# A higher-order generalization of an $A_{4}^{(1)}$-surface type $q$-Painlevé
equation with $\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\times
A_{1}^{(1)}\right)$ symmetry

Nobutaka Nakazono  Institute of Engineering, Tokyo University of Agriculture
and Technology, 2-24-16 Nakacho Koganei, Tokyo 184-8588, Japan.
<EMAIL_ADDRESS>
###### Abstract.

In [37], a birational representation of an extended affine Weyl group of
$(A_{2N}\rtimes A_{1})^{(1)}$-type, which gives a higher-order generalization
of an $A_{4}^{(1)}$-surface type $q$-Painlevé equation, was obtained. In this
paper, we extend it to a birational representation of an extended affine Weyl
group of $(A_{2N}\rtimes A_{1})^{(1)}\times A_{1}^{(1)}$-type. Moreover, we
provide conjectures on periodic reductions from systems of P$\Delta$Es with
the CAC property to Painlevé type $q$-O$\Delta$Es.

###### Key words and phrases:

discrete integrable systems; $q$-Painlevé system; integrable partial
difference equation; affine Weyl group; consistency around a cube property

###### 2020 Mathematics Subject Classification:

33E17, 35Q53, 37K10, 39A13, 39A14, 39A23, 39A36, 39A45

##  1\. Introduction

Fix an integer $N>0$. In this paper, we focus on the symmetry of the following
$2N$th-order $q$-O$\Delta$E[37]:

$\text{$q$P${}^{(2N)}(A_{4}^{(1)})$ :\hskip
5.0pt}\dfrac{{a_{i}}^{2N+1}\Big{(}\overline{F}_{i}F_{i}-1\Big{)}}{{a_{i}}^{2N+1}-c^{2(-1)^{i}}F_{i}}=\begin{cases}~{}\dfrac{\overline{F}_{i+1}F_{i+1}-1}{1-{a_{i+1}}^{2N+1}c^{2(-1)^{i}}\overline{F}_{i+1}}&\text{if
}i=1,\dots,2N-1,\\\\[15.00002pt]
~{}\dfrac{\left(\displaystyle\prod_{k=1}^{2N}{a_{k}}^{k}\right)bc}{\displaystyle\prod_{k=1}^{N}F_{2k-1}}&\text{if
}i=2N,\end{cases}$ (1.1)

where $b\in\mathbb{C}$ is an independent variable,
$F_{i}=F_{i}(b)\in\mathbb{C}$, $i=1,\dots,2N$, are dependent variables and
$a_{1},\dots,a_{2N},c,p\in\mathbb{C}$ are parameters. The symbol
$\overline{\rule{0.0pt}{5.0pt}~{}\,}$ denotes the discrete time evolution:

$\overline{b}=pb,\quad\overline{F}_{i}=F_{i}(pb),~{}i=1,\dots,2N,\quad\overline{a}_{j}=a_{j},~{}j=1,\dots,2N,\quad\overline{c}=c^{-1},$
(1.2)

where $p\in\mathbb{C}$ is a constant parameter. When $N=1$, the system (1.1)
is equivalent to a $q$-Painlevé V equation of $A_{4}^{(1)}$-surface type [60,
26, 27, 36]. (See [37] for details.)

######  Remark 1.1.

For a $q$-difference equation, it is common to use the symbol “$q$” for its
shift parameter. However, in this paper, the symbol “$q$” is used in the later
arguments, and the relations between the symbol “$p$” in the system (1.1) and
the symbol “$q$” in the later arguments will be given. Therefore, to avoid
confusion, we use the symbol “$p$” for the system (1.1) instead of the symbol
“$q$”.

The following property of the system (1.1) is known.

######  Theorem 1.2 ([37]).

The system (1.1) has an extended affine Weyl group symmetry of $(A_{2N}\rtimes
A_{1})^{(1)}$-type, that is, it can be derived from a birational
representation of an extended affine Weyl group of $(A_{2N}\rtimes
A_{1})^{(1)}$-type, denoted by $\widetilde{W}\left((A_{2N}\rtimes
A_{1})^{(1)}\right)$.

######  Remark 1.3.

An affine Weyl group of $(A_{2N}\rtimes A_{1})^{(1)}$-type means the semi-
direct product of an affine Weyl group of $A_{2N}^{(1)}$-type and that of
$A_{1}^{(1)}$-type. The extended affine Weyl group of $(A_{2N}\rtimes
A_{1})^{(1)}$-type in Theorem 1.2 indicates an affine Weyl group of
$(A_{2N}\rtimes A_{1})^{(1)}$-type extended by an automorphism of Dynkin
diagrams. For details, see §3.1. In particular, see (3.8) for the fundamental
relations of

$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)=\Big{(}\langle
s_{0},\dots,s_{2N}\rangle\rtimes\langle
w_{0},w_{1}\rangle\Big{)}\rtimes\langle r\rangle.$ (1.3)

It is known that Painlevé-type O$\Delta$Es are derived from birational actions
of the infinite-order elements of (extended) affine Weyl groups. For instance,
considering the transformation $T\in\widetilde{W}\left((A_{2N}\rtimes
A_{1})^{(1)}\right)$ acting on the parameters $\\{a_{1},\dots,a_{2N},b,c,p\\}$
such that

$T(a_{i})=a_{i},~{}i=1,\dots,2N,\quad T(b)=pb,\quad T(c)=c^{-1},\quad T(p)=p,$
(1.4)

as time evolution, we obtain the system (1.1). (See Remark 3.1 for the
transformation $T$.) In this context, the parameter $b$ can be regarded as the
independent variable of the system (1.1). The group
$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$ includes
transformations that regard not only the parameter $b$ but also the parameter
$a_{i}$ as independent variables. For instance, there exists a transformation
$\hat{T}\in\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$ satisfying
the following action on the parameters:

$\hat{T}(a_{1})=pa_{1},\quad\hat{T}(a_{i})=a_{i},~{}i=2,\dots,2N,\quad\hat{T}(b)=b,\quad\hat{T}(c)=c,\quad\hat{T}(p)=p.$
(1.5)

(See Remark 3.1 for the transformation $\hat{T}$.) However, focusing on the
action on the parameter $c$, it is observed that
$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$ contains only
transformations acting as $c\to c$ or $c\to c^{-1}$. This is attributed to the
fact that in the study of [37], the parameter $c$ is not an essential
parameter derived from the system of P$\Delta$Es considering reduction but
rather a parameter that can be set to $1$ through a gauge transformation
before reduction.

The purpose of this paper is to extend the transformation group
$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$ to include a
transformation $\widetilde{T}$ that regards the parameter $c$ as an
independent variable of an O$\Delta$E. To achieve this, we consider a
reduction of a different system of P$\Delta$Es from that treated in [37]. (See
Remark 3.6 for the transformation $\widetilde{T}$.)

The main theorem of this paper is the following.

######  Theorem 1.4.

The system (1.1) has an extended affine Weyl group symmetry of $(A_{2N}\rtimes
A_{1})^{(1)}\times A_{1}^{(1)}$-type, that is, it can be derived from a
birational representation of an extended affine Weyl group of $(A_{2N}\rtimes
A_{1})^{(1)}\times A_{1}^{(1)}$-type, denoted by
$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\times A_{1}^{(1)}\right)$.

###### Proof.

This theorem is proven in §3.2. ∎

######  Remark 1.5.

An affine Weyl group of $(A_{2N}\rtimes A_{1})^{(1)}\times A_{1}^{(1)}$-type
means the direct product of an affine Weyl group of $(A_{2N}\rtimes
A_{1})^{(1)}$-type and that of $A_{1}^{(1)}$-type. The extended affine Weyl
group of $(A_{2N}\rtimes A_{1})^{(1)}\times A_{1}^{(1)}$-type in Theorem 1.4
indicates an affine Weyl group of $(A_{2N}\rtimes A_{1})^{(1)}\times
A_{1}^{(1)}$-type extended by an automorphism of Dynkin diagrams. For details,
see §3.2. In particular, see (3.8) and (3.29) for the fundamental relations of

$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\times
A_{1}^{(1)}\right)=\bigg{(}\Big{(}\langle
s_{0},\dots,s_{2N}\rangle\rtimes\langle
w_{0},w_{1}\rangle\Big{)}\times\langle\mu_{0},\mu_{1}\rangle\bigg{)}\rtimes\langle
r\rangle.$ (1.6)

###  1.1. Background

#### Painlevé equations

In the early 20th century, to find a new class of special functions, Painlevé
and Gambier classified all the ordinary differential equations of the type

$y^{\prime\prime}=F(y^{\prime},y,t),$ (1.7)

where $y=y(t)$, ${}^{\prime}=d/dt$ and $F$ is a function meromorphic in $t$
and rational in $y$ and $y^{\prime}$, with the Painlevé property (solutions do
not have movable singularities other than poles)[52, 53, 54, 12]. As a result,
they obtained six new equations, and the resulting equations are now
collectively referred to as the Painlevé equations. Note that the Painlevé VI
equation was found by Fuchs [11] before Painlevé and Gambier.

After the discovery, the Painlevé equations withdrew from the stage of “modern
mathematics” for a while. The Painlevé equations regained attention after the
1970s because they appeared in mathematical physics research. For instance,
solutions of the Painlevé equations (Painlevé transcendents) were rediscovered
as scaling functions for the two-dimensional Ising model on a square lattice
[64] and as similarity solutions of the soliton equations describing specific
shallow water waves (solitons) [1].

In the late 20th century, Okamoto described a geometric framework (Okamoto’s
space of initial values) for studying the Painlevé equations [45, 44]. As a
result, the Painlevé III equation was classified into three types, which in
turn classified the Painlevé equations into eight types.

####  $q$-Painlevé equations

$q$-Painlevé equations are a family of second-order nonlinear $q$-O$\Delta$Es.
Historically, they have been obtained as $q$-discrete analogues of the
Painlevé equations (see, for example, [14, 22]). Similar to the Painlevé
equations, $q$-Painlevé equations are also known to describe special solutions
of various discrete soliton equations [15, 17, 10, 49, 18]. Particularly
famous is imposing periodic conditions on discrete soliton equations[16, 17,
50, 49, 51, 29, 62, 55, 26, 25, 24, 37, 4].

There are six (or eight) Painlevé equations, but an infinite number of
$q$-Painlevé equations are known to exist. By considering Sakai’s space of
initial values [58], which is an extension of Okamoto’s space of initial
values, $q$-Painlevé equations can be classified into nine surface types (see
Figure 1.1). In this paper, a $q$-Painlevé equation of $X$-surface type means
one $q$-Painlevé equation belonging to Sakai’s space of initial values of type
$X$. It should be noted that infinitely many $q$-Painlevé equations of the
same surface type exist.

$A_{0}^{(1)\ast}\to A_{1}^{(1)}\to A_{2}^{(1)}\to A_{3}^{(1)}\to
A_{4}^{(1)}\to A_{5}^{(1)}\to A_{6}^{(1)}~{}\begin{matrix}\nearrow\\\
\searrow\end{matrix}\begin{matrix}~{}A_{7}^{(1)}\\\\[16.00008pt]
~{}{A_{7}^{(1)}}^{\prime}\end{matrix}$ Figure 1.1.  Types of Sakai’s spaces of
initial values for $q$-Painlevé equations. The surface degenerates in the
direction of the arrow due to the specialization and confluence of the base
points that characterize the surface types.

#### (Extended) affine Weyl group symmetry

A transformation from an integrable system to an integrable system is called a
Bäcklund transformation. In the case of a Painlevé-type
differential/difference equation, self-Bäcklund transformations exist, meaning
transformations that map an equation to itself with varying parameters. These
collectively form an (extended) affine Weyl group. (See for example [30, 19,
58, 43].) In this sense, the equation is said to have an (extended) affine
Weyl group symmetry. Note that an (extended) affine Weyl group symmetry of a
Painlevé-type O$\Delta$E does not always refer solely to its Bäcklund
transformations, but it often means a large group of transformations,
including its time evolution. For example, see Theorems 1.2 and 1.4.

#### (Extended) KNY’s representation

In [29], Kajiwara-Noumi-Yamada showed a birational representation of an
extended affine Weyl group of $(A_{m-1}\times A_{n-1})^{(1)}$-type (KNY’s
representation), where $m$ and $n$ are integers greater than or equal to $2$,
except for $(m,n)=(2,2)$. Note that KNY’s representation is essentially the
same even if $m$ and $n$ are interchanged [35]. Recently, it has been reported
that KNY’s representation can be extended to a birational representation of an
extended affine Weyl group of $(A_{m-1}\times A_{n-1}\times
A_{g-1})^{(1)}$-type (extended KNY’s representation), where $g$ is the common
greatest divisor of $m$ and $n$ [32, 34].

It is well known that the $(A_{L}\times A_{1})^{(1)}$-type KNY’s
representation, where $L\in\mathbb{Z}_{\geq 2}$, gives the Painlevé-type
$q$-O$\Delta$Es, including $q$-Painlevé equations as the second-order
$q$-O$\Delta$Es. Indeed, the $(A_{2}\times A_{1})^{(1)}$-type KNY’s
representation gives $q$-Painlevé equations of $A_{5}^{(1)}$-surface type
[28], and the $(A_{3}\times A_{1})^{(1)}$-type gives $q$-Painlevé equations of
$A_{3}^{(1)}$-surface type [28, 59]. As mentioned above, the $(A_{2N+1}\times
A_{1})^{(1)}$-type KNY’s representation, where $N\in\mathbb{Z}_{\geq 1}$, is
extendable to the $(A_{2N+1}\times A_{1}\times A_{1})^{(1)}$-type extended
KNY’s representation. The paper [48] presents the explicit forms of the
Painlevé-type $q$-O$\Delta$Es obtained from the $(A_{2N+1}\times A_{1}\times
A_{1})^{(1)}$-type extended KNY’s representation.

#### Motivation for this study

In [37], a birational representation of an extended affine Weyl group of
$(A_{2N}\rtimes A_{1})^{(1)}$-type, where $N\in\mathbb{Z}_{\geq 1}$, was
constructed, and the system (1.1) was derived from its action. This
representation is presumed to be a degenerate version of KNY’s representation
(see Appendix B), and therefore, similar to KNY’s representation, it is
believed to be extendable. This is a motivation for this study. Note that the
methods for extending them are different. In this paper, the extension is
explored using the theory of consistency around the cube (CAC) property (see
[6, 38, 63, 41] for the CAC property), while the extension of KNY’s
representation was done using the theory of cluster algebra (see [5, 20, 46,
21, 42], for the theory of cluster algebra for discrete integrable systems).

###  1.2. Notation and Terminology

This paper will use the following notations and terminologies for conciseness.

  * •

An ordinary difference equation is written as O$\Delta$E and a partial
difference equation as P$\Delta$E. In particular, an ordinary multiplicative-
type difference ($q$-difference) equation is written as $q$-O$\Delta$E.

  * •

For transformations $s$ and $r$, the symbol $sr$ means the composite
transformation $s\circ r$.

  * •

In the context of transformations, the “$1$” signifies the identity
transformation.

  * •

For a transformation $s$, the relation $s^{\infty}=1$ means that there is no
positive integer $k$ such as $s^{k}=1$.

  * •

If the subscript number is greater than the superscript number in the product
symbol, $1$ is assumed. For example,

$\prod_{k=1}^{0}2^{k}=1.$ (1.8)

  * •

If the subscript number is greater than the superscript number in the
summation symbol, $0$ is assumed. For example,

$\sum_{k=1}^{0}2^{k}=0.$ (1.9)

  * •

When an equation number has a subscript such as ”$l=0$”, it signifies an
equation obtained by substituting $l=0$ into the equation corresponding to
that equation number. For instance, with reference to

$a_{l}+a_{l+1}+a_{l+2}=0,$ (1.10)

the equation (1.10)l=0 is equivalent to

$a_{0}+a_{1}+a_{2}=0.$ (1.11)

The same applies to the case where “$l\to l+1$” or other symbols are in the
subscripts. For instance, the equation (1.10)l→l+1 is equivalent to

$a_{l+1}+a_{l+2}+a_{l+3}=0.$ (1.12)

###  1.3. Outline of the paper

This paper is organized as follows. In §2, we introduce the system (2.1) and
the transformations $\\{\sigma,w,\mu\\}$ that remain the system invariant.
Then, imposing periodic conditions on the system (2.1), we obtain birational
actions of the transformations $\\{\sigma,w,\mu\\}$. In §3, using the
transformation group $\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$
given in [37] and the transformations $\\{\sigma,w,\mu\\}$, we prove Theorem
1.4. Some concluding remarks are given in §4. In Appendix A, we show that the
system (2.1) has the CAC and tetrahedron properties. In Appendix B, we provide
conjectures on periodic reductions from systems of P$\Delta$Es with the CAC
property to Painlevé type $q$-O$\Delta$Es.

##  2\. Transformations $\sigma$, $w$ and $\mu$

In this section, we define transformations $\\{\sigma,w,\mu\\}$ and obtain
their birational actions necessary for the proof of Theorem 1.4 through a
staircase reduction of a system of P$\Delta$Es.

###  2.1. A system of P$\Delta$Es with the CAC property

Let $(l,m)\in\mathbb{Z}^{2}$. We focus on the following system of P$\Delta$Es
for $U_{l,m},V_{l,m}\in\mathbb{C}$:

$\displaystyle\dfrac{U_{l+1,m+1}}{U_{l,m}}+{\gamma_{l+m}}^{4}\dfrac{U_{l,m+1}}{U_{l+1,m}}+\alpha_{l}\beta_{m}\gamma_{l+m}=0,$
(2.1a)
$\displaystyle\dfrac{V_{l+1,m+1}}{V_{l,m}}+{\delta_{l+m}}^{4}\dfrac{V_{l,m+1}}{V_{l+1,m}}+\alpha_{l}\beta_{m}\delta_{l+m}=0,$
(2.1b)
$\displaystyle\dfrac{U_{l,m+1}}{V_{l,m}}=\begin{cases}\dfrac{V_{l,m+1}}{U_{l,m}}&\text{(if
$(l+m)$ is even)},\\\\[10.00002pt]
\dfrac{\gamma^{4}}{\delta^{4}}\left(\dfrac{V_{l,m+1}}{U_{l,m}}+\dfrac{\beta_{m}\delta^{4}(\gamma-\delta)}{\epsilon^{l-2m-1}}\right)&\text{(if
$(l+m)$ is odd)},\end{cases}$ (2.1c)
$\displaystyle\dfrac{U_{l,m}}{V_{l+1,m}}=\begin{cases}\dfrac{\gamma^{3}}{\delta^{3}}\left(\dfrac{V_{l,m}}{U_{l+1,m}}-\dfrac{\delta^{3}(\gamma-\delta)}{\alpha_{l}\epsilon^{l-2m}}\right)&\text{(if
$(l+m)$ is even)},\\\\[10.00002pt]
\dfrac{\gamma}{\delta}\left(\dfrac{U_{l+1,m}}{V_{l,m}}+\dfrac{\gamma(\gamma-\delta)}{\alpha_{l}\epsilon^{l-2m-1}}\right)^{-1}&\text{(if
$(l+m)$ is odd)},\end{cases}$ (2.1d)

where
$\\{\alpha_{k},\beta_{k},\gamma_{k},\delta_{k},\gamma,\delta,\epsilon\\}_{k\in\mathbb{Z}}$
are complex parameters satisfying

$\gamma_{l}=\begin{cases}\gamma&\text{(if $l$ is even)},\\\\[5.0pt]
\gamma^{-1}&\text{(if $l$ is
odd)},\end{cases}\quad\delta_{m}=\begin{cases}\delta&\text{(if $m$ is
even)},\\\\[5.0pt] \delta^{-1}&\text{(if $m$ is
odd)},\end{cases}\quad\epsilon=\gamma\delta.$ (2.2)

The system (2.1) has the CAC and tetrahedron properties. See Appendix A for
details.

The transformations $\\{\sigma,w,\mu\\}$ are defined by the actions on the
parameters
$\\{\alpha_{k},\beta_{k},\gamma_{k},\delta_{k},\gamma,\delta,\epsilon\\}_{k\in\mathbb{Z}}$
and the variables $\\{U_{l,m},V_{l,m}\\}$ as follows.

$\displaystyle\sigma(\alpha_{k})=\epsilon^{2}\alpha_{k+2},\quad\sigma(\beta_{k})=\epsilon^{-2}\beta_{k},\quad\sigma(\gamma_{k})=\gamma_{k},\quad\sigma(\delta_{k})=\delta_{k},\quad\sigma(\gamma)=\gamma,$
$\displaystyle\sigma(\delta)=\delta,\quad\sigma(\epsilon)=\epsilon,\quad\sigma(U_{l,m})=U_{l+2,m},\quad\sigma(V_{l,m})=V_{l+2,m},$
(2.3a) $\displaystyle w(\alpha_{k})=\epsilon^{2N-4}\alpha_{-k+2N},\quad
w(\beta_{k})=\epsilon^{-2N+4}\beta_{-k-1},\quad
w(\gamma_{k})={\gamma_{k}}^{-1},\quad w(\delta_{k})={\delta_{k}}^{-1},$
$\displaystyle w(\gamma)={\gamma}^{-1},\quad w(\delta)={\delta}^{-1},\quad
w(\epsilon)={\epsilon}^{-1},\quad w(U_{l,m})=\dfrac{1}{U_{-l+2N+1,-m}},$
$\displaystyle w(V_{l,m})=\dfrac{1}{V_{-l+2N+1,-m}},$ (2.3b)
$\displaystyle\mu(\alpha_{k})=\alpha_{k},\quad\mu(\beta_{k})=\beta_{k},\quad\mu(\gamma_{k})=\delta_{k},\quad\mu(\delta_{k})=\gamma_{k},\quad\mu(\gamma)=\delta,\quad\mu(\delta)=\gamma,$
$\displaystyle\mu(\epsilon)=\epsilon,\quad\mu(U_{l,m})=V_{l,m},\quad\mu(V_{l,m})=U_{l,m}.$
(2.3c)

We can easily verify that the system (2.1) is invariant under the actions of
the transformations $\\{\sigma,w,\mu\\}$ and these transformations satisfy the
following relations:

$\sigma^{\,\infty}=w^{2}=\mu^{2}=1,\quad\sigma
w=w\sigma^{-1},\quad\sigma\mu=\mu\sigma,\quad w\mu=\mu w.$ (2.4)

###  2.2. A staircase reduction of the system (2.1)

Fix an integer $N>0$. We consider the $(2N+1,1)$-periodic conditions

$U_{l+2N+1,m+1}=U_{l,m},\quad V_{l+2N+1,m+1}=V_{l,m},$ (2.5)

which are equivalent to expressing $U_{l,m}$ and $V_{l,m}$ as

$U_{l,m}=U(l-(2N+1)m),\quad V_{l,m}=V(l-(2N+1)m).$ (2.6)

Substituting (2.6) into the system (2.1), we obtain the following system of
O$\Delta$Es:

$\displaystyle\dfrac{U(l)}{U(l+2N+2)}+{\gamma_{l}}^{4}\dfrac{U(l+1)}{U(l+2N+1)}+\dfrac{\alpha_{l}\beta_{0}{\gamma_{l}}^{3}}{\epsilon^{2N-1}}=0,$
(2.7a)
$\displaystyle\dfrac{V(l)}{V(l+2N+2)}+{\delta_{l}}^{4}\dfrac{V(l+1)}{V(l+2N+1)}+\dfrac{\alpha_{l}\beta_{0}{\delta_{l}}^{3}}{\epsilon^{2N-1}}=0,$
(2.7b)
$\displaystyle\dfrac{U(2l+1)}{V(2l+2N+2)}-\dfrac{V(2l+1)}{U(2l+2N+2)}=0,$
(2.7c)
$\displaystyle\dfrac{U(2l)}{V(2l+2N+1)}-\dfrac{\gamma^{4}V(2l)}{\delta^{4}U(2l+2N+1)}-\dfrac{\beta_{0}\gamma^{4}(\gamma-\delta)}{\epsilon^{2(l+N)}}=0,$
(2.7d)
$\displaystyle\dfrac{U(2l)}{V(2l+1)}-\dfrac{\gamma^{3}V(2l)}{\delta^{3}U(2l+1)}+\dfrac{\gamma^{3}(\gamma-\delta)}{\epsilon^{2l}\alpha_{2l}}=0,$
(2.7e) $\displaystyle\dfrac{U(2l+2)}{V(2l+1)}-\dfrac{\gamma V(2l+2)}{\delta
U(2l+1)}+\dfrac{\gamma(\gamma-\delta)}{\epsilon^{2l}\alpha_{2l+1}}=0,$ (2.7f)

together with the conditions for the parameters

$\alpha_{l+2N+1}=\epsilon^{-2N+1}\alpha_{l},\quad\beta_{m+1}=\epsilon^{2N-1}\beta_{m}.$
(2.8)

Define the parameters $\\{a_{0},\dots,a_{2N},b,c,q\\}$ and the variables
$\\{f(l),g(l)\\}$ by

$\displaystyle
a_{0}=\epsilon^{\frac{1-2N}{2N+1}}\left(\dfrac{\alpha_{0}}{\alpha_{2N}}\right)^{\frac{1}{2N+1}},\quad
a_{i}=\left(\dfrac{\alpha_{i}}{\alpha_{i-1}}\right)^{\frac{1}{2N+1}},~{}i=1,\dots,2N,\quad
b=\left(\prod_{k=0}^{2N}\alpha_{k}\right)^{\frac{1}{2N+1}}\beta_{0},$ (2.9a)
$\displaystyle c=\gamma,\quad q=\epsilon^{\frac{1-2N}{2N+1}},\quad
f(l)=\dfrac{U(l-1)}{U(l+1)},\quad g(l)=\dfrac{V(l-1)}{V(l+1)},$ (2.9b)

where the following holds:

$\prod_{i=0}^{2N}a_{i}=q.$ (2.10)

Then, the following lemma holds.

######  Lemma 2.1.

The following $2N$th-order $q$-O$\Delta$Es for $f(l)$ and $g(l)$ hold:

$\displaystyle
f(l+2N+1)f(l+1)\left(\prod_{k=1}^{N-1}f(l+2k+1)\right)+{\gamma_{l}}^{4}\left(\prod_{k=1}^{N}f(l+2k)\right)$
$\displaystyle=-\dfrac{q^{l+2N}a_{l}b{\gamma_{l}}^{3}}{\displaystyle\prod_{k=1}^{2N-1}{a_{l+k}}^{2N-k}},$
(2.11a) $\displaystyle
g(l+2N+1)g(l+1)\left(\prod_{k=1}^{N-1}g(l+2k+1)\right)+{\delta_{l}}^{4}\left(\prod_{k=1}^{N}g(l+2k)\right)$
$\displaystyle=-\dfrac{q^{l+2N}a_{l}b{\delta_{l}}^{3}}{\displaystyle\prod_{k=1}^{2N-1}{a_{l+k}}^{2N-k}},$
(2.11b)

where $a_{i+2N+1}=a_{i}$ for arbitrary $i\in\mathbb{Z}$. Here, $\gamma_{l}$
and $\delta_{l}$ are expressed using $c$ and $q$, respectively, as follows:

$\gamma_{l}=\begin{cases}c&\text{(if $l$ is even)},\\\\[10.00002pt]
\dfrac{1}{c}&\text{(if $l$ is
odd)},\end{cases}\qquad\delta_{l}=\begin{cases}\dfrac{\epsilon}{c}&\text{(if
$l$ is even)},\\\\[10.00002pt] \dfrac{c}{\epsilon}&\text{(if $l$ is
odd)},\end{cases}$ (2.12)

where $\epsilon=q^{\frac{2N+1}{1-2N}}$.

###### Proof.

Equations (2.11a) and (2.11b) are obtained from Equations (2.7a) and (2.7b),
respectively. ∎

Define

$f_{i}=f(i),\quad g_{i}=g(i),\quad i=1,\dots,2N.$ (2.13)

Then, the following lemma holds.

######  Lemma 2.2.

The following relations hold:

$\displaystyle\left(1-\dfrac{f_{1}}{{a_{1}}^{2N+1}c^{2}}\right)\dfrac{f_{1}-\epsilon^{2N-3}c^{6-4N}g_{1}}{f_{1}-q^{2N+1}\epsilon^{2N-3}c^{4}g_{1}}$
$\displaystyle\hskip
10.00002pt+\sum_{k=1}^{N-1}\dfrac{\displaystyle\prod_{i=1}^{k}f_{2i-1}}{\left(\displaystyle\prod_{i=1}^{2k}{a_{i}}^{2N+1}\right)c^{4k}\left(\displaystyle\prod_{i=1}^{k}f_{2i}\right)}\left(1-\dfrac{f_{2k+1}}{{a_{2k+1}}^{2N+1}c^{2}}\right)$
$\displaystyle\hskip
10.00002pt+\dfrac{{a_{0}}^{2N+1}}{q^{2N+1}c^{4N}\left(\displaystyle\prod_{i=1}^{N}f_{2i}\right)}\left(\rule{0.0pt}{25.00003pt}\right.\dfrac{q^{2N}bc}{\displaystyle\prod_{i=0}^{2N-1}{a_{i}}^{2N-i}}+\displaystyle\prod_{i=1}^{N}f_{2i-1}\left.\rule{0.0pt}{25.00003pt}\right)=0,$
(2.14a)
$\displaystyle\dfrac{g_{2l_{1}+3}\left(\epsilon^{2}f_{2l_{1}+1}-c^{4}g_{2l_{1}+1}\right)\left({a_{2l_{1}+3}}^{2N+1}c^{2}-f_{2l_{1}+3}\right)}{f_{2l_{1}+2}\left(\epsilon^{2}f_{2l_{1}+3}-c^{4}g_{2l_{1}+3}\right)\left({a_{2l_{1}+1}}^{2N+1}c^{2}-f_{2l_{1}+1}\right)}=\epsilon^{2}{a_{2l_{1}+2}}^{2N+1}{a_{2l_{1}+3}}^{2N+1},$
(2.14b)
$\displaystyle\dfrac{\left(c^{2}-\epsilon^{2}{a_{2l_{2}+2}}^{2N+1}g_{2l_{2}+2}\right)\left(\epsilon^{2}{a_{2l_{2}+1}}^{2N+1}-c^{2}g_{2l_{2}+1}\right)}{\left(1-{a_{2l_{2}+2}}^{2N+1}c^{2}f_{2l_{2}+2}\right)\left({a_{2l_{2}+1}}^{2N+1}c^{2}-f_{2l_{2}+1}\right)}=\epsilon^{2},$
(2.14c)

where

$l_{1}=0,\dots,N-2,\quad
l_{2}=0,\dots,N-1,\quad\epsilon=q^{\frac{2N+1}{1-2N}}.$ (2.15)

###### Proof.

For simplicity, let

$A(l)=\dfrac{\epsilon^{2l-3}V(2l)}{\gamma^{4l-3}(\gamma-\delta)U(2l)},$ (2.16)

which satisfies

$\dfrac{A(l)}{A(l+1)}=\dfrac{\gamma^{4}g(2l+1)}{\epsilon^{2}f(2l+1)}.$ (2.17)

Eliminating $V(2l+1)$ from Equations (2.7e) and (2.7f), we obtain

$A(l+1)-A(l)=-\dfrac{U(2l+1)}{\alpha_{2l}\gamma^{4l}U(2l)}\left(1-\dfrac{\alpha_{2l}U(2l)}{\alpha_{2l+1}\gamma^{2}U(2l+2)}\right).$
(2.18)

Then, from Equation (2.18), we obtain the following relations:

$\displaystyle
A(N)=A(l+1)-\sum_{k=l+1}^{N-1}\dfrac{U(2k+1)}{\alpha_{2k}\gamma^{4k}U(2k)}\left(1-\dfrac{\alpha_{2k}U(2k)}{\alpha_{2k+1}\gamma^{2}U(2k+2)}\right),$
(2.19) $\displaystyle
A(0)=A(l)+\sum_{k=0}^{l-1}\dfrac{U(2k+1)}{\alpha_{2k}\gamma^{4k}U(2k)}\left(1-\dfrac{\alpha_{2k}U(2k)}{\alpha_{2k+1}\gamma^{2}U(2k+2)}\right),$
(2.20)

where $0\leq l\leq N-1$. Eliminating $V(2N+1)$ from Equations (2.7d)l=0 and
(2.7e)l=N, we obtain

$A(N)-\dfrac{\epsilon^{2N-1}}{\gamma^{4N-2}}A(0)=\dfrac{U(2N+1)}{\alpha_{2N}\gamma^{4N}U(0)}\left(\alpha_{2N}\beta_{0}\gamma+\dfrac{U(0)}{U(2N)}\right).$
(2.21)

Eliminating $A(0)$ and $A(N)$ from Equations (2.19), (2.20) and (2.21), we
obtain

$\displaystyle A(l+1)-\dfrac{\epsilon^{2N-1}}{\gamma^{4N-2}}A(l)=$
$\displaystyle\sum_{k=l+1}^{N-1}\dfrac{U(2k+1)}{\alpha_{2k}\gamma^{4k}U(2k)}\left(1-\dfrac{\alpha_{2k}U(2k)}{\alpha_{2k+1}\gamma^{2}U(2k+2)}\right)$
$\displaystyle+\dfrac{\epsilon^{2N-1}}{\gamma^{4N-2}}\sum_{k=0}^{l-1}\dfrac{U(2k+1)}{\alpha_{2k}\gamma^{4k}U(2k)}\left(1-\dfrac{\alpha_{2k}U(2k)}{\alpha_{2k+1}\gamma^{2}U(2k+2)}\right)$
$\displaystyle+\dfrac{U(2N+1)}{\alpha_{2N}\gamma^{4N}U(0)}\left(\alpha_{2N}\beta_{0}\gamma+\dfrac{U(0)}{U(2N)}\right).$
(2.22)

From (2.18) and (2.2), we obtain

$\displaystyle\left(1-\dfrac{\alpha_{2l}U(2l)}{\alpha_{2l+1}\gamma^{2}U(2l+2)}\right)\dfrac{A(l+1)-\dfrac{\epsilon^{2N-1}}{\gamma^{4N-2}}A(l)}{A(l+1)-A(l)}$
$\displaystyle=-\sum_{k=l+1}^{N-1}\dfrac{\alpha_{2l}U(2k+1)U(2l)}{\alpha_{2k}\gamma^{4(k-l)}U(2l+1)U(2k)}\left(1-\dfrac{\alpha_{2k}U(2k)}{\alpha_{2k+1}\gamma^{2}U(2k+2)}\right)$
$\displaystyle\hskip
10.00002pt-\dfrac{\epsilon^{2N-1}}{\gamma^{4N-2}}\sum_{k=0}^{l-1}\dfrac{\alpha_{2l}\gamma^{4(l-k)}U(2k+1)U(2l)}{\alpha_{2k}U(2l+1)U(2k)}\left(1-\dfrac{\alpha_{2k}U(2k)}{\alpha_{2k+1}\gamma^{2}U(2k+2)}\right)$
$\displaystyle\hskip
10.00002pt-\dfrac{\alpha_{2l}U(2N+1)U(2l)}{\alpha_{2N}\gamma^{4(N-l)}U(2l+1)U(0)}\left(\alpha_{2N}\beta_{0}\gamma+\dfrac{U(0)}{U(2N)}\right),$
(2.23)

where $0\leq l\leq N-1$. Equation (2.2)l=0 gives (2.14a).

From Equations (2.2) and (2.2)l→l+1, we obtain

$\dfrac{A(l+1)-A(l)}{A(l+2)-A(l+1)}=\dfrac{\alpha_{2l+2}\gamma^{4}U(2l+2)U(2l+1)}{\alpha_{2l}U(2l)U(2l+3)}\left(\cfrac{1-\dfrac{\alpha_{2l}U(2l)}{\alpha_{2l+1}\gamma^{2}U(2l+2)}}{1-\dfrac{\alpha_{2l+2}U(2l+2)}{\alpha_{2l+3}\gamma^{2}U(2l+4)}}\right),$
(2.24)

where $0\leq l\leq N-2$, which gives Equation (2.14b).

By solving Equation (2.7e) with $V(2l)$ and Equation (2.7e)l→l+1 with
$V(2l+3)$, and substituting the results into Equation (2.14c), we obtain

$\displaystyle\left(\dfrac{U(2l+2)}{V(2l+1)}-\dfrac{\gamma V(2l+2)}{\delta
U(2l+1)}+\dfrac{\gamma(\gamma-\delta)}{\epsilon^{2l}\alpha_{2l+1}}\right)$
$\displaystyle\times\left(\dfrac{U(2l)}{V(2l+1)}-\dfrac{\alpha_{2l+2}\gamma^{5}V(2l+2)}{\alpha_{2l}\delta
U(2l+3)}+\dfrac{\gamma^{3}(\gamma-\delta)}{\epsilon^{2l}\alpha_{2l}}\right)=0.$
(2.25)

The equation above holds due to Equation (2.7f). Therefore, Equation (2.14c)
holds. ∎

Let $\mathcal{K}$ be a field of rational functions over $\mathbb{C}$ defined
as

${\mathcal{K}}=\mathbb{C}(a_{1},\dots,a_{2N},b,c,q^{\frac{1}{2N-1}}).$ (2.26)

Then, from Lemmas 2.1 and 2.2, the following lemma holds.

######  Lemma 2.3.

The variables $f(l)$ and $g(l)$ are expressed as rational functions of the
variables $\\{f_{1},\dots,f_{2N}\\}$ over ${\mathcal{K}}$.

###### Proof.

From Equations (2.11a) and (2.11b), the variables $f(l)$ and $g(l)$ are
expressed as a rational function of $\\{f_{1},\dots,f_{2N}\\}$ and that of
$\\{g_{1},\dots,g_{2N}\\}$ over ${\mathcal{K}}$, respectively.

The variables $g_{i}$, $i=1,\dots,2N$, are shown to be rational functions of
$\\{f_{1},\dots,f_{2N}\\}$ over ${\mathcal{K}}$, as described below. From
(2.14a), it is evident that $g_{1}$ is a rational function of the variables
$\\{f_{1},\dots,f_{2N}\\}$. From (2.14b), $g_{2i+3}$, $i=0,\dots,N-2$, are
sequentially shown to be rational functions of the variables
$\\{f_{1},\dots,f_{2N}\\}$. Finally, $g_{2j+2}$, $j=0,\dots,N-1$, are shown to
be a rational function of the variables $\\{f_{1},\dots,f_{2N}\\}$ by using
(2.14c). ∎

We can easily verify that the conditions of the reduction (2.5) and (2.8) are
also invariant under the actions of the transformations $\\{\sigma,w,\mu\\}$.
Therefore, even after reduction, it is possible to use these transformations.
The actions of these transformations can be obtained from (2.3), (2.6), (2.8),
(2.9) and (2.13). The resulting actions are in the following lemma.

######  Lemma 2.4.

The actions of the transformations $\\{\sigma,w,\mu\\}$ on the prameters
$\\{a_{0},\dots,a_{2N},b,c,q\\}$ are given by

$\displaystyle\sigma(a_{i})=a_{i+2},\quad\sigma(b)=q^{2}b,\quad\sigma(c)=c,\quad\sigma(q)=q,$
(2.27a) $\displaystyle w(a_{i})=\dfrac{1}{a_{2N+1-i}},\quad
w(b)=q^{2N+1}b,\quad w(c)=c^{-1},\quad w(q)=q^{-1},$ (2.27b)
$\displaystyle\mu(a_{i})=a_{i},\quad\mu(b)=b,\quad\mu(c)=\dfrac{\epsilon}{c},\quad\mu(q)=q,$
(2.27c)

where $i\in\mathbb{Z}/(2N+1)\mathbb{Z}$ and $\epsilon=q^{\frac{2N+1}{1-2N}}$,
while their actions on the variables $\\{f_{1},\dots,f_{2N}\\}$ are given by

$\displaystyle\sigma(f_{i})=f_{i+2},~{}i=1,\dots,2N-2,\quad\sigma(f_{2N-1})=f(2N+1),\quad\sigma(f_{2N})=f(2N+2),$
(2.28a) $\displaystyle w(f_{j})=f_{2N+1-j},\quad\mu(f_{j})=g_{j},\quad
j=1,\dots,2N.$ (2.28b)

Here, the variables $f(2N+1)$ and $f(2N+2)$ are given by (2.11a), and the
variables $\\{g_{1},\dots,g_{2N}\\}$ are given by (2.14).

##  3\. Proof of Theorem 1.4

In this section, we first review the transformation group
$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$ in [37]. Then, using
the transformations $\\{\sigma,w,\mu\\}$ given in §2, we extend it to the
transformation group $\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\times
A_{1}^{(1)}\right)$.

###  3.1. Review of the transformation group
$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$

This subsection reviews the result in [37].

Let us define the action of the transformations
$\\{s_{1},\dots,s_{2N},w_{1},\pi\\}$ on the complex parameters
$\\{a_{0},\dots,a_{2N},b,c,q\\}$ and complex variables
$\\{f_{1},\dots,f_{2N}\\}$. Note that the parameters satisfy the following
relation:

$\prod_{i=0}^{2N}a_{i}=q.$ (3.1)

The actions of the transformations $\\{s_{1},\dots,s_{2N},w_{1},\pi\\}$ on the
prameters $\\{a_{0},\dots,a_{2N},b,c,q\\}$ are given by

$\displaystyle s_{i}(a_{j})=\begin{cases}{a_{i}}^{-1}&\text{if }j=i,\\\
a_{i}a_{j}&\text{if }j=i\pm 1,\\\ a_{j}&\text{otherwise},\end{cases}\qquad
i=1,\dots,2N-1,$ (3.2a) $\displaystyle
s_{2N}(a_{j})=\begin{cases}{a_{2N}}^{-1}&\text{if }j=2N,\\\
a_{2N}a_{j}&\text{if }j=0,2N-1,\\\ a_{j}&\text{otherwise},\end{cases}$ (3.2b)
$\displaystyle s_{k}(b)=b,\quad s_{k}(c)=c,\quad s_{k}(q)=q,\quad
k=1,\dots,2N,$ (3.2c) $\displaystyle w_{1}(a_{j})=\dfrac{1}{a_{2N+1-j}},\quad
w_{1}(b)=q^{2N+1}b,\quad w_{1}(c)=c^{-1},\quad w_{1}(q)=q^{-1},$ (3.2d)
$\displaystyle\pi(a_{j})=a_{j+1},\quad\pi(b)=qb,\quad\pi(c)=c^{-1},\quad\pi(q)=q,$
(3.2e)

where $j\in\mathbb{Z}/(2N+1)\mathbb{Z}$, while their actions on the variables
$\\{f_{1},\dots,f_{2N}\\}$ are given by

$\displaystyle
s_{i}(f_{j})=\begin{cases}f_{i-1}\dfrac{\lambda(i-1)^{2}-{a_{i}}^{2N+1}f_{i}}{{a_{i}}^{2N+1}\lambda(i-1)^{2}-f_{i}}&\text{if
}j=i-1,\\\
f_{i+1}\dfrac{{a_{i}}^{2N+1}\lambda(i-1)^{2}-f_{i}}{\lambda(i-1)^{2}-{a_{i}}^{2N+1}f_{i}}&\text{if
}j=i+1,\\\ f_{j}&\text{otherwise},\end{cases}\qquad i,j=1,\dots,2N,$ (3.3a)
$\displaystyle w_{1}(f_{j})=f_{2N+1-j},\quad j=1,\dots,2N,$ (3.3b)
$\displaystyle\pi(f_{j})=\begin{cases}f_{j+1}&\text{if }j=1,\dots,2N-1,\\\
-\dfrac{c^{4}}{\displaystyle\prod_{k=1}^{N}f_{2k-1}}\left(\displaystyle\prod_{k=1}^{N}f_{2k}+\dfrac{q^{2N}a_{0}b}{\left(\displaystyle\prod_{k=1}^{2N-1}{a_{k}}^{2N-k}\right)c}\right)&\text{if
}j=2N,\end{cases}$ (3.3c)

where

$\lambda(l)=\begin{cases}c&\text{(if $l$ is even)},\\\ c^{-1}&\text{(if $l$ is
odd)}.\end{cases}$ (3.4)

Define the transformation $s_{0}$ by

$s_{0}=\pi^{-1}s_{1}\pi.$ (3.5)

Then, the transformations $\\{s_{0},\dots,s_{2N},w_{1},\pi\\}$ satisfy the
following relations:

$\displaystyle{s_{i}}^{2}=(s_{i}s_{i\pm 1})^{3}=(s_{i}s_{j})^{2}=1,~{}j\neq
i\pm 1,\quad{w_{1}}^{2}=1,$ (3.6a) $\displaystyle
w_{1}s_{k}=s_{2N-k+1}w_{1},\quad\pi^{\infty}=1,\quad\pi
s_{k}=s_{k+1}\pi,\quad\pi w_{1}=w_{1}\pi^{-1},$ (3.6b)

where $i,j,k\in\mathbb{Z}/(2N+1)\mathbb{Z}$. Moreover, by defining the
transformations $w_{0}$ and $r$ by

$w_{0}=\pi^{2}w_{1},\quad r=\pi w_{1},$ (3.7)

the transformations $\\{s_{0},\dots,s_{2N},w_{0},w_{1},r\\}$ satisfy the
following relations:

$\displaystyle{s_{i}}^{2}=(s_{i}s_{i\pm 1})^{3}=(s_{i}s_{j})^{2}=1,~{}j\neq
i\pm 1,\quad{w_{0}}^{2}={w_{1}}^{2}=(w_{0}w_{1})^{\infty}=1,$ (3.8a)
$\displaystyle w_{0}s_{i}=s_{2N-i+3}w_{0},\quad w_{1}s_{i}=s_{2N-i+1}w_{1},$
(3.8b) $\displaystyle r^{2}=1,\quad rs_{i}=s_{2N-i+2}r,\quad
rw_{0}=w_{1}r,\quad rw_{1}=w_{0}r,$ (3.8c)

where $i,j\in\mathbb{Z}/(2N+1)\mathbb{Z}$. From the relation (3.8), the
following hold.

  * •

Transformation groups $\langle s_{0},\dots,s_{2N}\rangle$ and $\langle
w_{0},w_{1}\rangle$ form the affine Weyl group of type $A_{2N}^{(1)}$ and that
of type $A_{1}^{(1)}$, respectively.

  * •

The transformation group $\langle s_{0},\dots,s_{2N},w_{0},w_{1}\rangle$ is a
semidirect product of $\langle s_{0},\dots,s_{2N}\rangle$ and $\langle
w_{0},w_{1}\rangle$, that is,

$\langle s_{0},\dots,s_{2N},w_{0},w_{1}\rangle=\langle
s_{0},\dots,s_{2N}\rangle\rtimes\langle w_{0},w_{1}\rangle.$ (3.9)

We denote it as $W\left((A_{2N}\rtimes A_{1})^{(1)}\right)$.

  * •

The transformation $r$ corresponds to a reflection of the Dynkin diagram of
type $A_{2N}^{(1)}$ associated with $\langle s_{0},\dots,s_{2N}\rangle$ and to
that of type $A_{1}^{(1)}$ associated with $\langle w_{0},w_{1}\rangle$.
Therefore, we refer to the transformation group $\langle
s_{0},\dots,s_{2N},w_{0},w_{1},r\rangle$ as an extended affine Weyl group of
type $(A_{2N}\rtimes A_{1})^{(1)}$ and denote it as
$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$, that is,

$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)=\Big{(}\langle
s_{0},\dots,s_{2N}\rangle\rtimes\langle
w_{0},w_{1}\rangle\Big{)}\rtimes\langle r\rangle.$ (3.10)

######  Remark 3.1.

Define the transformations $T_{0},T_{1}\in\widetilde{W}\left((A_{2N}\rtimes
A_{1})^{(1)}\right)$ as

$T_{0}=\pi^{-2N-1},\quad T_{1}=\pi s_{2N}s_{2N-1}\cdots s_{1},$ (3.11)

whose actions on the parameters $\\{a_{0},\dots,a_{2N},b,c,q\\}$ are given by

$\displaystyle T_{0}(a_{i})=a_{i},~{}i=0,\dots,2N,\quad
T_{0}(b)=q^{-2N-1}b,\quad T_{0}(c)=c^{-1},\quad T_{0}(q)=q,$ (3.12a)
$\displaystyle T_{1}(a_{j})=\begin{cases}qa_{0}&\text{if }j=0,\\\
q^{-1}a_{1}&\text{if }j=1,\\\ a_{j}&\text{otherwise},\end{cases}\quad
T_{1}(b)=qb,\quad T_{1}(c)=c^{-1},\quad T_{1}(q)=q.$ (3.12b)

The system (1.1) can be obtained from the action of $T_{0}$ with the following
correspondence (see [37] for details):

$\overline{\rule{0.0pt}{5.0pt}~{}\,}=T_{0},\quad F_{i}=f_{i},\quad
p=q^{-2N-1}.$ (3.13)

Thus, the transformation $T$ satisfying (1.4) is given by

$T=T_{0}.$ (3.14)

Also, the transformation $\hat{T}$ satisfying (1.5) is given by

$\hat{T}={T_{1}}^{2N+1}T_{0}.$ (3.15)

Note that in §1, we omit the parameter $a_{0}$ by using the relation (3.1),
and furthermore, we use $p=q^{-2N-1}$ instead of $q$.

######  Remark 3.2.

The transformation $\pi\in\widetilde{W}\left((A_{2N}\rtimes
A_{1})^{(1)}\right)$ gives the following relation:

$\pi^{2N}(f_{1})f_{1}\left(\prod_{k=1}^{N-1}\pi^{2k}(f_{1})\right)+c^{4}\left(\prod_{k=1}^{N}\pi^{2k-1}(f_{1})\right)+\dfrac{q^{2N}a_{0}bc^{3}}{\displaystyle\prod_{k=1}^{2N-1}{a_{k}}^{2N-k}}=0.$
(3.16)

Applying $\pi^{l}$ on the equation above and setting

$f(k)=\pi^{k-1}(f_{1}),$ (3.17)

we obtain the $2N$th-order $q$-O$\Delta$E (2.11a).

###  3.2. Proof of Theorem 1.4

In this subsection, using the transformations $\\{\sigma,w,\mu\\}$ given in §2
and the transformation group $\widetilde{W}\left((A_{2N}\rtimes
A_{1})^{(1)}\right)$ shown in §3.1, we prove Theorem 1.4. For this purpose, in
this subsection, we will discuss the proof under the following setup.

  * •

The parameters $\\{a_{0},\dots,a_{2N},b,c,q\\}$ and variables
$\\{f_{1},\dots,f_{2N}\\}$ defined in §2.2 are considered identical to those
in §3.1.

  * •

We consider that the transformations $\\{\sigma,w,\mu\\}$ have actions defined
only on the parameters $\\{a_{0},\dots,a_{2N},b,c,q\\}$ and variables
$\\{f_{1},\dots,f_{2N}\\}$.

Comparing the actions of the transformations $\\{\sigma,w\\}$ (2.27) and
(2.28) with those of $\\{\pi,w_{1}\\}$ (3.2) and (3.3), we obtain

$\sigma=\pi^{2},\quad w=w_{1},$ (3.18)

which mean $\sigma,w\in\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$.
We note Remark 3.2 for the action of $\pi$. Next, let us focus on the
transformation $\mu$. It is trivially evident from actions on the parameter
$c$ that the transformation $\mu$ is not contained within
$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\right)$. Therefore, in the
following, we consider what type of extended affine Weyl group is formed by
extending the transformation group $\widetilde{W}\left((A_{2N}\rtimes
A_{1})^{(1)}\right)$ through the addition of the transformation $\mu$.

######  Lemma 3.3.

The following relations hold:

$s_{1}\mu=\mu s_{1},\quad w_{1}\mu=\mu
w_{1},\quad\pi^{2}\mu=\mu\pi^{2},\quad(\mu\pi)^{\infty}=1.$ (3.19)

###### Proof.

Due to the correspondence (3.18) and the relation (2.4), the relations
$w_{1}\mu=\mu w_{1}$ and $\pi^{2}\mu=\mu\pi^{2}$. For positive integer $k$,
the following holds:

$(\mu\pi)^{k}(c)=\epsilon^{-k}c,$ (3.20)

which gives $(\mu\pi)^{\infty}=1$.

Let us prove that the relation $s_{1}\mu=\mu s_{1}$ holds. Since the relation
obviously holds for the action on the parameters, we here consider the action
on the $f$-variables. Applying $s_{1}$ to Equation (2.14a), we obtain

$\displaystyle\left(1-\dfrac{{a_{1}}^{2N+1}f_{1}}{c^{2}}\right)\dfrac{f_{1}-\epsilon^{2N-3}c^{6-4N}s_{1}(g_{1})}{f_{1}-q^{2N+1}\epsilon^{2N-3}c^{4}s_{1}(g_{1})}$
$\displaystyle+\dfrac{{a_{1}}^{2N+1}(c^{2}-{a_{1}}^{2N+1}f_{1})}{{a_{1}}^{2N+1}c^{2}-f_{1}}\left\\{\,\sum_{k=1}^{N-1}\dfrac{\displaystyle\prod_{i=1}^{k}f_{2i-1}}{\left(\displaystyle\prod_{i=1}^{2k}{a_{i}}^{2N+1}\right)c^{4k}\left(\displaystyle\prod_{i=1}^{k}f_{2i}\right)}\left(1-\dfrac{f_{2k+1}}{{a_{2k+1}}^{2N+1}c^{2}}\right)\right.$
$\displaystyle\left.\hskip
100.00015pt+\dfrac{{a_{0}}^{2N+1}}{q^{2N+1}c^{4N}\left(\displaystyle\prod_{i=1}^{N}f_{2i}\right)}\left(\rule{0.0pt}{25.00003pt}\right.\dfrac{q^{2N}bc}{\displaystyle\prod_{i=0}^{2N-1}{a_{i}}^{2N-i}}+\displaystyle\prod_{i=1}^{N}f_{2i-1}\left.\rule{0.0pt}{25.00003pt}\right)\,\right\\}=0.$
(3.21)

Eliminating the second and third terms of Equation (2.14a) by using the
equation above, we obtain

$\displaystyle\left(1-\dfrac{f_{1}}{{a_{1}}^{2N+1}c^{2}}\right)\dfrac{f_{1}-\epsilon^{2N-3}c^{6-4N}g_{1}}{f_{1}-q^{2N+1}\epsilon^{2N-3}c^{4}g_{1}}$
$\displaystyle=\dfrac{{a_{1}}^{2N+1}c^{2}-f_{1}}{{a_{1}}^{2N+1}(c^{2}-{a_{1}}^{2N+1}f_{1})}\,\left(1-\dfrac{{a_{1}}^{2N+1}f_{1}}{c^{2}}\right)\dfrac{f_{1}-\epsilon^{2N-3}c^{6-4N}s_{1}(g_{1})}{f_{1}-q^{2N+1}\epsilon^{2N-3}c^{4}s_{1}(g_{1})},$
(3.22)

which, upon simplification, gives

$s_{1}(g_{1})=g_{1}.$ (3.23)

Furthermore, comparing the equation obtained by applying $s_{1}$ to Equation
(2.14b)${}_{l_{1}=0}$ with Equation (2.14b)${}_{l_{1}=0}$, we obtain

$s_{1}(g_{3})=g_{3}.$ (3.24)

Similarly, we can prove

$s_{1}(g_{2l_{1}+3})=g_{2l_{1}+3},\quad l_{1}=1,\dots,N-2,$ (3.25)

sequentially from Equation (2.14b). Finally, comparing the equation obtained
by applying $s_{1}$ to Equation (2.14c) with Equation (2.14c), we obtain

$s_{1}(g_{2l_{2}+2})=g_{2l_{2}+2},\quad l_{2}=0,\dots,N-1.$ (3.26)

Therefore, we obtain

$s_{1}\mu(f_{l})=s_{1}(g_{l})=g_{l}=\mu(f_{l})=\mu s_{1}(f_{l}),\quad
l=1,\dots,2N.$ (3.27)

Thus, $s_{1}\mu=\mu s_{1}$ holds. ∎

Defining the transformation $\mu_{0}$ and $\mu_{1}$ as

$\mu_{0}=\pi^{-1}\mu\pi,\quad\mu_{1}=\mu,$ (3.28)

the following lemma holds.

######  Lemma 3.4.

The following relations hold:

${\mu_{i}}^{2}=(\mu_{0}\mu_{1})^{\infty}=1,\quad\mu_{i}s_{j}=s_{j}\mu_{i},\quad\mu_{i}w_{k}=w_{k}\mu_{i},\quad
r\mu_{0}=\mu_{1}r,\quad r\mu_{1}=\mu_{0}r,$ (3.29)

where

$i=0,1,\quad j=0,\dots,2N,\quad k=0,1.$ (3.30)

###### Proof.

Relation $(\mu_{0}\mu_{1})^{\infty}=1$ holds becuse for positive integer $k$,
the following holds:

$(\mu_{0}\mu_{1})^{k}(c)=\epsilon^{2k}c.$ (3.31)

The other relations are shown by using Equations (2.4), (3.6), (3.7), (3.8)
and (3.19) as the following:

$\displaystyle{\mu_{0}}^{2}=(\pi^{-1}\mu\pi)(\pi^{-1}\mu\pi)=\pi^{-1}\mu^{2}\pi=1,$
(3.32a) $\displaystyle{\mu_{1}}^{2}=\mu^{2}=1,$ (3.32b) $\displaystyle
r\mu_{0}=(\pi w_{1})(\pi^{-1}\mu\pi)=w_{1}\pi^{-2}\mu\pi=w_{1}\mu\pi^{-1}=\mu
w_{1}\pi^{-1}=\mu(\pi w_{1})=\mu_{1}r,$ (3.32c) $\displaystyle
r\mu_{1}=r\mu_{1}r^{2}=r(\mu_{1}r)r=r(r\mu_{0})r=r^{2}\mu_{0}r=\mu_{0}r,$
(3.32d) $\displaystyle\mu_{1}w_{1}=\mu w_{1}=w_{1}\mu=w_{1}\mu_{1},$ (3.32e)
$\displaystyle\mu_{1}w_{0}=\mu(\pi^{2}w_{1})=(\pi^{2}w_{1})\mu=w_{0}\mu_{1},$
(3.32f)
$\displaystyle\mu_{0}w_{1}=\mu_{0}r^{2}w_{1}=(\mu_{0}r)(rw_{1})=r(\mu_{1}w_{0})r=r(w_{0}\mu_{1})r=(w_{1}r)(r\mu_{0})=w_{1}\mu_{0},$
(3.32g)
$\displaystyle\mu_{0}w_{0}=\mu_{0}r^{2}w_{0}=(\mu_{0}r)(rw_{0})=r(\mu_{1}w_{1})r=r(w_{1}\mu_{1})r=(w_{0}r)(r\mu_{0})=w_{0}\mu_{0},$
(3.32h) $\displaystyle\mu_{1}s_{1}=\mu s_{1}=s_{1}\mu=s_{1}\mu_{1},$ (3.32i)
$\displaystyle\mu_{1}s_{2i+1}=\mu(\pi^{2i}s_{1}\pi^{-2i})=(\pi^{2i}s_{1}\pi^{-2i})\mu=s_{2i+1}\mu_{1},$
(3.32j)
$\displaystyle\mu_{1}s_{2N}=\mu_{1}{w_{1}}^{2}s_{2N}=(\mu_{1}w_{1})(w_{1}s_{2N})=w_{1}(\mu_{1}s_{1})w_{1}=w_{1}(s_{1}\mu_{1})w_{1}=s_{2N}\mu_{1},$
(3.32k)
$\displaystyle\mu_{1}s_{2j}=\mu_{1}(\pi^{2(j-N)}s_{2N}\pi^{2(N-j)})=(\pi^{2(j-N)}s_{2N}\pi^{2(N-j)})\mu_{1}=s_{2j}\mu_{1},$
(3.32l)
$\displaystyle\mu_{0}s_{k}=(\pi^{-1}\mu_{1}\pi)s_{k}=\pi^{-1}\mu_{1}s_{k+1}\pi=\pi^{-1}s_{k+1}\mu_{1}\pi=s_{k}(\pi^{-1}\mu_{1}\pi)=s_{k}\mu_{0},$
(3.32m)

where

$i=1,\dots,N-1,\quad j=0,\dots,N-1,\quad k\in\mathbb{Z}/(2N+1)\mathbb{Z}.$
(3.33)

∎

From the relation (3.29), we find that the transformation group
$\langle\mu_{0},\mu_{1}\rangle$ forms the affine Weyl group of type
$A_{1}^{(1)}$, and it orthogonal to the transformation group

$W\left((A_{2N}\rtimes A_{1})^{(1)}\right)=\langle
s_{0},\dots,s_{2N}\rangle\rtimes\langle w_{0},w_{1}\rangle.$ (3.34)

The transformation $r$ corresponds not only to a reflection of the Dynkin
diagram of type $A_{2N}^{(1)}$ associated with $\langle
s_{0},\dots,s_{2N}\rangle$ and that of type $A_{1}^{(1)}$ associated with
$\langle w_{0},w_{1}\rangle$, but also to a reflection of the Dynkin diagram
of type $A_{1}^{(1)}$ associated with $\langle\mu_{0},\mu_{1}\rangle$.
Therefore, we refer to the transformation group

$\langle
s_{0},\dots,s_{2N},w_{0},w_{1},\mu_{0},\mu_{1},r\rangle=\left(W\left((A_{2N}\rtimes
A_{1})^{(1)}\right)\times\langle\mu_{0},\mu_{1}\rangle\right)\rtimes\langle
r\rangle$ (3.35)

as an extended affine Weyl group of type $(A_{2N}\rtimes A_{1})^{(1)}\times
A_{1}^{(1)}$ and denote it as

$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\times A_{1}^{(1)}\right),$
(3.36)

that is,

$\widetilde{W}\left((A_{2N}\rtimes A_{1})^{(1)}\times
A_{1}^{(1)}\right)=\bigg{(}\Big{(}\langle
s_{0},\dots,s_{2N}\rangle\rtimes\langle
w_{0},w_{1}\rangle\Big{)}\times\langle\mu_{0},\mu_{1}\rangle\bigg{)}\rtimes\langle
r\rangle.$ (3.37)

Thus, we have completed the proof of Theorem 1.4.

######  Remark 3.5.

In the case $N=1$, the group $\widetilde{W}\left((A_{2}\rtimes
A_{1})^{(1)}\times A_{1}^{(1)}\right)$ can be reconstructed as a subgroup of
the extended affine Weyl symmetry group of type $A_{4}^{(1)}$ for
$A_{4}^{(1)}$-surface type $q$-Painlevé equations [58, 61, 26]. Indeed, all
elements of $\widetilde{W}\left((A_{2}\rtimes A_{1})^{(1)}\times
A_{1}^{(1)}\right)$ are given by elements of the transformation group in [26]:

$\widetilde{W}\left(A_{4}^{(1)}\right)=\langle{\bm{s}_{0}},{\bm{s}_{1}},{\bm{s}_{2}},{\bm{s}_{3}},{\bm{s}_{4}}\rangle\rtimes\langle{\bm{\sigma}},{\bm{\iota}}\rangle$
(3.38)

as the following:

$\displaystyle s_{0}={\bm{s}_{0}},\quad
s_{1}={\bm{s}_{3}}{\bm{s}_{4}}{\bm{s}_{3}},\quad
s_{2}={\bm{s}_{1}}{\bm{s}_{2}}{\bm{s}_{1}},\quad
w_{0}={\bm{\sigma}}{\bm{\iota}}{\bm{s}_{2}}{\bm{s}_{4}},\quad
w_{1}={\bm{\iota}},$ (3.39a)
$\displaystyle\mu_{0}={\bm{s}_{4}}{\bm{s}_{0}}{\bm{s}_{1}}{\bm{s}_{0}}{\bm{s}_{4}},\quad\mu_{1}={\bm{s}_{2}}{\bm{s}_{3}}{\bm{s}_{2}},\quad
r={\bm{\sigma}^{3}}{\bm{\iota}}{\bm{s}_{4}}.$ (3.39b)

In this case, the correspondence between the parameters
$\\{a_{0},a_{1},a_{2},b,c,q\\}$ and variables $\\{f_{1},f_{2}\\}$ on which
$\widetilde{W}\left((A_{2}\rtimes A_{1})^{(1)}\times A_{1}^{(1)}\right)$ acts
and the parameters
$\\{{\bm{a}_{0}},{\bm{a}_{1}},{\bm{a}_{2}},{\bm{a}_{3}},{\bm{a}_{4}},{\bm{q}}\\}$
and variables $\\{{\bm{f}}_{1}^{(i)},{\bm{f}}_{2}^{(i)}\\}_{i=1,\dots,5}$ on
which $\widetilde{W}\left(A_{4}^{(1)}\right)$ acts is given below.

$\displaystyle a_{0}={\bm{a}_{0}}^{-1/6},\quad
a_{1}={\bm{a}_{3}}^{-1/6}{\bm{a}_{4}}^{-1/6},\quad
a_{2}={\bm{a}_{1}}^{-1/6}{\bm{a}_{2}}^{-1/6},$ (3.40a) $\displaystyle
b=-\dfrac{{\bm{a}_{0}}^{1/4}{\bm{a}_{1}}^{7/12}{\bm{a}_{2}}^{1/12}{\bm{a}_{3}}^{5/12}}{{\bm{a}_{4}}^{1/12}},\quad
c=\dfrac{{\bm{a}_{0}}^{1/4}{\bm{a}_{1}}^{1/4}{\bm{a}_{4}}^{1/4}}{{\bm{a}_{2}}^{1/4}{\bm{a}_{3}}^{1/4}},\quad
q={\bm{q}}^{-1/6},$ (3.40b) $\displaystyle
f_{1}=\dfrac{{\bm{a}_{0}}^{1/2}{\bm{a}_{1}}^{1/2}}{{\bm{a}_{2}}^{1/2}}\left(1-\dfrac{{\bm{a}_{0}}{\bm{a}_{4}}}{{\bm{a}_{2}}{\bm{a}_{3}}}{\bm{f}}_{1}^{(2)}{\bm{f}}_{1}^{(3)}\right),\quad
f_{2}=-\dfrac{{\bm{a}_{0}}^{1/2}}{{\bm{a}_{3}}^{1/2}{\bm{a}_{4}}^{1/2}}{\bm{f}}_{1}^{(3)}.$
(3.40c)

######  Remark 3.6.

The transformation $\widetilde{T}$ in §1 is given by, for example,

$\widetilde{T}=\mu_{0}\mu_{1}\in\widetilde{W}\left((A_{2N}\rtimes
A_{1})^{(1)}\times A_{1}^{(1)}\right)$ (3.41)

whose action on the parameters $\\{a_{0},\dots,a_{2N},b,c,q\\}$ is

$\widetilde{T}(a_{i})=a_{i},~{}i=0,\dots,2N,\quad\widetilde{T}(b)=b,\quad\widetilde{T}(c)=\epsilon^{2}c,\quad\widetilde{T}(q)=q,$
(3.42)

where $\epsilon=q^{\frac{2N+1}{1-2N}}$.

##  4\. Concluding remarks

In this paper, we extend the birational representation of an extended affine
Weyl group of $(A_{2N}\rtimes A_{1})^{(1)}$-type, the symmetry of the system
(1.1) obtained in [37], to a birational representation of an extended affine
Weyl group of $(A_{2N}\rtimes A_{1})^{(1)}\times A_{1}^{(1)}$-type. For this
purpose, we used the system (2.1) having the CAC property.

It is surprising that a Painlevé-type O$\Delta$E is related to a system of
P$\Delta$E such as the system (2.1), which consists of an alternation of two
types of CAC-cubes (see Appendix A). This finding suggests that the theory of
the CAC property is deeply connected to the theory of Painlevé type
O$\Delta$Es. We give conjectures on periodic reductions from systems of
P$\Delta$Es with the CAC property to Painlevé type O$\Delta$Es in Appendix B.

### Acknowledgment

This work was supported by JSPS KAKENHI Grant Numbers JP19K14559 and
JP23K03145.

##  Appendix A CAC and tetrahedron properties of the system (2.1)

This Appendix shows that the system (2.1) has the CAC and tetrahedron
properties. (See [6, 38, 63, 41] for the CAC and tetrahedron properties.)

Let us consider the system (2.1) as a system of equations around cubes as
follows. Assign the variables $U_{l,m}$, $V_{l,m}$ to the vertices
$(l,m,0),(l,m,1)\in\mathbb{Z}^{3}$, respectively. Then, for fixed
$l,m\in\mathbb{Z}$, the system (2.1) can be regarded as equations on the faces
(face equations) of the cube whose vertices are given by

$\displaystyle(l,m,0),\quad(l+1,m,0),\quad(l,m+1,0),\quad(l+1,m+1,0),$
$\displaystyle(l,m,1),\quad(l+1,m,1),\quad(l,m+1,1),\quad(l+1,m+1,1).$ (A.1)

We here label the six faces of the cube by ${\mathcal{A}}$,
${\mathcal{A}}^{\prime}$, ${\mathcal{B}}$, ${\mathcal{B}}^{\prime}$,
${\mathcal{C}}$, ${\mathcal{C}}^{\prime}$ as illustrated in Figure A.1. Then,
from the system (2.1) the six face equations are given by

$\displaystyle{\mathcal{A}}:~{}\dfrac{~{}\widetilde{\overline{U}}~{}}{U}+\dfrac{\gamma^{4}\widetilde{U}}{\overline{U}}+\alpha_{l}\,\beta_{m}\gamma=0,$
$\displaystyle{\mathcal{A}}^{\prime}:~{}\dfrac{~{}\widetilde{\overline{V}}~{}}{V}+\dfrac{\delta^{4}\widetilde{V}}{\overline{V}}+\alpha_{l}\,\beta_{m}\delta=0,$
(A.2a)
$\displaystyle{\mathcal{B}}:~{}\dfrac{~{}\widetilde{U}~{}}{V}=\dfrac{~{}\widetilde{V}~{}}{U},$
$\displaystyle{\mathcal{B}}^{\prime}:~{}\dfrac{~{}\widetilde{\overline{U}}~{}}{\overline{V}}=\dfrac{\gamma^{4}}{\delta^{4}}\left(\dfrac{~{}\widetilde{\overline{V}}~{}}{\overline{U}}+\dfrac{\beta_{m}\delta^{4}(\gamma-\delta)}{\epsilon^{l-2m}}\right),$
(A.2b)
$\displaystyle{\mathcal{C}}:~{}\dfrac{~{}U~{}}{\overline{V}}=\dfrac{\gamma^{3}}{\delta^{3}}\left(\dfrac{~{}V~{}}{\overline{U}}-\dfrac{\delta^{3}(\gamma-\delta)}{\alpha_{l}\epsilon^{l-2m}}\right),$
$\displaystyle{\mathcal{C}}^{\prime}:~{}\dfrac{~{}\widetilde{U}~{}}{\widetilde{\overline{V}}}=\dfrac{\gamma}{\delta}\left(\dfrac{~{}\widetilde{\overline{U}}~{}}{\widetilde{V}}+\dfrac{\gamma(\gamma-\delta)}{\alpha_{l}\epsilon^{l-2m-3}}\right)^{-1},$
(A.2c)

if $(l+m)$ is even, and by

$\displaystyle{\mathcal{A}}:~{}\dfrac{~{}\widetilde{\overline{U}}~{}}{U}+\dfrac{\widetilde{U}}{\gamma^{4}\overline{U}}+\dfrac{\alpha_{l}\,\beta_{m}}{\gamma}=0,$
$\displaystyle{\mathcal{A}}^{\prime}:~{}\dfrac{~{}\widetilde{\overline{V}}~{}}{V}+\dfrac{\widetilde{V}}{\delta^{4}\overline{V}}+\dfrac{\alpha_{l}\,\beta_{m}}{\delta}=0,$
(A.3a)
$\displaystyle{\mathcal{B}}:~{}\dfrac{~{}\widetilde{U}~{}}{V}=\dfrac{\gamma^{4}}{\delta^{4}}\left(\dfrac{~{}\widetilde{V}~{}}{U}+\dfrac{\beta_{m}\delta^{4}(\gamma-\delta)}{\epsilon^{l-2m-1}}\right),$
$\displaystyle{\mathcal{B}}^{\prime}:~{}\dfrac{~{}\widetilde{\overline{U}}~{}}{\overline{V}}=\dfrac{~{}\widetilde{\overline{V}}~{}}{\overline{U}},$
(A.3b)
$\displaystyle{\mathcal{C}}:~{}\dfrac{~{}U~{}}{\overline{V}}=\dfrac{\gamma}{\delta}\left(\dfrac{~{}\overline{U}~{}}{V}+\dfrac{\gamma(\gamma-\delta)}{\alpha_{l}\epsilon^{l-2m-1}}\right)^{-1},$
$\displaystyle{\mathcal{C}}^{\prime}:~{}\dfrac{~{}\widetilde{U}~{}}{\widetilde{\overline{V}}}=\dfrac{\gamma^{3}}{\delta^{3}}\left(\dfrac{~{}\widetilde{V}~{}}{\widetilde{\overline{U}}}-\dfrac{\delta^{3}(\gamma-\delta)}{\alpha_{l}\epsilon^{l-2m-2}}\right),$
(A.3c)

if $(l+m)$ is odd. Here,

$\displaystyle
U=U_{l,m},\quad\overline{U}=U_{l+1,m},\quad\widetilde{U}=U_{l,m+1},\quad\widetilde{\overline{U}}=U_{l+1,m+1},$
(A.4a) $\displaystyle
V=V_{l,m},\quad\overline{V}=V_{l+1,m},\quad\widetilde{V}=V_{l,m+1},\quad\widetilde{\overline{V}}=V_{l+1,m+1}.$
(A.4b)

Denote the cubes given by (A.2) and (A.3) as $C^{\rm(e)}_{l,m}$ and
$C^{\rm(o)}_{l,m}$, respectively. We can verify by direct calculation that
both cubes $C^{\rm(e)}_{l,m}$ and $C^{\rm(o)}_{l,m}$ have the CAC and
tetrahedron properties. Since all cubes composing the system (2.1) have the
CAC and tetrahedron properties, the system (2.1) is said to have the CAC and
tetrahedron properties. Note that the tetrahedron relations of the cube
$C^{\rm(e)}_{l,m}$ are given by

$\displaystyle\overline{U}\,\widetilde{U}-\dfrac{\alpha_{l}\epsilon^{l-2m-3}}{\delta(\gamma-\delta)}\left(\overline{U}\,\widetilde{\overline{V}}+\gamma^{2}\epsilon\,\widetilde{U}\,V+\alpha_{l}\,\beta_{m}\,\delta\,\overline{U}\,V\right)=0,$
(A.5a)
$\displaystyle\overline{V}\,\widetilde{V}+\dfrac{\alpha_{l}\epsilon^{l-2m-3}}{\gamma(\gamma-\delta)}\left(\widetilde{\overline{U}}\,\overline{V}+\delta^{2}\epsilon\,U\,\widetilde{V}+\alpha_{l}\,\beta_{m}\,\gamma\,U\,\overline{V}\right)=0,$
(A.5b)

while those of the cube $C^{\rm(o)}_{l,m}$ are given by

$\displaystyle
V\,\widetilde{\overline{V}}+\dfrac{\alpha_{l}\epsilon^{l-2m-2}}{\gamma^{3}(\gamma-\delta)}\left(\widetilde{U}\,V+\gamma^{2}\epsilon\,\overline{U}\,\widetilde{\overline{V}}+\alpha_{l}\,\beta_{m}\,\gamma^{3}\overline{U}\,V\right)=0,$
(A.6a) $\displaystyle
U\,\widetilde{\overline{U}}-\dfrac{\alpha_{l}\epsilon^{l-2m-2}}{\delta^{3}(\gamma-\delta)}\left(U\,\widetilde{V}+\delta^{2}\epsilon\,\widetilde{\overline{U}}\,\overline{V}+\alpha_{l}\,\beta_{m}\,\delta^{3}U\,\overline{V}\right)=0.$
(A.6b)

Figure A.1. A cube with 8 variables labelled by
$\\{U,\overline{U},\widetilde{U},\widetilde{\overline{U}},V,\overline{V},\widetilde{V},\widetilde{\overline{V}}\\}$
and 6 faces labelled by
$\\{{\mathcal{A}},{\mathcal{A}}^{\prime},{\mathcal{B}},{\mathcal{B}}^{\prime},{\mathcal{C}},{\mathcal{C}}^{\prime}\\}$.
Bottom: ${\mathcal{A}}$, top: ${\mathcal{A}}^{\prime}$, left: ${\mathcal{B}}$,
right: ${\mathcal{B}}^{\prime}$, front: ${\mathcal{C}}$, back:
${\mathcal{C}}^{\prime}$.

######  Remark A.1.

Adler-Bobenko-Suris [2, 3] and Boll [9, 7, 8] classified six face equations of
cubes having the CAC and tetrahedron properties. The face equations given by
the systems (A.2) and (A.3) all belong to $D_{4}$ of type $H^{6}$ in Boll’s
classification.

######  Remark A.2.

Conversely, the system (2.1) can be regarded as a system of P$\Delta$Es
obtained from space-filling two types of cubes $C^{\rm(e)}_{l,m}$ and
$C^{\rm(o)}_{l,m}$ into the sublattice of the lattice $\mathbb{Z}^{3}$ whose
vertices are given by (A). The cubes $C^{\rm(e)}_{l,m}$ and $C^{\rm(o)}_{l,m}$
are arranged on quadrilaterals of the lattice $\mathbb{Z}^{2}$ as illustrated
in Figure A.2.

Figure A.2. Arrangement of the cubes $C^{\rm(e)}_{l,m}$ and
$C^{\rm(o)}_{l,m}$. These cubes are alternately positioned on quadrilaterals
of the lattice $\mathbb{Z}^{2}$.

##  Appendix B Conjectures on periodic reductions from systems of P$\Delta$Es
with the CAC property to Painlevé type $q$-O$\Delta$Es

Fix integers $n\geq 1$ and $L\geq 0$. In this appendix, we label a pair, the
system of P$\Delta$Es

$\displaystyle\lambda(l_{0\cdots
n})^{2}\dfrac{U_{\overline{ij}}}{U}=\dfrac{\alpha^{(i)}(l_{i})U_{\overline{i}}-\alpha^{(j)}(l_{j})U_{\overline{j}}}{\alpha^{(j)}(l_{j})U_{\overline{i}}-\alpha^{(i)}(l_{i})U_{\overline{j}}},$
$\displaystyle i<j,\quad i,j\in\\{1,\dots,n\\},$ (B.1a)
$\displaystyle\dfrac{U_{\overline{0\,k}}}{U}+\lambda(l_{0\cdots
n})^{4}\dfrac{U_{\overline{0}}}{U_{\overline{k}}}+\alpha^{(k)}(l_{k})\kappa(l_{0})\lambda(l_{0\cdots
n})=0,$ $\displaystyle k=1,\dots,n,$ (B.1b)

and the periodic condition

$U(l_{1}+1,\dots,l_{n}+1,l_{0}+L)=U(l_{1},\dots,l_{n},l_{0}),$ (B.2)

as an $(n,L)$-pair. Note that the $(n,0)$\- and $(1,L)$-pair are exceptions
and are defined as follows. The $(n,0)$-pair is defined by a pair of the
system (B.1) and the condition

$\displaystyle
U(l_{1},\dots,l_{n},l_{0})=G(l_{1},\dots,l_{n},l_{0})\omega(l_{1},\dots,l_{n},l_{0}),$
(B.3a)
$\displaystyle\omega(l_{1}+1,\dots,l_{n}+1,l_{0})=\omega(l_{1},\dots,l_{n},l_{0}),$
(B.3b)

where, depending on the value of $n$, it is necessary to specify an
appropriate gauge function $G(l_{1},\dots,l_{n},l_{0})$. (See [23] for the
$(3,0)$-pair.) On the other hand, the $(1,L)$-pair is defined by the following
pair:

$\displaystyle\dfrac{U(l_{1}+1,l_{0}+1)}{U(l_{1},l_{0})}+\lambda(l_{0}+l_{1})^{4}\dfrac{U(l_{1},l_{0}+1)}{U(l_{1}+1,l_{0})}+\alpha^{(1)}(l_{1})\kappa(l_{0})\lambda(l_{0}+l_{1})=0,$
(B.4) $\displaystyle U(l_{1}+1,l_{0}+L)=U(l_{1},l_{0}).$ (B.5)

Here, $l_{0},\dots,l_{n}\in\mathbb{Z}$ are lattice parameters,
$\\{\alpha^{(1)}(l),\dots,\alpha^{(n)}(l),\kappa(l),\lambda_{0}\\}_{l\in\mathbb{Z}}$
are complex parameters and

$\displaystyle U=U(l_{1},\dots,l_{n},l_{0}),\quad
U_{\overline{i}}=U|_{\,l_{i}\to l_{i}+1},\quad
U_{\overline{ij}}=U|_{(l_{i},l_{j})\to(l_{i}+1,l_{j}+1)},$ (B.6a)
$\displaystyle l_{0\cdots
n}:=\sum_{i=0}^{n}l_{i},\quad\lambda(l)=\begin{cases}\lambda_{0}&l\in
2\mathbb{Z},\\\\[5.0pt] \dfrac{1}{\lambda_{0}}&\text{otherwise}.\end{cases}$
(B.6b)

######  Remark B.1.

The system (B.1) has the CAC and tetrahedron properties. Equation (B.1a) with
fixed $i$ and $j$ is known as the lattice modified KdV equation [39, 40] and
belongs to H3 in ABS’s classification [2, 3], while Equation (B.1b) with fixed
$k$ belongs to $D_{4}$ of type $H^{6}$ in Boll’s classification [7, 9, 8].

###  B.1. Conjectures on the $(n,L)$-pair

This subsection presents conjectures on the $(n,L)$-pair.

Before providing conjectures on the $(n,L)$-pair, let us explain its results.

  * •

In [37], we obtained a birational representation of an extended affine Weyl
group of $(A_{n-1}\rtimes A_{1})^{(1)}$-type from the $(n,1)$-pair, where
$n\geq 2$. Note that the $(2,1)$-pair and the $(3,1)$-pair were studied
earlier in [25] and [26], respectively. In particular, the birational
representation for the $(2N+1,1)$-pair, where $N\geq 1$, has been extended in
this paper to a birational representation of an extended affine Weyl group of
type $(A_{2N}\rtimes A_{1})^{(1)}\times A_{1}^{(1)}$.

  * •

From the $(1,L)$-pair, where $L\geq 2$, we obtain the Painlevé type
$q$-O$\Delta$Es given in [47, 33]. See §B.3 for details.

  * •

We can derive $q$-Painlevé equations of $A_{5}^{(1)}$-surface type from the
$(3,0)$-pair (see [23, 25]). Moreover, we can obtain $q$-Painlevé equations of
$A_{6}^{(1)}$\- and $A_{4}^{(1)}$-surface type from the $(2,1)$\- and
$(3,1)$-pair, respectively (see [26]) and $q$-Painlevé equations of
$A_{7}^{(1)}$\- and $A_{6}^{(1)}$-surface type from the $(1,2)$\- and
$(1,3)$-pair, respectively (see §B.3).

The following are conjectures.

  * •

We expect that the $(A_{n-1}\times A_{1})^{(1)}$-type KNY’s representation can
be derived from the $(n,0)$-pair, where $n\geq 3$, in the same way as in [37].
(It has been shown in [23, 25, 24] that this conjecture is correct for the
$(3,0)$-pair.) In particular, the birational representation for the
$(2N+2,0)$-pair, where $N\geq 1$, can be extended to the $(A_{2N+1}\times
A_{1}\times A_{1})^{(1)}$-type extended KNY’s representation by using a
different pair of a system of P$\Delta$Es and a periodic condition from the
$(2N+2,0)$-pair, as demonstrated in this paper.

  * •

We expect to obtain $q$-Painlevé equations of $A_{5}^{(1)}$\- and
$A_{3}^{(1)}$-surface type from the $(2,2)$\- and $(4,0)$-pair, respectively.

  * •

Through a limit operation, the $(n,L)$-pair can be degenerate to the
$(n-1,L+1)$-pair. As illustrations, limit operations from the $(3,L)$-pair,
where $L\neq 0$, to the $(2,L+1)$-pair and from the $(2,L)$-pair, where $L\neq
0$, to the $(1,L+1)$-pair are demonstrated in §B.4.

The results and conjectures mentioned above and predictions for other pairs
are summarized in Table B.1. Note that everything not mentioned in the results
above is a conjecture. The following items briefly explain Table B.1.

  * •

$q$-O$\Delta$Es appearing at the positions marked with “$\times$” in the table
are linearizable. Moreover, there exist $q$-Painlevé equations of
$A_{i}^{(1)}$-surface type at the positions marked with “$A_{i}^{(1)}$”, and
there exist $j$ th-order $q$-O$\Delta$Es at the positions marked with “$j$
th”.

  * •

The number $k$ in the symbol “$[k]$” represents the number of parameters of an
appearing $q$-O$\Delta$E including an independent variable but excluding a
shift parameter. For example, let us consider the system (1.1) obtained from
the $(2N+1,1)$-pair, where $N\geq 1$. The parameters involved in the system
are $\\{a_{1},\dots,a_{2N},b,c,p\\}$. Thus, the number of parameters,
including the independent variable “$b$” but excluding the shift parameter
“$p$”, is $2N+2$.

  * •

Moving to the right by two steps (i.e., $(n,L)\to(n+2,L)$) increases the order
of $q$-O$\Delta$Es by $2$ and also increases the number of their parameters by
$2$. On the other hand, moving downward by two steps (i.e., $(n,L)\to(n,L+2)$)
increases the order of appearing $q$-O$\Delta$Es by $2$, but the number of
their parameters remains unchanged.

  * •

An appropriate limit operation induces degeneration in the direction of the
bottom-left of the table (i.e., $(n,L)\to(n-1,L+1)$).

$\begin{array}[]{c||c|c|c|c|c|c|c}\text{\diagbox[]{{\shortstack[l]{$L$}}}{{\shortstack[r]{$n$}}}{}}&1&2&3&4&5&6&\cdots\\\
\hline\cr
0&\times&\times&A_{5}^{(1)}{}_{[3]}&A_{3}^{(1)}{}_{[5]}&\text{$4$th}~{}{}_{[5]}&\text{$4$th}~{}{}_{[7]}&\cdots\\\
\hline\cr
1&\times&A_{6}^{(1)}{}_{[2]}&A_{4}^{(1)}{}_{[4]}&\text{$4$th}~{}{}_{[4]}&\text{$4$th}~{}{}_{[6]}&\text{$6$th}~{}{}_{[6]}&\cdots\\\
\hline\cr
2&A_{7}^{(1)}{}_{[1]}&A_{5}^{(1)}{}_{[3]}&\text{$4$th}~{}{}_{[3]}&\text{$4$th}~{}{}_{[5]}&\text{$6$th}~{}{}_{[5]}&\text{$6$th}~{}{}_{[7]}&\cdots\\\
\hline\cr
3&A_{6}^{(1)}{}_{[2]}&\text{$4$th}~{}{}_{[2]}&\text{$4$th}~{}{}_{[4]}&\text{$6$th}~{}{}_{[4]}&\text{$6$th}~{}{}_{[6]}&\text{$8$th}~{}{}_{[6]}&\cdots\\\
\hline\cr
4&\text{$4$th}~{}{}_{[1]}&\text{$4$th}~{}{}_{[3]}&\text{$6$th}~{}{}_{[3]}&\text{$6$th}~{}{}_{[5]}&\text{$8$th}~{}{}_{[5]}&\text{$8$th}~{}{}_{[7]}&\cdots\\\
\hline\cr
5&\text{$4$th}~{}{}_{[2]}&\text{$6$th}~{}{}_{[2]}&\text{$6$th}~{}{}_{[4]}&\text{$8$th}~{}{}_{[4]}&\text{$8$th}~{}{}_{[6]}&\text{$10$th}\,{}_{[6]}&\cdots\\\
\hline\cr\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}$

Table B.1. Table presenting results and conjectures for the $(n,L)$-pair.

###  B.2. A conjecture on linear problems for $q$-O$\Delta$Es obtained from
the $(n,L)$-pair

In this subsection, we describe a conjecture concerning a linear problem of
Painlevé type $q$-O$\Delta$Es derived from the $(n,L)$-pair, where $L\geq 1$.
We provide an explanation for the pair (B.1) and (B.2), but the same applies
to the pair (B.4) and (B.5).

Let $L\geq 1$. In [37], a Lax representation of the system (B.1) is given by

$\displaystyle\phi_{\overline{i}}=\begin{pmatrix}\dfrac{\mu
U_{\overline{i}}}{\alpha^{(i)}(l_{i})U}&-\dfrac{{U_{\overline{i}}}^{2}}{\lambda(l_{0\cdots
n})}\\\\[10.00002pt] \dfrac{\lambda(l_{0\cdots n})}{U^{2}}&-\dfrac{\mu
U_{\overline{i}}}{\alpha^{(i)}(l_{i})U}\end{pmatrix}.\phi,\quad i=1,\dots,n,$
(B.7a)
$\displaystyle\phi_{\overline{0}}=\begin{pmatrix}-\dfrac{\mu\kappa(l_{0})U_{\overline{0}}}{U}&-\lambda(l_{0\cdots
n})^{2}{U_{\overline{0}}}^{2}\\\\[10.00002pt] \dfrac{1}{\lambda(l_{0\cdots
n})^{2}U^{2}}&0\end{pmatrix}.\phi,$ (B.7b)

where $\mu\in\mathbb{C}$ is a spectral variable,
$\phi=\phi(l_{1},\dots,l_{n},l_{0})$ is a column vector of length two and

$\phi_{\overline{i}}=\phi|_{\,l_{i}\to l_{i}+1}.$ (B.8)

Indeed, we can easily verify that the compatibility conditions

$\Big{(}\phi_{\overline{i}}\Big{)}_{\overline{j}}=\Big{(}\phi_{\overline{j}}\Big{)}_{\overline{i}}~{},\quad
0\leq i<j\leq n,$ (B.9)

give the system (B.1). Imposing the condition (B.2) on the system (B.1), we
obtain the following conditions:

$\dfrac{\alpha^{(1)}(l_{1})}{\alpha^{(1)}(l_{1}+1)}=\cdots=\dfrac{\alpha^{(n)}(l_{n})}{\alpha^{(n)}(l_{n}+1)}=\dfrac{\kappa(l_{0}+L)}{\kappa(l_{0})}=q,$
(B.10)

where $q\in\mathbb{C}$. Note that if $(n+L)$ is odd, then the following
additional condition is needed:

$\lambda_{0}=1.$ (B.11)

Define the shift operators $T_{i}$, $i=0,\dots,n$, by the actions on the
parameters
$\\{\alpha^{(1)}(l),\dots,\alpha^{(n)}(l),\kappa(l),\lambda_{0}\\}_{l\in\mathbb{Z}}$
and $\mu$, the variable $U=U(l_{1},\dots,l_{n},l_{0})$ and the vector
$\phi=\phi(l_{1},\dots,l_{n},l_{0})$ as the following:

$\displaystyle T_{i}(\alpha^{(j)}(l))=\begin{cases}\alpha^{(i)}(l+1)&\text{if
}j=i,\\\ \alpha^{(j)}(l)&\text{otherwise},\end{cases}\quad
T_{i}(\kappa(l))=\begin{cases}\kappa(l+1)&\text{if }i=0,\\\
\kappa(l)&\text{otherwise},\end{cases}$ (B.12a) $\displaystyle
T_{i}(\lambda_{0})=\dfrac{1}{\lambda_{0}},\quad T_{i}(\mu)=\mu,\quad
T_{i}(U)=U_{\overline{i}},\quad T_{i}(\phi)=\phi_{\overline{i}}.$ (B.12b)

Moreover, define the parameter $x$ and the operator $T_{x}$ by

$x=\dfrac{\mu}{\prod_{k=1}^{n}\alpha^{(k)}(0)^{1/n}},\quad
T_{x}=T_{1}T_{2}\cdots T_{n}{T_{0}}^{L}.$ (B.13)

Then, $x$ and $T_{x}$ can be regarded as a spectral variable and a spectral
operator of a Lax pair of a Painlevé type $q$-O$\Delta$E obtained from the
$(n,L)$-pair, respectively. The operator $T_{x}$ acts on $x$ as

$T_{x}(x)=qx.$ (B.14)

On the other hand, Painlevé parameters (an independent variable and parameters
of a Painlevé type $q$-O$\Delta$E derived from the $(n,L)$-pair) and Painlevé
variable (a dependent variable of the Painlevé type $q$-O$\Delta$E) are
invariant under the action of $T_{x}$. Note that in general the Painlevé
parameters are expressed as rational functions of the parameters
$\\{\alpha^{(1)}(l),\dots,\alpha^{(n)}(l),\kappa(l),\lambda_{0}\\}_{l\in\mathbb{Z}}$
(or their rational number powers), and the Painlevé variable is expressed as a
rational function of the $U$-variable. A spectral direction of linear problem
for Painlevé type $q$-O$\Delta$Es derived from the $(n,L)$-pair takes the
following form:

$T_{x}(\phi)=\begin{pmatrix}\ast\,x&\ast\\\
\ast&0\end{pmatrix}\dots\begin{pmatrix}\ast\,x&\ast\\\
\ast&0\end{pmatrix}.\begin{pmatrix}\ast\,x&\ast\\\
\ast&\ast\,x\end{pmatrix}\dots\begin{pmatrix}\ast\,x&\ast\\\
\ast&\ast\,x\end{pmatrix}.\phi,$ (B.15)

where the coefficient matrices above are given by the products of $L$ matrices
of the form

$\begin{pmatrix}\ast\,x&\ast\\\ \ast&0\end{pmatrix}$ (B.16)

and $n$ matrices of the form

$\begin{pmatrix}\ast\,x&\ast\\\ \ast&\ast\,x\end{pmatrix}.$ (B.17)

Here, the entries denoted by “$\ast$” are expressed as rational functions of
the parameters
$\\{\alpha^{(1)}(l),\dots,\alpha^{(n)}(l),\kappa(l),\lambda_{0}\\}_{l\in\mathbb{Z}}$
and the $U$-variables, and the entries remain invariant under the action of
$T_{x}$. Expressing the “$\ast$” entries solely in terms of Painlevé
parameters and variables may require careful gauge transformations of the
vector $\phi$. For specific examples, see [37] for the $(n,1)$-pair. (See also
[26] for the $(3,1)$-pair.)

###  B.3. Painlevé type $q$-O$\Delta$Es obtained from the $(1,L)$-pair

In this subsection, we show that the $(1,L)$-pair gives the Painlevé type
$q$-O$\Delta$Es in [47, 33]. We consider the cases separately based on the
parity of $L$.

####  B.3.1. The case $L$ is even

Consider the $(1,2h)$-pair, where $h\in\mathbb{Z}_{\geq 1}$. In this case, the
condition (B.5) is equivalent to expressing $U(l_{1},l_{0})$ as

$U(l_{1},l_{0})=U(l_{0}-2hl_{1}).$ (B.18)

Substituting (B.18) into Equation (B.4), we obtain

$\dfrac{U(l+2h+1)}{U(l)}+\dfrac{U(l+1)}{U(l+2h)}+p\alpha^{(1)}(0)\kappa(l)=0,$
(B.19)

with the conditions of the parameters

$\dfrac{\alpha(l_{1})}{\alpha(l_{1}+1)}=\dfrac{\kappa(l_{0}+2h)}{\kappa(l_{0})}=p,\quad\lambda_{0}=1,$
(B.20)

where $p\in\mathbb{C}$. Setting

$x_{l}=\dfrac{U(l+1)}{U(l)},$ (B.21)

from (B.19) we obtain

$x_{l+2h}x_{l}=-\dfrac{1}{\prod_{k=1}^{2h-1}x_{l+k}}\left(\dfrac{1}{\prod_{k=1}^{2h-1}x_{l+k}}+p\alpha^{(1)}(0)\kappa(l)\right).$
(B.22)

Set

$y_{l}=p\alpha^{(1)}(0)\kappa(l)\dfrac{U(l+2h)}{U(l+1)}=p\alpha^{(1)}(0)\kappa(l)\prod_{k=1}^{2h-1}x_{l+k}.$
(B.23)

Then, from Equation (B.22) the following relation holds:

$x_{l+2h}x_{l}=-p^{2}{\alpha^{(1)}(0)}^{2}{\kappa(l)}^{2}\dfrac{y_{l}+1}{{y_{l}}^{2}}.$
(B.24)

Therefore, we obtain

$\displaystyle y_{l+2h}y_{l}$
$\displaystyle=p^{3}{\alpha^{(1)}(0)}^{2}{\kappa(l)}^{2}\prod_{k=1}^{2h-1}x_{l+2h+k}x_{l+k}$
$\displaystyle=p^{3}{\alpha^{(1)}(0)}^{2}{\kappa(l)}^{2}\left(\prod_{k=1}^{2h-1}-p^{2}{\alpha^{(1)}(0)}^{2}{\kappa(l+k)}^{2}\dfrac{y_{l+k}+1}{{y_{l+k}}^{2}}\right)$
$\displaystyle=p^{4h+3}{\alpha^{(1)}(0)}^{4h+2}\left(\prod_{k=0}^{2h-1}{\kappa(l+k)}^{2}\right)\left(\prod_{k=1}^{2h-1}\dfrac{y_{l+k}+1}{{y_{l+k}}^{2}}\right).$
(B.25)

Finally, defining $q$ and $t_{l}$ by

$q=p^{2},\quad
t_{l}=q^{4h+3}{\alpha^{(1)}(0)}^{4h+2}\left(\prod_{k=0}^{2h-1}{\kappa(l+k)}^{2}\right),$
(B.26)

we obtain the following one kind of Painlevé type $q$-O$\Delta$Es in [47, 33]:

$y_{l+2h}y_{l}=t_{l}\left(\prod_{k=1}^{2h-1}\dfrac{y_{l+k}+1}{{y_{l+k}}^{2}}\right),$
(B.27)

where $t_{l}=q^{l}t_{0}$.

######  Remark B.2.

Equation (B.27) has an extended affine Weyl group symmetry of
$A_{1}^{(1)}$-type (see [33] for details).

######  Remark B.3.

When $h=1$, Equation (B.27) is equivalent to a $q$-Painlevé I equation of
$A_{7}^{(1)}$-surface type[13, 58] (see [47] for details).

####  B.3.2. The case $L$ is odd

Consider the $(1,2h+1)$-pair, where $h\in\mathbb{Z}_{\geq 1}$. In this case,
the condition (B.5) is equivalent to expressing $U(l_{1},l_{0})$ as

$U(l_{1},l_{0})=U(l_{0}-(2h+1)l_{1}).$ (B.28)

Substituting (B.28) into Equation (B.4), we obtain

$\dfrac{U(l+2h+2)}{U(l)}+\lambda(l)^{4}\dfrac{U(l+1)}{U(l+2h+1)}+q\alpha^{(1)}(0)\kappa(l)\lambda(l)^{3}=0,$
(B.29)

with the conditions of the parameters

$\dfrac{\alpha(l_{1})}{\alpha(l_{1}+1)}=\dfrac{\kappa(l_{0}+2h+1)}{\kappa(l_{0})}=q,$
(B.30)

where $q\in\mathbb{C}$. Setting

$x_{l}=\dfrac{U(l+2)}{U(l)},$ (B.31)

from (B.29) we obtain

$x_{l+2h}x_{l}=-\dfrac{\lambda(l)^{3}}{\prod_{k=1}^{h-1}x_{l+2k}}\left(\dfrac{\lambda(l)}{\prod_{k=0}^{h-1}x_{l+2k+1}}+q\alpha^{(1)}(0)\kappa(l)\right).$
(B.32)

Set

$y_{l}=q\alpha^{(1)}(0)\kappa(l-1)\lambda(l)\dfrac{U(l+2h)}{U(l)}=q\alpha^{(1)}(0)\kappa(l-1)\lambda(l)\prod_{k=0}^{h-1}x_{l+2k}.$
(B.33)

Then, from Equation (B.32) the following relation holds:

$x_{l+2h}=-q^{2}{\alpha^{(1)}(0)}^{2}\kappa(l-1)\kappa(l)\lambda(l)^{4}\dfrac{y_{l+1}+1}{y_{l}y_{l+1}}.$
(B.34)

Therefore, we obtain

$\displaystyle y_{l+2h}$
$\displaystyle=q\alpha^{(1)}(0)\kappa(l+2h-1)\lambda(l)\prod_{k=0}^{h-1}x_{l+2h+2k}$
$\displaystyle=(-1)^{h}q^{2h}{\alpha^{(1)}(0)}^{2h+1}\lambda(l)^{4h+1}\left(\prod_{k=0}^{2h}\kappa(l+k)\right)\left(\prod_{k=0}^{h-1}\dfrac{y_{l+2k+1}+1}{y_{l+2k}y_{l+2k+1}}\right)$
$\displaystyle=(-1)^{h}q^{2h}{\alpha^{(1)}(0)}^{2h+1}\lambda(l)^{4h+1}\left(\prod_{k=0}^{2h}\kappa(l+k)\right)\dfrac{\prod_{k=0}^{h-1}\left(y_{l+2k+1}+1\right)}{y_{l}\left(\prod_{k=1}^{2h-1}y_{l+k}\right)}.$
(B.35)

Finally, defining $c$ and $t_{l}$ by

$c={\lambda_{0}}^{4h+1},\quad
t_{l}=(-1)^{h}q^{2h}{\alpha^{(1)}(0)}^{2h+1}\left(\prod_{k=0}^{2h}\kappa(l+k)\right),$
(B.36)

we obtain the following another kind of Painlevé type $q$-O$\Delta$Es in [47,
33]:

$y_{l+2h}y_{l}=c^{(-1)^{l}}t_{l}\dfrac{\prod_{k=0}^{h-1}\left(y_{l+2k+1}+1\right)}{\prod_{k=1}^{2h-1}y_{l+k}},$
(B.37)

where $t_{l}=q^{l}t_{0}$.

######  Remark B.4.

Equation (B.37) has an extended affine Weyl group symmetry of $(A_{1}\times
A_{1})^{(1)}$-type (see [33] for details).

######  Remark B.5.

When $h=1$, Equation (B.37) is equivalent to a $q$-Painlevé II equation of
$A_{6}^{(1)}$-surface type[56, 31, 57, 58] (see [47] for details).

###  B.4. Degenerations of the $(3,L)$\- and $(2,L)$-pair through limit
operations

In this subsection, we demonstrate degenerations from the $(3,L)$-pair, where
$L\geq 1$, to the $(2,L+1)$-pair and from the $(2,L)$-pair, where $L\geq 1$,
to the $(1,L+1)$-pair through limit operations.

####  B.4.1. Degeneration from the $(3,L)$-pair to the $(2,L+1)$-pair

Let $L\geq 1$. Then, the $(3,L)$-pair is given by the system of P$\Delta$Es

$\displaystyle\lambda(l_{0\cdots
3})^{2}\dfrac{U_{\overline{12}}}{U}=\dfrac{\alpha^{(1)}(l_{1})U_{\overline{1}}-\alpha^{(2)}(l_{2})U_{\overline{2}}}{\alpha^{(2)}(l_{2})U_{\overline{1}}-\alpha^{(1)}(l_{1})U_{\overline{2}}},$
(B.38a) $\displaystyle\lambda(l_{0\cdots
3})^{2}\dfrac{U_{\overline{13}}}{U}=\dfrac{\alpha^{(1)}(l_{1})U_{\overline{1}}-\alpha^{(3)}(l_{3})U_{\overline{3}}}{\alpha^{(3)}(l_{3})U_{\overline{1}}-\alpha^{(1)}(l_{1})U_{\overline{3}}},$
(B.38b) $\displaystyle\lambda(l_{0\cdots
3})^{2}\dfrac{U_{\overline{23}}}{U}=\dfrac{\alpha^{(2)}(l_{2})U_{\overline{2}}-\alpha^{(3)}(l_{3})U_{\overline{3}}}{\alpha^{(3)}(l_{3})U_{\overline{2}}-\alpha^{(2)}(l_{2})U_{\overline{3}}},$
(B.38c) $\displaystyle\dfrac{U_{\overline{0\,1}}}{U}+\lambda(l_{0\cdots
3})^{4}\dfrac{U_{\overline{0}}}{U_{\overline{1}}}+\alpha^{(1)}(l_{1})\kappa(l_{0})\lambda(l_{0\cdots
3})=0,$ (B.38d)
$\displaystyle\dfrac{U_{\overline{0\,2}}}{U}+\lambda(l_{0\cdots
3})^{4}\dfrac{U_{\overline{0}}}{U_{\overline{2}}}+\alpha^{(2)}(l_{2})\kappa(l_{0})\lambda(l_{0\cdots
3})=0,$ (B.38e)
$\displaystyle\dfrac{U_{\overline{0\,3}}}{U}+\lambda(l_{0\cdots
3})^{4}\dfrac{U_{\overline{0}}}{U_{\overline{3}}}+\alpha^{(3)}(l_{3})\kappa(l_{0})\lambda(l_{0\cdots
3})=0,$ (B.38f)

where $U=U(l_{1},l_{2},l_{3},l_{0})$, and the periodic condition

$U(l_{1}+1,l_{2}+1,l_{3}+1,l_{0}+L)=U(l_{1},l_{2},l_{3},l_{0}).$ (B.39)

Due to the periodic condition (B.39), the parameters satisfy the following
conditions:

$\dfrac{\alpha^{(1)}(l_{1})}{\alpha^{(1)}(l_{1}+1)}=\dfrac{\alpha^{(2)}(l_{2})}{\alpha^{(2)}(l_{2}+1)}=\dfrac{\alpha^{(3)}(l_{3})}{\alpha^{(3)}(l_{3}+1)}=\dfrac{\kappa(l_{0}+L)}{\kappa(l_{0})}=q,$
(B.40)

where $q\in\mathbb{C}$, which give

$\alpha^{(1)}(l_{1})=q^{-l_{1}}\alpha^{(1)}(0),\quad\alpha^{(2)}(l_{2})=q^{-l_{2}}\alpha^{(2)}(0),\quad\alpha^{(3)}(l_{3})=q^{-l_{3}}\alpha^{(3)}(0).$
(B.41)

If $L$ is even, then the following additional condition is needed:

$\lambda_{0}=1.$ (B.42)

Therefore, regardless of the parity of $L$, the following relation holds:

$\lambda(l+L)=\dfrac{1}{\lambda(l)}.$ (B.43)

In the following, we show that the pair (B.38) and (B.39) is reduced to the
$(2,L+1)$-pair through a limit operation. The condition (B.39) is equivalent
to expressing $U(l_{1},l_{2},l_{3},l_{0})$ as

$U(l_{1},l_{2},l_{3},l_{0})=\omega(l_{1}-l_{3},l_{2}-l_{3},l_{0}-Ll_{3}).$
(B.44)

Set

$\displaystyle\omega(l_{1},l_{2},l_{0})=\xi^{-l_{0}}A(l_{0})\lambda(l_{0}+l_{1}+l_{2})^{\frac{3(2l_{0}-l_{1}-l_{2})}{2L}}u(l_{1},l_{2},l_{0}),$
(B.45a)
$\displaystyle\alpha^{(1)}(0)=\xi^{L}a^{(1)}(0),\quad\alpha^{(2)}(0)=\xi^{L}a^{(2)}(0),\quad\kappa(l_{0})=\xi^{-L-1}K(l_{0}),$
(B.45b)

where $\xi$ is a positive real number and $A(l)\in\mathbb{C}$ is a function
satisfying

$A(l+L+1)=-qK(l)\alpha^{(3)}(0)A(l).$ (B.46)

Then, taking $\xi\to 0$, we obtain a pair of the system of P$\Delta$Es

$\displaystyle\Lambda(l_{0}+l_{1}+l_{2})^{2}\dfrac{u_{\overline{12}}}{u}=\dfrac{a^{(1)}(l_{1})u_{\overline{1}}-a^{(2)}(l_{2})u_{\overline{2}}}{a^{(2)}(l_{2})u_{\overline{1}}-a^{(1)}(l_{1})u_{\overline{2}}},$
(B.47a)
$\displaystyle\dfrac{u_{\overline{0\,1}}}{u}+\Lambda(l_{0}+l_{1}+l_{2})^{4}\dfrac{u_{\overline{0}}}{u_{\overline{1}}}+a^{(1)}(l_{1})k(l_{0})\Lambda(l_{0}+l_{1}+l_{2})=0,$
(B.47b)
$\displaystyle\dfrac{u_{\overline{0\,2}}}{u}+\Lambda(l_{0}+l_{1}+l_{2})^{4}\dfrac{u_{\overline{0}}}{u_{\overline{2}}}+a^{(2)}(l_{2})k(l_{0})\Lambda(l_{0}+l_{1}+l_{2})=0,$
(B.47c)

where

$u=u(l_{1},l_{2},l_{0}),\quad
k(l):=\dfrac{A(l)K(l)}{A(l+1)},\quad\Lambda(l):=\lambda(l)^{\frac{2L-3}{2L}},$
(B.48)

and the periodic condition

$u(l_{1}+1,l_{2}+1,l_{0}+L+1)=u(l_{1},l_{2},l_{0}),$ (B.49)

which is equivalent to the $(2,L+1)$-pair. Indeed, Equation (B.38a) is reduced
to Equation (B.47a), Equation (B.38d) is reduced to Equation (B.47b), Equation
(B.38e) is reduced to Equation (B.47c), and Equation (B.38f) is reduced to the
condition (B.49). Moreover, the reduced equations obtained from Equations
(B.38b) and (B.38c) can be derived from Equation (B.47b) with the condition
(B.49) and Equations (B.47c) with the condition (B.49), respectively. Note
that the parameters in the system (B.47) satisfy the following relations:

$\dfrac{a^{(1)}(l_{1})}{a^{(1)}(l_{1}+1)}=\dfrac{a^{(2)}(l_{2})}{a^{(2)}(l_{2}+1)}=\dfrac{k(l_{0}+L+1)}{k(l_{0})}=q,\quad\Lambda(l+1)=\dfrac{1}{\Lambda(l)}.$
(B.50)

####  B.4.2. Degeneration from the $(2,L)$-pair to the $(1,L+1)$-pair

Let $L\geq 1$. Then, the $(2,L)$-pair is given by the system of P$\Delta$Es

$\displaystyle\lambda(l_{0\cdots
2})^{2}\dfrac{U_{\overline{12}}}{U}=\dfrac{\alpha^{(1)}(l_{1})U_{\overline{1}}-\alpha^{(2)}(l_{2})U_{\overline{2}}}{\alpha^{(2)}(l_{2})U_{\overline{1}}-\alpha^{(1)}(l_{1})U_{\overline{2}}},$
(B.51a) $\displaystyle\dfrac{U_{\overline{0\,1}}}{U}+\lambda(l_{0\cdots
2})^{4}\dfrac{U_{\overline{0}}}{U_{\overline{1}}}+\alpha^{(1)}(l_{1})\kappa(l_{0})\lambda(l_{0\cdots
2})=0,$ (B.51b)
$\displaystyle\dfrac{U_{\overline{0\,2}}}{U}+\lambda(l_{0\cdots
2})^{4}\dfrac{U_{\overline{0}}}{U_{\overline{2}}}+\alpha^{(2)}(l_{2})\kappa(l_{0})\lambda(l_{0\cdots
2})=0,$ (B.51c)

where $U=U(l_{1},l_{2},l_{0})$, and the periodic condition

$U(l_{1}+1,l_{2}+1,l_{0}+L)=U(l_{1},l_{2},l_{0}).$ (B.52)

Due to the periodic condition (B.52), the parameters satisfy the following
conditions:

$\dfrac{\alpha^{(1)}(l_{1})}{\alpha^{(1)}(l_{1}+1)}=\dfrac{\alpha^{(2)}(l_{2})}{\alpha^{(2)}(l_{2}+1)}=\dfrac{\kappa(l_{0}+L)}{\kappa(l_{0})}=q,$
(B.53)

where $q\in\mathbb{C}$, which give

$\alpha^{(1)}(l_{1})=q^{-l_{1}}\alpha^{(1)}(0),\quad\alpha^{(2)}(l_{2})=q^{-l_{2}}\alpha^{(2)}(0).$
(B.54)

If $L$ is odd, then the following additional condition is needed:

$\lambda_{0}=1.$ (B.55)

Therefore, regardless of the parity of $L$, the following relation holds:

$\lambda(l+L)=\lambda(l).$ (B.56)

In the following, we show that the pair (B.51) and (B.52) is reduced to the
$(1,L+1)$-pair through a limit operation. The condition (B.52) is equivalent
to expressing $U(l_{1},l_{2},l_{0})$ as

$U(l_{1},l_{2},l_{0})=\omega(l_{1}-l_{2},l_{0}-Ll_{2}).$ (B.57)

Set

$\displaystyle\omega(l_{1},l_{0})=\xi^{-l_{0}}A(l_{0})\lambda(l_{0}+l_{1})^{\frac{3(2l_{0}-l_{1})}{2L+1}}u(l_{1},l_{0}),$
(B.58a)
$\displaystyle\alpha^{(1)}(0)=\xi^{L}a^{(1)}(0),\quad\kappa(l_{0})=\xi^{-L-1}K(l_{0}),$
(B.58b)

where $\xi$ is a positive real number and $A(l)\in\mathbb{C}$ is a function
satisfying

$A(l+L+1)=-qK(l)\alpha^{(2)}(0)A(l).$ (B.59)

Then, taking $\xi\to 0$, we obtain the following pair:

$\displaystyle\dfrac{u_{\overline{0\,1}}}{u}+\Lambda(l_{0}+l_{1})^{4}\dfrac{u_{\overline{0}}}{u_{\overline{1}}}+a^{(1)}(l_{1})k(l_{0})\Lambda(l_{0}+l_{1})=0,$
(B.60) $\displaystyle u(l_{1}+1,l_{0}+L+1)=u(l_{1},l_{0}),$ (B.61)

where

$u=u(l_{1},l_{0}),\quad
k(l):=\dfrac{A(l)K(l)}{A(l+1)},\quad\Lambda(l):=\lambda(l)^{\frac{2(L-1)}{2L+1}},$
(B.62)

which is equivalent to the $(1,L+1)$-pair. Indeed, Equations (B.51b) and
(B.51c) are reduced to Equations (B.60) and (B.61), respectively. Moreover,
the reduced equation obtained from Equation (B.51a) can be derived from
Equation (B.60) with the condition (B.61). Note that the parameters in
Equation (B.60) satisfy the following relations:

$\dfrac{a^{(1)}(l_{1})}{a^{(1)}(l_{1}+1)}=\dfrac{k(l_{0}+L+1)}{k(l_{0})}=q,\quad\Lambda(l+1)=\dfrac{1}{\Lambda(l)}.$
(B.63)

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