s_{j}x_{j}}{y_{j}}\right)^{n_{j}}q^{\frac{1}{2}n_{j}^{2}}T_{q,x_{i}}^{\pm
1}$
$\displaystyle\hphantom{\text{LHS}}{}=\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\sum_{i=1}^{N}A_{N,i}\big{(}x|t^{\pm
1}\big{)}\left(\frac{y_{i}}{x_{i}}\right)^{\pm 1}q^{-1/2}T_{q,x_{i}}^{\pm
1}=\text{RHS}$

since

$\displaystyle\big{[}e_{1}\big{(}x^{\mp 1}\big{)},E^{\pm}_{N}(x|y|q,t)\big{]}$
$\displaystyle\qquad{}=\sum_{i=1}^{N}A_{N,i}\big{(}x|t^{\pm
1}\big{)}y_{i}^{\pm 1}\big{[}x_{i}^{\mp 1},T_{q,x_{i}}^{\pm
1}\big{]}=\big{(}1-q^{-1}\big{)}\sum_{i=1}^{N}A_{N,i}\big{(}x|t^{\pm
1}\big{)}\left(\frac{y_{i}}{x_{i}}\right)^{\pm 1}T_{q,x_{i}}^{\pm 1}.$

Next,

$\displaystyle\big{[}e_{1}\big{(}x^{\mp
1}\big{)},E^{\pm}_{N}(x|y|q,t)\big{]}\chi_{N}(x|y|q,t)=\big{[}e_{1}\big{(}y^{\pm
1}\big{)},E^{\mp}_{N}(y|x|q,q/t)\big{]}\chi_{N}(x|y|q,t),$ (C.10)

which is proved by

$\displaystyle\text{LHS}=\left(e_{1}\big{(}x^{\mp 1}\big{)}e_{1}\big{(}y^{\pm
1}\big{)}-E^{\pm}_{N}(x|y|q,t)E^{\mp}_{N}(y|x|q,q/t)\right)\chi_{N}(x|y|q,t)$
$\displaystyle\hphantom{\text{LHS}}{}=\left(e_{1}\big{(}y^{\pm
1}\big{)}e_{1}\big{(}x^{\mp
1}\big{)}-E^{\mp}_{N}(y|x|q,q/t)E^{\pm}_{N}(x|y|q,t)\right)\chi_{N}(x|y|q,t)=\text{RHS}$

using (LABEL:cDpsi) and

$\displaystyle\big{[}E^{\pm}_{N}(x|y|q,t),E^{\mp}_{N}(y|x|q,q/t)\big{]}=0;$

the latter is verified by a simple computation using the definition in
(C.2)–(C.3). Third,

$\displaystyle
E^{\mp}_{N}\big{(}y|s^{-1}|q,q/t\big{)}\prod_{i=1}^{N}\\!\vartheta_{3}(s_{i}x_{i}/y_{i}|q)$
$\displaystyle\qquad{}=\frac{q^{-1/2}}{1-q^{-1}}\prod_{i=1}^{N}\\!\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\big{[}e_{1}\big{(}y^{\pm
1}\big{)},E^{\mp}_{N}(y|x|q,q/t)\big{]},$ (C.11)

which is proved similarly as (C.9):

$\displaystyle\text{LHS}=\sum_{i=1}^{N}A_{N,i}\big{(}y|(q/t)^{\mp
1}\big{)}s^{\pm 1}_{i}T_{q,y_{i}}^{\mp
1}\sum_{n\in{\mathbb{Z}}^{N}}\prod_{j=1}^{N}\left(\frac{s_{j}x_{j}}{y_{j}}\right)^{n_{j}}q^{\frac{1}{2}n_{j}^{2}}$
$\displaystyle\hphantom{\text{LHS}}{}=\sum_{n\in{\mathbb{Z}}^{N}}\sum_{i=1}^{N}A_{N,i}\big{(}y|(q/t)^{\mp
1}\big{)}\left(\frac{y_{i}}{x_{i}}\right)^{\pm
1}\left(\frac{s_{i}x_{i}}{y_{i}}\right)^{n_{i}\pm 1}q^{\frac{1}{2}n_{i}^{2}\pm
n_{i}}\prod_{\genfrac{}{}{0.0pt}{}{j=1}{j\neq
i}}^{N}\left(\frac{s_{j}x_{j}}{y_{j}}\right)^{n_{j}}q^{\frac{1}{2}n_{j}^{2}}T_{q,x_{i}}^{\mp
1}$
$\displaystyle\hphantom{\text{LHS}}{}=\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\sum_{i=1}^{N}A_{N,i}\big{(}y|(q/t)^{\mp
1}\big{)}\left(\frac{y_{i}}{x_{i}}\right)^{\pm 1}q^{-1/2}T_{q,y_{i}}^{\mp
1}=\text{RHS}$

since

$\displaystyle\big{[}e_{1}\big{(}y^{\pm
1}\big{)},E^{\mp}_{N}(y|x|q,q/t)\big{]}$
$\displaystyle\qquad{}=\sum_{i=1}^{N}A_{N,i}\big{(}y|(q/t)^{\mp
1}\big{)}x_{i}^{\mp 1}\big{[}y_{i}^{\pm 1},T_{q,y_{i}}^{\mp
1}\big{]}=\big{(}1-q^{-1}\big{)}\sum_{i=1}^{N}A_{N,i}\big{(}y|(q/t)^{\mp
1}\big{)}\left(\frac{y_{i}}{x_{i}}\right)^{\pm 1}T_{q,y_{i}}^{\mp 1}.$

We are now ready to prove (C.7): we insert (C.9) into the LHS in (C.7), use
(C.10) and (C.11), and obtain the RHS in (C.7).

To conclude our proof of (4.3), we prove (C.8) by the following computation,
using the definitions in (C.2)–(C.3) and the basic property
$\left[\left(T_{q,y_{i}}^{\mp
1}g_{1}(y)\right)g_{2}(y)\right]_{1,y}=\left[g_{1}(y)\left(T_{q,y_{i}}^{\pm
1}g_{2}(y)\right)\right]_{1,y}$ of the constant term:

$\displaystyle\text{LHS}\,{=}\,\Biggl{[}\sum_{i=1}^{N}\\!A_{N,i}\big{(}y|(q/t)^{\mp
1}\big{)}s_{i}^{\pm 1}\\!\Bigg{(}\\!T_{q,y_{i}}^{\mp
1}\prod_{i=1}^{N}\\!\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\chi_{N}(y|x|q,t)\\!\Bigg{)}\\!\\!\\!\prod_{1\leq
i<j\leq N}\\!\\!\\!\\!\\!(1-y_{j}/y_{i})f(y)\\!\Biggr{]}_{1,y}$
$\displaystyle\hphantom{\text{LHS}}{}\,{=}\,\Biggl{[}\prod_{i=1}^{N}\vartheta_{3}(s_{i}x_{i}/y_{i}|q)\chi_{N}(y|x|q,t)\\!\sum_{i=1}^{N}\\!\Bigg{(}\\!T_{q,y_{i}}^{\pm
1}A_{N,i}\big{(}y|(q/t)^{\mp 1}\big{)}s_{i}^{\pm 1}\\!\\!\\!\prod_{1\leq
i<j\leq N}\\!\\!\\!\\!\\!(1-y_{j}/y_{i})f(y)\\!\Bigg{)}\\!\Biggr{]}_{1,y}$
$\displaystyle\hphantom{\text{LHS}}{}=\text{RHS}$

provided

$\displaystyle\frac{T_{q,y_{i}}^{\pm 1}A_{N,i}\big{(}y|(q/t)^{\mp
1}\big{)}\prod_{1\leq i<j\leq N}(1-y_{j}/y_{i})}{\prod_{1\leq i<j\leq
N}(1-y_{j}/y_{i})}=A_{N,i}\big{(}y|t^{\pm}\big{)};$

the latter holds true for the coefficients $A_{N,i}(x|t)$ in (C.3), as is
easily verified:

$\displaystyle\text{LHS}=\prod_{j=1}^{i-1}\frac{1-(q/t)^{\mp 1}q^{\pm
1}y_{i}/y_{j}}{1-q^{\pm 1}y_{i}/y_{j}}\prod_{k=i+1}^{N}\frac{1-(q/t)^{\pm
1}y_{k}/q^{\pm 1}y_{i}}{1-y_{k}/q^{\pm 1}y_{i}}\prod_{j=1}^{i-1}\frac{1-q^{\pm
1}y_{i}/y_{j}}{1-y_{i}/y_{j}}$
$\displaystyle\hphantom{\text{LHS}=}{}\times\prod_{k=i+1}^{N}\frac{1-y_{k}/q^{\pm
1}y_{i}}{1-y_{k}/y_{i}}=\text{RHS}.$

###  C.2 Eigenfunction property

The eigenfunctions $x^{\lambda}f_{N}(x|s|q,t)$ of $D^{\pm}_{N}(x|s|q,t)$ are
unique, and (4.3) therefore implies that $x^{\lambda}f_{N}(x|s|q,t)$ also are
eigenfunction of ${\mathcal{T}}_{N}(x|q,t)$.

We are left to determine the eigenvalues. For that, we introduce some
notation: the space of formal power series
${\mathbb{C}}[[x_{2}/x_{1},\dots,x_{N}/x_{N-1}]]$ is spanned by monomials

$\displaystyle(x_{2}/x_{1})^{\alpha_{1}}\cdots(x_{N}/x_{N-1})^{\alpha_{N_{1}}}$

with $\alpha=(\alpha_{1},\dots,\alpha_{N-1})\in{\mathbb{Z}}_{\geq 0}^{N-1}$.
Any such monomial can be written as $x^{\mu}=x_{1}^{\mu_{1}}\cdots
x_{N}^{\mu_{N}}$ with $\mu=\mu(\alpha)$ given by

$\displaystyle\mu_{1}(\alpha)=-\alpha_{1},\qquad\mu_{i}(\alpha)=\alpha_{i-1}-\alpha_{i}\qquad(i=2,\dots,N-1),\qquad\mu_{N}(\alpha)=\alpha_{N-1}.$

One can verify that the action of the operator in (4.2) is triangular on this
basis in the following sense,

$\displaystyle{\mathcal{T}}_{N}(x|q,t)x^{\lambda}x^{\mu(\alpha)}=\varepsilon(\lambda+\mu(\alpha))x^{\lambda}\Bigg{(}x^{\mu(\alpha)}+\sum_{\beta>\alpha}v_{\alpha\beta}x^{\mu(\beta)}\Bigg{)}$

for some coefficients $v_{\alpha\beta}$, where $\beta\geq\alpha$ means that
$\beta_{i}\geq\alpha_{i}$ for all $i=1,\dots,N-1$ (this follows from

$\displaystyle
q^{\frac{1}{2}\Delta}x^{\lambda}x^{\mu}=\varepsilon(\lambda+\mu)x^{\lambda+\mu},\qquad\varepsilon(\lambda+\mu)=q^{\frac{1}{2}\sum_{i=1}^{N}(\lambda_{i}+\mu_{i}+(N-i)\beta)^{2}}$

used already in the main text, and the fact that all functions of $x$
appearing in the definition of ${\mathcal{T}}_{N}(x|q,t)$ in (4.2) can be
expanded as power series in
${\mathbb{C}}[[x_{2}/x_{1},\dots,x_{N}/x_{N-1}]]$). Since

$\displaystyle
x^{\lambda}f_{N}(x|s|q,t)=x^{\lambda}\Bigg{(}1+\sum_{\alpha>0}b_{\alpha}x^{\mu(\alpha)}\Bigg{)}$

for some coefficients $b_{\alpha}$, the eigenvalue is
$\varepsilon(\lambda)=\varepsilon(s|q)$ in (4.5).

##  Appendix D Identities

For clarify, and for the convenience of the reader, we state and prove two
identities used in the main text.

First, the identity

$\displaystyle\sum_{\lambda\in\mathsf{P}}\alpha^{|\lambda|}=\frac{1}{(\alpha;\alpha)_{\infty}}\qquad(|\alpha|<1),$
(D.1)

which goes back to Euler, is important in our proof of Theorem 3.7. It is
proved by the following elementary computation making absolute convergence of
the series manifest,

$\displaystyle\text{LHS}=\lim_{M\to\infty}\sum_{\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{M}\geq
0}\alpha^{\lambda_{1}+\lambda_{2}+\dots+\lambda_{M}}$
$\displaystyle\phantom{\text{LHS}}{}=\lim_{M\to\infty}\sum_{\lambda_{1}=\lambda_{2}}^{\infty}\sum_{\lambda_{2}=\lambda_{3}}^{\infty}\cdots\sum_{\lambda_{M-1}=\lambda_{M}}^{\infty}\sum_{\lambda_{M}=0}^{\infty}\alpha^{\lambda_{1}+\lambda_{2}+\dots+\lambda_{M}}$
$\displaystyle\phantom{\text{LHS}}{}=\lim_{M\to\infty}\sum_{\lambda_{2}=\lambda_{3}}^{\infty}\cdots\sum_{\lambda_{M-1}=\lambda_{M}}^{\infty}\sum_{\lambda_{M}=0}^{\infty}\frac{1}{1-\alpha}\alpha^{2\lambda_{2}+\lambda_{3}+\dots+\lambda_{M}}$
$\displaystyle\phantom{\text{LHS}}{}=\lim_{M\to\infty}\sum_{\lambda_{3}=\lambda_{4}}^{\infty}\cdots\sum_{\lambda_{M-1}=\lambda_{M}}^{\infty}\sum_{\lambda_{M}=0}^{\infty}\frac{1}{(1-\alpha)(1-\alpha^{2})}\alpha^{3\lambda_{3}+\lambda_{4}+\dots+\lambda_{M}}=\cdots$
$\displaystyle\phantom{\text{LHS}}{}=\lim_{M\to\infty}\frac{1}{(1-\alpha)}\frac{1}{(1-\alpha^{2})}\cdots\frac{1}{(1-\alpha^{M})}=\text{RHS},$
(D.2)

summing repeatedly the geometric series.

Second, we state and prove an identity used in the proof of Fact 3.6 and
Remark 4.6.

######  Lemma D.1.

For $p\in{\mathbb{C}}$, let $x_{i}\in{\mathbb{C}}$ be given for $i=1,\dots,N$,
and extend this definition to all $i\geq 1$ by $x_{i+N}=px_{i}$. Then

$\displaystyle\prod_{i=1}^{N}\prod_{j>i}\frac{(ax_{j}/x_{i};q)_{\infty}}{(bx_{j}/x_{i};q)_{\infty}}=\prod_{1\leq
i<j\leq
N}\frac{(ax_{j}/x_{i};q,p)_{\infty}}{(bx_{j}/x_{i};q,p)_{\infty}}\prod_{1\leq
i\leq j\leq
N}\frac{(pax_{i}/x_{j};q,p)_{\infty}}{(pbx_{i}/x_{j};q,p)_{\infty}}$

for all $a,b,q\in{\mathbb{C}}$.

###### Proof.

We note that $(z;q,p)=\prod_{m=0}^{\infty}(p^{m}x;q)_{\infty}$, and thus

$\displaystyle\text{LHS}=\prod_{1\leq i<j\leq
N}\prod_{m=0}^{\infty}\frac{(ax_{j+mN}/x_{i};q)_{\infty}}{(bx_{j+mN}/x_{i};q)_{\infty}}\prod_{1\leq
j\leq i\leq
N}\prod_{m=1}^{\infty}\frac{(ax_{j+mN}/x_{i};q)_{\infty}}{(bx_{j+mN}/x_{i};q)_{\infty}}$
$\displaystyle\phantom{\text{LHS}}{}=\prod_{1\leq i<j\leq
N}\prod_{m=0}^{\infty}\frac{(p^{m}ax_{j}/x_{i};q)_{\infty}}{(p^{m}bx_{j}/x_{i};q)_{\infty}}\prod_{1\leq
j\leq i\leq
N}\prod_{m=1}^{\infty}\frac{(p^{m}ax_{j}/x_{i};q)_{\infty}}{(p^{m}bx_{j}/x_{i};q)_{\infty}}=\text{RHS},$

inserting $x_{j+mN}=p^{m}x_{j}$. ∎

### Acknowledgements

We would like to thank F. Atai, A. Negut, and V. Pasquier for useful
discussions. We thank J. Lamers for a helpful comment on the manuscript. We
are grateful to careful referees for remarks helping us to improve our paper.
This work is supported by VR Grant No 2016-05167 (E.L.) and by JSPS Kakenhi
Grants (B) 15H03626 (M.N.), (C) 19K03512 (J.S.). MN gratefully acknowledges
financial support by the Knut and Alice Wallenberg foundation (KAW 2019.0525).
We are grateful to the Stiftelse Olle Engkvist Byggmästare, Contract 184-0573,
for financial support.

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