#  A $3\times 3$ Lax form for the $q$-Painlevé equation of type $E_{6}$

Kanam Park
National Institude of Technology, Toba College,
1-1, Ikegami, Toba, Mie, 517-0012, Japan
Email<EMAIL_ADDRESS>
###### Abstract

For the $q$-Painlevé equation with affine Weyl group symmetry of type
$E_{6}^{(1)}$, a $2\times 2$ matrix Lax form and a second order scalar lax
form were known. We give a new $3\times 3$ matrix Lax form and a third order
scalar equation related to it. Continuous limit is also discussed.

Key Words and Phrases : Lax formalism, $q$-Painlevé equation,

2020 Mathematics Subject Classification Numbers: 14H70, 34M56, 39A13.

##  1 Introduction

The $q$-Painlevé equation with affine Weyl group symmetry of type
$E_{6}^{(1)}$ was first discovered in [8]. The well-known form of it is as
follows

$\begin{array}[]{c}\cfrac{(\overline{f}g-1)(fg-1)}{f\overline{f}}=\cfrac{(g-1/a_{5})(g-1/a_{6})(g-1/a_{7})(g-1/a_{8})}{(g-a_{3})(g-a_{4})},\vspace{12pt}\\\
\cfrac{(\overline{f}g-1)(\overline{f}\overline{g}-1)}{g\overline{g}}=\cfrac{(\overline{f}-a_{5})(f-a_{6})(f-a_{7})(f-a_{8})}{(\overline{f}-a_{1}/q)(\overline{f}-a_{2}/q)},\vspace{12pt}\\\
q=\cfrac{a_{1}a_{2}}{a_{3}a_{4}a_{5}a_{6}a_{7}a_{8}},\end{array}$ ( 1.1)

where $f$ and $g$ are dependent variables, $a_{1}$, $a_{2}$, $\cdots$, $a_{8}$
are parameters, $q$ is a constant and the overline symbol $"\ \bar{}\ "$
denotes the discrete time evolution.

In previous works, the following Lax forms for the $q$-Painlevé equation for
type $E_{6}^{(1)}$ has been obtained. In [11], a $2\times 2$ matrix Lax pair
was derived as a reduction of the $q$-Garnier system. In [15], a second order
scalar Lax form was obtained as a reduction from the $q$-Painlevé equation of
type $E_{8}$. The relation between these Lax forms was given in [14].

In this article, we give a new Lax form with $3\times 3$ matrix Lax pair. We
derive such a Lax form as a special case of the system investigated in our
previous work [7]. As a result, we derive an equation which is equivalent to
the $q$-Painlevé equation of type $E_{6}^{(1)}$ [3], [9].

In the previous work [7], we defined a nonlinear $q$-difference system as a
connection preserving deformation of the following linear equation

$\begin{array}[]{l}\Psi(qz)=\Psi(z)A(z),\quad
A(z)=DX_{1}^{\varepsilon_{1}}(z)X_{2}^{\varepsilon_{2}}(z)\cdots
X_{M}^{\varepsilon_{M}}(z),\end{array}$ ( 1.2)
$\begin{array}[]{c}D=\mathrm{diag}[d_{1},d_{2},\cdots,d_{N}],\\\
X_{i}(z)=\mathrm{diag}[u_{1,i},u_{2,i},\cdots
u_{N,i}]+\Lambda,\quad\Lambda=\begin{bmatrix}0&1&&{\bf O}\\\
&\ddots&\ddots&\\\ &&&1\\\ z&&&0\end{bmatrix},\end{array}$ ( 1.3)

where the exponents are $\varepsilon_{i}=\pm 1$ $(1\leq i\leq M)$, $u_{j,i}$
($1\leq j\leq N$) are dependent variables and $c_{i},d_{j}$ are parameters
which satisfy

$\prod_{j=1}^{N}u_{j,i}=c_{i}.$ ( 1.4)

Since one can exchange the order of matrices $X_{i}^{\pm 1}$ by suitable
rational transformations of variables $u_{j,i}$, the equation (1.2)
essentially depends on $M_{+}$, $M_{-}$, where
$M_{\pm}=\\#\\{\varepsilon_{i}|\varepsilon_{i}=\pm 1\\}$.

The contents of this paper is as follows. In section 2, we set up a linear
$q$-difference equation (2.1), which is a case of $(M_{+},M_{-},N)=(3,0,3)$
for the equation (1.2). And we discuss about its two deformations. One
deformation gives rise to a well known form of the $q$-Painlevé equation of
type $E_{6}^{(1)}$, and the other deformation gives an equation for a non-
standard direction. In section 3, we derive a scalar equation from the
$3\times 3$ matrix equation (2.1) and consider its characteristic properties.
In section 4, we study continuous limit of our constructions and its relation
to the Boalch’s Lax pair [1].

######  Remark 1.1.

The equation (1.2) in case $(M_{+},M_{-},N)=(2n+2,0,2)$ is known that it is
equivalent to the linear $q$-difference equation related to the $2n$
dimensional $q$-Garnier system [10]. And in case of
$(M_{+},M_{-},N)=(2,0,2n+2)$, the equation (1.2) is also equivalent to the
linear $q$-difference equation related to the $2n$ dimensional system
$q$-$P_{(n+1,n+1)}$ [12]. In both cases, when $n=1$, they give rise to the
$q$-$P_{\rm VI}$ equation [4]. We note that the equation (2.1) is coincide to
the $q$-difference linear equation of the Lax form in [13] in case of ($m$,
$n$)=($3$, $1$).

##  2 A $3\times 3$ matrix Lax form

In this section, we consider two types of deformations for the linear
$q$-difference equation (1.2) in a case $(M_{+},M_{-},N)=(3,0,3)$.

We consider the connection preserving deformation for the following
$q$-difference equation for an unknown function
$\Psi(z)=[\Psi_{1}(z),\Psi_{2}(z),\Psi_{3}(z)]$,

$\begin{array}[]{c}\Psi(qz)=\Psi(z)A(z)=\Psi(z)A(z,t),\vspace{6pt}\\\
A(z)=DX_{1}(z)X_{2}(z)X_{3}(z)=\begin{bmatrix}b_{1}d_{1}&*&*\\\
0&b_{2}d_{2}&*\\\ 0&0&b_{3}d_{3}\end{bmatrix}+\begin{bmatrix}d_{1}&0&0\\\
*&d_{2}&0\\\ *&*&d_{3}\end{bmatrix}z,\end{array}$ ( 2.1)

where the matrices $D$ and $X_{i}(z)$ $(1\leq i\leq 3)$ stand for

$\begin{array}[]{c}D=\mathrm{diag}[d_{1},d_{2},d_{3}],\vspace{6pt}\\\
X_{i}(z)=\mathrm{diag}[u_{1,i},u_{2,i},u_{3,i}]+\Lambda,\vspace{6pt}\\\
\Lambda=\begin{bmatrix}0&1&0\\\ 0&0&1\\\ z&0&0\end{bmatrix},\end{array}$ (
2.2)

and $u_{j,i}$ ($1\leq i$, $j\leq 3$) are dependent variables and $c_{i},d_{j}$
are parameters which satisfy

$\begin{array}[]{c}\prod_{i=1}^{3}u_{j,i}=b_{j}\quad(1\leq j\leq
3),\quad\prod_{j=1}^{3}u_{j,i}=c_{i}\quad(1\leq i\leq 3),\vspace{6pt}\\\
b_{1}b_{2}b_{3}=c_{1}c_{2}c_{3}.\end{array}$ ( 2.3)

The first equation in (2.3) is equivalent to that the characteristic exponents
at $z=0$ of the equation (2.1) are $b_{j}d_{j}$. The second equation in (2.3)
is equivalent to the following condition

$|A(z)|=d_{1}d_{2}d_{3}(z+c_{1})(z+c_{2})(z+c_{3}).$ ( 2.4)

Through a gauge transformation by a $3\times 3$ diagonal matrix we can take
two components in (2.1) as 1. In the following, we use this kind of gauge
fixings in case by case. By the condition (2.4), two of the remaining four
components are determined by other components and parameters
$b_{j},c_{i},d_{j}$.

In this article, we will consider two deformations $T_{1}$ and $T_{2}$ for the
equation (2.1) which act on parameters $b_{j},\ c_{i},\ d_{j}$ as:

$\begin{array}[]{l}T_{1}:({b_{1}},{b_{2}},{b_{3}},{c_{1}},{c_{2}},{c_{3}},{d_{1}},{d_{2}},{d_{3}})\to(b_{1},qb_{2},\cfrac{b_{3}}{q},c_{1},c_{2},c_{3},qd_{1},d_{2},qd_{3}),\vspace{6pt}\\\
T_{2}:({b_{1}},{b_{2}},{b_{3}},{c_{1}},{c_{2}},{c_{3}},{d_{1}},{d_{2}},{d_{3}})\to(\cfrac{b_{1}d_{1}}{d_{2}},\cfrac{b_{2}d_{2}}{d_{3}},b_{3},c_{1},c_{2},c_{3}q,d_{2},d_{3},\cfrac{d_{1}}{q}).\end{array}$
( 2.5)

###  2.1  The deformation $T_{1}$

We will show that the deformation $T_{1}$ gives rise to the standard form of
the $q$-$E_{6}^{(1)}$. We consider the following $q$-difference linear
equation

$\begin{array}[]{c}\Psi(qz)=\Psi(z)A(z),\quad
A(z)=\begin{bmatrix}b_{1}d_{1}&1&v_{1}\\\ 0&b_{2}d_{2}&1\\\
0&0&b_{3}d_{3}\end{bmatrix}+\begin{bmatrix}d_{1}&0&0\\\ v_{2}&d_{2}&0\\\
v_{3}&v_{4}&d_{3}\end{bmatrix}z,\end{array}$ ( 2.6)
$|A(z)|=d_{1}d_{2}d_{3}(z+c_{1})(z+c_{2})(z+c_{3}).$ ( 2.7)

We take a deformation equation for the equation (2.6), (2.7) as follows:

$\begin{array}[]{c}T_{1}\Psi(z)=\Psi(z)B(z),\quad
B(z)=\begin{bmatrix}0&0&w_{1}\\\ 0&0&w_{2}\\\
0&0&w_{3}\end{bmatrix}+\begin{bmatrix}w_{4}&0&0\\\ w_{5}&0&0\\\
w_{6}&1&w_{7}\end{bmatrix}z,\vspace{6pt}\\\
|B(z)|=(w_{1}w_{5}-w_{2}w_{4})z^{2},\end{array}$ ( 2.8)
$T_{1}:({b_{1}},{b_{2}},{b_{3}},{c_{1}},{c_{2}},{c_{3}},{d_{1}},{d_{2}},{d_{3}};v_{1},v_{4})\to(b_{1},qb_{2},\cfrac{b_{3}}{q},c_{1},c_{2},c_{3},qd_{1},d_{2},qd_{3};\overline{v_{1}},\overline{v_{4}}),$
( 2.9)

######  Theorem 2.1.

Through a compatibility condition of the equations (2.6), (2.9), (2.8)

$B(z)\overline{A(z)}=A(z)B(qz),$ ( 2.10)

we obtain the following equations

$\begin{array}[]{c}\cfrac{(fg-1)(\overline{f}g-1)}{f\overline{f}}=\cfrac{(g-1/c_{1})(g-1/c_{2})(g-1/c_{3})(g-d_{2}/b_{3}d_{3})}{(g-1/b_{2})(g-d_{2}/b_{1}d_{1})},\vspace{6pt}\\\
\cfrac{(\overline{f}g-1)(\overline{f}\overline{g}-1)}{g\overline{g}}=\cfrac{(\overline{f}-c_{1})(\overline{f}-c_{2})(\overline{f}-c_{3})(\overline{f}-b_{3}d_{3}/d_{2})}{(\overline{f}-b_{3}d_{3}/d_{1})(\overline{f}-b_{3}/q)},\end{array}$
( 2.11)

where,

$\begin{array}[]{c}f=\cfrac{b_{3}d_{3}}{d_{3}-v_{1}v_{4}},\quad
g=\cfrac{d_{2}v_{1}}{b_{2}d_{2}v_{1}-1},\end{array}$ ( 2.12)

and $\overline{*}$ stands for $T_{1}(*)$.

The equation (2.11) is the well known form of the $q$-Painlevé equation of
type $E_{6}$ [3], [9] (See also [5]).

###### Proof.

The result is obtained by a direct computation of the compatibility condition
of (2.10). Since the computation is rather heavy, we will give a comment how
to do it efficiently. Though the $2$ variables $v_{2}$, $v_{3}$ can be
represented by the rational functions of the remaining two variables $v_{1}$,
$v_{4}$ by the relation (2.7), it is more efficient to do this elimination
after the calculation of the compatibility condition (2.10) in $4$ variables,
and then reduce it to 2 variables. In this way, we get the following time
evolutions for $v_{1}$, $v_{4}$ as rational functions of $v_{1}$, $v_{4}$:

$\begin{array}[]{c}\overline{v_{1}}=\cfrac{C_{1}(v_{1},v_{4})C_{2}(v_{1},v_{4})}{D_{1}(v_{1},v_{4})D_{2}(v_{1},v_{4})},\quad\overline{v_{4}}=\cfrac{C_{3}(v_{1},v_{4})}{D_{3}(v_{1},v_{4})D_{4}(v_{1},v_{4})},\end{array}$
( 2.13)

where

$\begin{array}[]{rl}C_{3}(v_{1},v_{4}),D_{2}(v_{1},v_{4})&:\text{polynomials
in }v_{1},v_{4}\text{ of degree }(4,\ 3),\\\ C_{2}(v_{1},v_{4})&:\text{a
polynomial in }v_{1},v_{4}\text{ of degree }(3,\ 1),\\\
C_{1}(v_{1},v_{4}),D_{3}(v_{1},v_{4}),D_{4}(v_{1},v_{4})&:\text{polynomials in
}v_{1},v_{4}\text{ of degree }(2,\ 1),\\\ D_{1}(v_{1},v_{4})&:\text{a
polynomial in }v_{1},v_{4}\text{ of degree }(1,\ 0).\end{array}$ ( 2.14)

The remaining task is to rewrite the equation (2.13) to (2.11). A useful way
to solve it is to look at the singularities of the equations [2]. Namely, we
investigate the points at which the right hand side of the equations (2.13)
are indeterminate. We focus on the equation $D_{3}(v_{1},\ v_{4})$ of them

$D_{3}(v_{1},v_{4})=d_{3}\left(-b_{2}d_{2}v_{1}+b_{3}d_{2}v_{1}+1\right)+v_{1}v_{4}\left(b_{2}d_{2}v_{1}-1\right).$
( 2.15)

Investigating common zero points of the equation (2.15) and the other
polynomials $C_{k}(v_{1},\ v_{4})$, $D_{l}(v_{1},\ v_{4})$, we find 4
indeterminate points as follows:

$\begin{array}[]{c}(v_{1},v_{4})=\big{(}-\cfrac{1}{(u^{-1}-b_{2})d_{2}},-\cfrac{(u^{-1}-b_{3})(u^{-1}-b_{2})d_{3}d_{2}}{u^{-1}}\big{)},\vspace{6pt}\\\
(u=c_{1}^{-1},c_{2}^{-1},c_{3}^{-1},d_{2}/b_{3}d_{3}).\end{array}$ ( 2.16)

In view of the form of the points in (2.16), we define the variables $f$, $g$
as follows

$(v_{1},v_{1}v_{4})=\big{(}-\cfrac{1}{(g^{-1}-b_{2})d_{2}},\cfrac{(f-b_{3})d_{3}}{f}\big{)}.$
( 2.17)

By the transformation (2.17), the equation $D_{3}(v_{1},v_{4})=0$ is
transformed to an equation $fg=1$. Through the correspondence (2.17) and the
time evolution equations for the variables $v_{1}$, $v_{4}$ (2.13), we have
time evolution equations for $f$, $g$ (2.11). ∎

###  2.2 The deformation $T_{2}$

In this subsection, we take a deformation equation which is one of that
considered in the previous work [7]. It corresponded to the permutations of
the matrices $X_{i}(z)^{\pm 1}$. We consider the following Lax pair

$\begin{array}[]{c}\Psi(qz)=\Psi(z)A(z),\quad
A(z)=DX_{1}(z)X_{2}(z)X_{3}(z)=\begin{bmatrix}b_{1}d_{1}&1&*\\\
0&b_{2}d_{2}&1\\\ 0&0&b_{3}d_{3}\end{bmatrix}+\begin{bmatrix}d_{1}&0&0\\\
*&d_{2}&0\\\ *&*&d_{3}\end{bmatrix}z,\end{array}$ ( 2.18)
$\begin{array}[]{c}T_{2}{\Psi}(z)=\Psi(z)B(z),\quad
B(z)=X_{3}(z/q)^{-1},\vspace{6pt}\\\
|B(z)|=\cfrac{q}{z+qc_{3}},\vspace{6pt}\\\
T_{2}:({b_{1}},{b_{2}},{b_{3}},{c_{1}},{c_{2}},{c_{3}},{d_{1}},{d_{2}},{d_{3}};x,y)\to(\cfrac{b_{1}d_{1}}{d_{2}},\cfrac{b_{2}d_{2}}{d_{3}},b_{3},c_{1},c_{2},c_{3}q,d_{2},d_{3},\cfrac{d_{1}}{q};\overline{x},\overline{y}),\end{array}$
( 2.19)

where we define variables $x$, $y$ and gauge freedom $w_{1}$, $w_{2}$ with the
variables $u_{j,i}$ in (2.18) as follows:

$x=\frac{u_{1,1}u_{1,2}u_{2,1}u_{2,2}}{u_{1,1}+u_{2,2}},\quad
y=\frac{1}{u_{1,1}u_{1,2}\left(u_{2,1}+u_{3,2}\right)},\quad
w_{1}=u_{1,1},\quad w_{2}=u_{1,3}.$ ( 2.20)

######  Theorem 2.2.

Through a compatibility condition of the equations (2.18), (2.19), (2.20),

$B(z)\overline{A(z)}=A(z)B(qz),$ ( 2.21)

we obtain the following equations

$\begin{array}[]{c}\cfrac{\overline{x}+c_{1}}{\overline{x}+c_{2}}=\cfrac{E_{1}E_{2}}{E_{3}E_{4}},\quad\cfrac{c_{1}\overline{y}+1}{c_{2}\overline{y}+1}=\cfrac{F_{1}F_{2}}{F_{3}F_{4}},\end{array}$
( 2.22)

where,

$\begin{array}[]{l}E_{1}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{1}d_{3}y\left(c_{2}+x\right)\right)+c_{1}d_{2}d_{3}x\left(c_{2}y+1\right),\\\
E_{2}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{3}d_{3}qy\left(c_{1}+x\right)\right)+c_{3}d_{2}d_{3}qx\left(c_{1}y+1\right),\\\
E_{3}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{2}d_{3}y\left(c_{1}+x\right)\right)+c_{2}d_{2}d_{3}x\left(c_{1}y+1\right),\\\
E_{4}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{3}d_{3}qy\left(c_{2}+x\right)\right)+c_{3}d_{2}d_{3}qx\left(c_{2}y+1\right),\\\
F_{1}=E_{1},F_{2}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{2}d_{3}y\left(c_{1}+x\right)\right)+c_{3}d_{2}d_{3}qx\left(c_{1}y+1\right),\\\
F_{3}=E_{3},F_{4}=b_{1}d_{1}\left(b_{2}d_{2}(1-xy)+c_{1}d_{3}y\left(c_{2}+x\right)\right)+c_{3}d_{2}d_{3}qx\left(c_{2}y+1\right),\end{array}$
( 2.23)

and $\overline{*}$ stands for $T_{2}(*)$.

###### Proof.

Solving a compatibility condition (2.21) with (2.20) for the variables
$\overline{x}$, $\overline{y}$, we obtain the following equations

$\overline{x}=\cfrac{(xy-1)G_{1}(x,y)}{H_{1}(x,y)},\quad\overline{y}=\cfrac{xG_{2}(x,y)}{H_{2}(x,y)},$
( 2.24)

where $G_{k}(x,y)$, $H_{k}(x,y)$ $(k=1,2)$ are polynomials in variables $x$,
$y$. The polynomial $G_{1}(x,y)$ is of degree $(1,1)$, $G_{2}(x,y)$ is of
degree $(1,2)$ and $H_{k}(x,y)$ are of degree $(2,2)$. Using a method [5] for
finding point configuration, a configuration of points for the equation (2.22)
is as follows:

$\begin{array}[]{rl}(x,y)=&(-c_{1},-\cfrac{1}{c_{1}}),(-c_{2},-\cfrac{1}{c_{2}}),(-\cfrac{b_{1}d_{1}}{d_{2}},-\cfrac{d_{2}}{b_{1}d_{1}}),(-\cfrac{b_{1}b_{2}}{c_{3}},-\cfrac{c_{3}}{b_{1}b_{2}}),\\\
&(-b_{2},0),(-\cfrac{b_{1}b_{2}d_{1}}{c_{3}d_{3}q},0),\\\
&(0,-\cfrac{1}{b_{1}}),(0,-\cfrac{b_{2}d_{2}}{c_{1}c_{2}d_{3}}).\end{array}$ (
2.25)

Calculating $\cfrac{\overline{x}+c_{1}}{\overline{x}+c_{2}}$ and
$\cfrac{c_{1}\overline{y}+1}{c_{2}\overline{y}+1}$ respectively, we have
equations (2.22), (2.23). ∎

######  Remark 2.1.

Here, the time evolution equations $\overline{u_{j,i}}$ are derived by solving
the compatibility condition (2.21). For another derivation using the
transformations which correspond to permutations of the matrices
$X_{i}^{\varepsilon_{i}}(z)$, see [7] Proposition 2.1.

Before ending this subsection, we show a relation between pairs of the
variables $(f,g)$ in (2.11) and $(x,y)$ in (2.22).

######  Proposition 2.1.

The Lax equations (2.6) with (2.12) and (2.18) with (2.20) are equivalent with
each other if the variables $f$, $g$ and $x$, $y$ are related as:

$\begin{array}[]{l}f=\cfrac{c_{1}c_{2}c_{3}\left(b_{2}+x\right)(xy-1)}{b_{1}b_{2}y\left(c_{1}+x\right)\left(c_{2}+x\right)+c_{3}x\left(b_{2}(1-xy)+c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)},\vspace{6pt}\\\
g=-\cfrac{y\left(b_{1}b_{2}+c_{3}x\right)}{c_{3}\left(b_{2}(1-xy)+x\right)+b_{1}b_{2}xy}\end{array}$
( 2.26)

###### Proof.

Comparing the coefficient matrix $A(z)$ of the equation (2.6), (2.12) to the
coefficient matrix $A(z)$ of the equation (2.18), (2.20), we can solve for the
variables $x$, $y$ and gauge freedom $w_{1}$, $w_{2}$ in terms of variables
$f$, $g$ and parameters $b_{j}$, $c_{i}$, $d_{j}$ ($1\leq i,j\leq 3$). Then we
have the desired relation between pairs of variables $(f,g)$ and $(x,y)$
(2.26). ∎

##  3 A scalar equation related to the $q$-$E_{6}^{(1)}$

In this section, we derive a scalar $q$-difference equation from the matrix
$q$-difference equation (2.18), (2.20) for an unknown function
$\Psi(z)=[\Psi_{1}(z),\Psi_{2}(z),\Psi_{3}(z)]$ and its geometric properties.
Eliminating functions $\Psi_{2}(z)$, $\Psi_{3}(z)$ in the equation (2.18),
(2.20), we obtain the following third linear $q$-difference equation for
$\Psi_{1}(z)=:\Phi(z)$

$L(z):=P_{3}(z)\Phi(q^{3}z)+P_{2}(z)\Phi(q^{2}z)+P_{1}(z)\Phi(qz)+P_{0}(z)\Phi(z)=0,$
( 3.1)

where

$\begin{array}[]{l}P_{3}(z)=p_{31}(z-u),\\\
P_{2}(z)=p_{22}z^{2}+p_{21}z+p_{20},\\\
P_{1}(z)=p_{13}z^{3}+p_{12}z^{2}+p_{11}z+p_{10},\\\
P_{0}(z)=-P_{3}(qz)d_{1}d_{2}d_{3}(z+c_{1})(z+c_{2})(z+c_{3}).\end{array}$ (
3.2)

Here, the coefficients $p_{k,l}$ $(1\leq k\leq 3,0\leq l\leq 3)$ in the
polynomials $P_{k}(z)$ (3.2) depend on parameters $b_{j}$, $c_{i}$, $d_{j}$,
and a variable $u$ defined as the zero of $P_{3}(z)$. The variable $u$ is
expressed in terms of $x$, $y$ as follows

$\begin{array}[]{l}u=\cfrac{y\left(b_{1}\left(c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)+c_{1}c_{2}(1-xy)\right)}{b_{2}\left(b_{1}y+1\right)(xy-1)}\vspace{6pt}\\\
\cfrac{\left(b_{2}d_{3}y\left(b_{1}\left(c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)+c_{1}c_{2}(1-xy)\right)+b_{2}^{2}d_{2}\left(b_{1}y+1\right)(1-xy)+c_{3}d_{3}x\left(c_{1}y+1\right)\left(c_{2}y+1\right)\right)}{\left(d_{2}\left(b_{2}\left(b_{1}y+1\right)(xy-1)-x\left(c_{1}y+1\right)\left(c_{2}y+1\right)\right)-d_{3}y\left(b_{1}\left(c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)+c_{1}c_{2}(1-xy)\right)\right)}.\end{array}$
( 3.3)

Explicit forms of the polynomials $P_{j}(z)$ ($0\leq j\leq 3$) (3.2) are given
in Appendix.

Then we have

######  Lemma 3.1.

The equation $L(z)=0$ (3.1) has the following properties:

  1. (i)

It is a linear four term equation between $\Phi(q^{j}z)$ ($0\leq j\leq 3$) and
its coefficients $P_{j}(z)$ are polynomials for $z$ of degree $4-j$,

  2. (ii)

A polynomial $P_{0}(z)$ has four zero points at $z=-c_{i}$ ($1\leq i\leq 3$),
$u/q$,

  3. (iii)

The exponents of solutions $\Phi(z)$ are $b_{1}d_{1}$, $qb_{2}d_{2}$,
$qb_{3}d_{3}$ (at $z=0$) and $d_{1}$, $d_{2}$, $d_{3}$ (at $z=\infty$),

  4. (iv)

A point $z=u$ such that $P_{3}(z)=0$ is an apparent singularity, namely we
have

$v:=\cfrac{P_{0}(u)}{P_{1}(u/q)}=\cfrac{P_{1}(u)}{P_{2}(u/q)}=\cfrac{P_{2}(u)}{P_{3}(u/q)}.$
( 3.4)

Conversely, the equation $L(z)=0$ (3.1) is uniquely characterized by these
properties (i)-(iv) up to normalization.

###### Proof.

The properties (i)-(iv) follows by computation through eliminating
$\Psi_{2}(z)$, $\Psi_{3}(z)$ in (2.18), (2.20). The converse can be confirmed
that coefficients $P_{j}(z)$ are defined uniquely by (i)-(iv) up to a
normalization. To see this, we consider the following equation which satisfies
the properties (i), (ii):

$L^{\prime}(z)=P_{3}^{\prime}(z)\Phi(q^{3}z)+P_{2}^{\prime}(z)\Phi(q^{2}z)+P_{1}^{\prime}(z)\Phi(qz)+P_{0}^{\prime}(z)\Phi(z)=0,$
( 3.5) $\begin{array}[]{l}P_{3}^{\prime}(z)=p_{31}^{\prime}(z-u),\\\
P_{2}^{\prime}(z)=p_{22}^{\prime}z^{2}+p_{21}^{\prime}z+p_{20}^{\prime},\\\
P_{1}^{\prime}(z)=p_{13}^{\prime}z^{3}+p_{12}^{\prime}z^{2}+p_{11}^{\prime}z+p_{10}^{\prime},\\\
P_{0}^{\prime}(z)=p_{04}^{\prime}(z-u/q)(z+c_{1})(z+c_{2})(z+c_{3}).\end{array}$
( 3.6)

From the property (iii), the condition of the exponents of solutions $\Phi(z)$
at $z=0$ determines the coefficients $p_{31}^{\prime}$, $p_{20}^{\prime}$,
$p_{10}^{\prime}$ and the condition of the exponents of solutions $\Phi(z)$ at
$z=\infty$ determines the coefficients $p_{22}^{\prime}$, $p_{13}^{\prime}$.
The remaining coefficients except for $p_{04}^{\prime}$ are determined by the
property (iv). If we put the normalization factor $p_{04}^{\prime}$ as
$q^{5}uvc_{1}c_{2}d_{1}d_{2}d_{3}$, the function $L^{\prime}(z)$ (3.5) equals
to the function $L(z)$ (3.1). ∎

In the following, viewing $\Phi(q^{i}z)$ $(0\leq i\leq 3)$ as parameters, we
regard the scalar $q$-difference equation $L(z)=0$ (3.1) as an algebraic curve
in variables $(u,v)$ $\in$ $\mathbb{P}^{1}\times\mathbb{P}^{1}$. We represent
as the curve as $P(u,v)=0$. The features of the curve are the following.

######  Lemma 3.2.

The algebraic curve $P(u,v)=0$ has the following properties:

  1. (i)

The polynomial $P(u,v)$ has the following form

$P(u,v)=\displaystyle\sum_{0\leq j\leq 3\atop 0\leq i+j\leq
4}c_{i,j}u^{i}v^{j},\quad c_{0,0}:=c_{0}z^{2}\Phi(qz).$ ( 3.7)

The coefficients $c_{i,j}$ depend on $b_{i}$, $c_{i}$, $d_{i}$, $q$, $z$,
$\Phi(q^{i}z)$.

  2. (ii)

It passes the following $8$ points

$\begin{array}[]{c}(u,v)=(0,qb_{1}d_{1}),\quad(0,q^{2}b_{2}d_{2}),\quad(0,q^{2}b_{3}d_{3}),\quad(qz,\infty),\vspace{6pt}\\\
(z,0),\quad(-c_{1},0),\quad(-c_{2},0),\quad(-c_{3},0),\end{array}$ ( 3.8)

and 3 points in the coordinate $(r,s)=(u,v/u)$

$(r,s)=(\infty,q^{2}d_{1}),\quad(\infty,q^{2}d_{2}),\quad(\infty,q^{2}d_{3}).$
( 3.9)

  3. (iii)

At $u=z$ the equation $P(u,v)=0$ has the following property

$\begin{array}[]{c}\displaystyle\big{(}\sum_{0\leq i\leq
3-j}c_{i,j+1}z^{i}\big{)}\Big{|}_{z\to
qz,\Phi(q^{k}z)\to\Phi(q^{k+1}z)}=\sum_{0\leq i\leq
4-j}c_{i,j}(qz)^{i}\quad(j=0,1,2).\end{array}$ ( 3.10)

Conversely, the equation $P(u,v)=0$ is uniquely characterized by these
properties (i)-(iii) up to normalization factor $c_{0}$.

###### Proof.

The properties (i)-(iii) follow for the polynomial $P(u,v)$. Conversely, we
consider a polynomial $P^{\prime}(u,v):=\displaystyle\sum_{0\leq j\leq 3\atop
0\leq i+j\leq 4}c_{i,j}^{\prime}u^{i}v^{j}$,
$c_{0,0}^{\prime}(z)=c_{0}^{\prime}\Phi(qz)$. The polynomial $P^{\prime}(u,v)$
has $14$ coefficients. From the property (ii), $10$ coefficients are described
in terms of parameters $b_{j}$, $c_{j}$, $d_{j}$ and the coefficient
$c_{0}^{\prime}$.

$\begin{array}[]{l}P^{\prime}(u,v)=-\cfrac{c_{0}^{\prime}z(qz-u)\Phi(qz)}{b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{6}}v^{3}\vspace{6pt}\\\
+\left(uc_{1,2}^{\prime}+\cfrac{c_{0}^{\prime}z\Phi(qz)\left(b_{2}d_{2}qz+b_{3}d_{3}qz+b_{1}d_{1}z-d_{1}u^{2}-d_{2}u^{2}-d_{3}u^{2}\right)}{b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{4}}\right)v^{2}\vspace{6pt}\\\
+\left(u^{2}c_{2,1}^{\prime}+uc_{1,1}^{\prime}-\cfrac{c_{0}^{\prime}z\Phi(qz)\left(b_{2}b_{3}d_{2}d_{3}qz+b_{1}b_{2}d_{1}d_{2}z+b_{1}b_{3}d_{1}d_{3}z-d_{1}d_{2}u^{3}-d_{1}d_{3}u^{3}-d_{2}d_{3}u^{3}\right)}{b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{2}}\right)v\vspace{6pt}\\\
+\cfrac{c_{0}^{\prime}z\left(c_{1}+u\right)\left(c_{2}+u\right)(z-u)\left(c_{3}+u\right)\Phi(qz)}{b_{1}b_{2}b_{3}}.\end{array}$
( 3.11)

The remaining $3$ parameters $c_{1,1}^{\prime}$, $c_{1,2}^{\prime}$,
$c_{2,1}^{\prime}$ are determined in terms of parameters $b_{i}$, $c_{i}$,
$d_{i}$, $q$, $z$, $\Phi(q^{i}z)$, $c_{0}^{\prime}$ by the property (iii).
Namely the property (iii) gives $3$ linear inhomogeneous equations among
$c_{1,1}^{\prime}$, $c_{1,2}^{\prime}$, $c_{2,1}^{\prime}$,
$c_{1,1}^{\prime}\big{|}_{z\to qz}$, $c_{1,2}^{\prime}\big{|}_{z\to qz}$,
$c_{2,1}^{\prime}\big{|}_{z\to qz}$. Though these relations are apparently
$q$-difference equations, we can solve them algebraically. For example, in the
equation (3.10), we solve $c_{12}^{\prime}$ when $j=2$. Then when $j=0$, we
solve $\left.c_{21}^{\prime}\right|_{z\to qz,\Phi(q^{k}z)\to\Phi(q^{k+1}z)}$.
And finally solving $c_{11}^{\prime}$ when $j=1$, they algebraically can be
solved. ∎

Explicit forms of the coefficients $c_{i,j}(z)$ of the polynomial $P(u,v)$ are
in Appendix.

###  3.1 Relations among pairs of variables $(f,g)$, $(x,y)$ and $(u,v)$

######  Proposition 3.1.

Under the relations among variables ($f$, $g$), ($x$, $y$) and ($u$, $v$):

$\begin{array}[]{l}(f,g)=\vspace{6pt}\\\
\left(\cfrac{c_{1}c_{2}c_{3}\left(b_{2}+x\right)(xy-1)}{b_{1}b_{2}y\left(c_{1}+x\right)\left(c_{2}+x\right)+c_{3}x\left(b_{2}(1-xy)+c_{2}xy+c_{1}y\left(c_{2}+x\right)+x\right)},-\cfrac{y\left(b_{1}b_{2}+c_{3}x\right)}{c_{3}\left(b_{2}(1-xy)+x\right)+b_{1}b_{2}xy}\right),\end{array}$
( 3.12) $\begin{array}[]{l}(x,y)=\vspace{6pt}\\\
\Big{(}\Big{(}b_{2}v\left(-b_{2}d_{2}q^{2}-d_{3}q^{2}u+v\right)\\\
\quad\quad\left(c_{1}c_{2}d_{3}q^{2}\left(c_{3}\left(b_{2}d_{2}q^{2}-v\right)+b_{2}d_{2}q^{2}u\right)+b_{1}b_{2}\left(d_{2}q^{2}\left(-b_{2}v+d_{3}q^{2}u\left(c_{3}+u\right)-uv\right)+v\left(v-d_{3}q^{2}u\right)\right)\right)\Big{)}\\\
\cdot\Big{(}q^{2}(b_{2}^{2}d_{2}(d_{2}q^{2}\left(d_{3}q^{2}\left(c_{1}+u\right)\left(c_{3}+u\right)-uv\right)\left(d_{3}q^{2}\left(c_{2}+u\right)\left(c_{3}+u\right)-uv\right)\\\
\quad-v\left(-d_{3}q^{2}v\left(b_{1}\left(2c_{3}+u\right)+u\left(c_{3}+2u\right)\right)+d_{3}^{2}q^{4}u\left(b_{1}+u\right)\left(c_{3}+u\right)+uv^{2}\right))\\\
\quad+b_{2}^{3}d_{2}^{2}\left(-q^{2}\right)v\left(d_{3}q^{2}\left(b_{1}+u\right)\left(c_{3}+u\right)-uv\right)-b_{2}c_{3}d_{3}v(b_{1}v\left(v-d_{3}q^{2}u\right)\\\
\quad+d_{2}q^{2}\left(c_{2}u\left(d_{3}q^{2}\left(c_{3}+u\right)-v\right)+c_{1}\left(d_{3}q^{2}\left(2c_{2}+u\right)\left(c_{3}+u\right)-uv\right)\right))+c_{1}c_{2}c_{3}^{2}d_{3}^{2}q^{2}v^{2})\Big{)}^{-1},\\\
\cfrac{b_{2}d_{2}q^{2}u\left(d_{3}q^{2}\left(c_{3}+u\right)-v\right)}{c_{1}c_{2}d_{3}q^{2}\left(c_{3}\left(b_{2}d_{2}q^{2}-v\right)+b_{2}d_{2}q^{2}u\right)+b_{1}b_{2}v\left(-b_{2}d_{2}q^{2}-d_{3}q^{2}u+v\right)}\Big{)},\end{array}$
( 3.13)

the corresponding Lax equations (2.6) with (2.17), (2.18) with (2.20) and
(2.18) with (3.3), (3.4) are equivalent. Conversely, such relations among
$(f,g)$, $(x,y)$ and $(u,v)$ are uniquely determined as (3.12), (3.13).

###### Proof.

Using the relation (3.12), we can check that the equation (2.6) with (2.17) is
equivalent to (2.18) with (2.20). Similarly, using the relation (3.13), we can
check that the equation (2.18) with (2.20) is equivalent to (2.18) with with
(3.3), (3.4). The converse is obvious from the form of the Lax matrix. ∎

##  4 Continuous limit

In this section, we describe a relation between our result and the result of
Boalch [1]. in [1], a Lax pair for the additional-difference Painlevé equation
with affine Weyl symmetry group of type $E_{6}$ was described. The linear
differential equation of the Lax pair is as follows

$\begin{array}[]{c}\cfrac{d}{dz}\Psi(z)=\Psi(z)\left(\cfrac{A_{1}^{b}}{z}+\cfrac{A_{2}^{b}}{z-1}\right),\quad
A_{3}^{b}:=-(A_{1}^{b}+A_{2}^{b}),\\\ \end{array}$ ( 4.1)

where the matrices $A_{i}^{b}$ ($1\leq i\leq 3$) are $3\times 3$ matrices with
different eigenvalues.

We show that the linear $q$-difference equation (2.1) reduces to the equation
(4.1) via a continuous limit $q\to 1$. The equation (2.1) takes the following
form after a scale transformation $z\to-z$ and gauge transformations

$\begin{array}[]{c}(1-z)\Psi(qz)=\Psi(z)\mathcal{A}(z),\quad\mathcal{A}(z)=\begin{bmatrix}b_{1}d_{1}&k_{1}&v_{1}\\\
0&b_{2}d_{2}&k_{2}\\\ 0&0&b_{3}d_{3}\end{bmatrix}-\begin{bmatrix}d_{1}&0&0\\\
v_{2}&d_{2}&0\\\ v_{3}&v_{4}&d_{3}\end{bmatrix}z,\vspace{6pt}\\\
|\mathcal{A}(z)|=d_{1}d_{2}d_{3}(z-c_{1})(z-c_{2})(z-c_{3}),\end{array}$ (
4.2)

where $k_{j}$ ($j=1$, $2$) are constants. We put $q=e^{h}$ and consider the
limit $h\to 0$. We set

$\begin{array}[]{c}b_{i}=q^{\beta_{i}},\quad c_{i}=q^{\gamma_{i}},\quad
d_{i}=q^{\delta_{i}}\quad(1\leq i\leq 3),\vspace{6pt}\\\
k_{j}=hl_{j}\quad(j=1,2),\quad v_{m}=hu_{m}\quad(1\leq m\leq 4),\end{array}$ (
4.3)

where $l_{j}$ are constants. By using Taylor’s expansion for (4.2), (4.3)

$\begin{array}[]{rl}(LHS)&=(1-z)\Psi(z)+h(1-z)z\cfrac{d}{dz}\Psi(z)+\mathcal{O}(h^{2}),\vspace{6pt}\\\
(RHS)&=(1-z)\Psi(z)+h\Psi(z)\begin{bmatrix}\beta_{1}-\delta_{1}(z-1)&l_{1}&u_{1}\\\
-u_{2}z&\beta_{2}-\delta_{2}(z-1)&l_{2}\\\
-u_{3}z&-u_{4}z&\beta_{3}-\delta_{3}(z-1)\end{bmatrix}+\mathcal{O}(h^{2}),\end{array}$
( 4.4)

we find the following limit as $h\to 0$

$\begin{array}[]{c}\cfrac{d}{dz}\Psi(z)=\Psi(z)\left(\cfrac{A_{1}}{z}+\cfrac{A_{2}}{z-1}\right),\vspace{6pt}\\\
A_{1}=\begin{bmatrix}\beta_{1}+\delta_{1}&l_{1}&u_{1}\\\
0&\beta_{2}+\delta_{2}&l_{2}\\\ 0&0&\beta_{3}+\delta_{3}\end{bmatrix},\quad
A_{2}=\begin{bmatrix}-\beta_{1}&-l_{1}&-u_{1}\\\ u_{2}&-\beta_{2}&-l_{2}\\\
u_{3}&u_{4}&-\beta_{3}\end{bmatrix},\vspace{6pt}\\\
A_{3}:=-(A_{1}+A_{2})=\begin{bmatrix}\delta_{1}&0&0\\\ u_{2}&\delta_{2}&0\\\
u_{3}&u_{4}&\delta_{3}\end{bmatrix},\end{array}$ ( 4.5)

where eigenvelues of the matrix $A_{2}$ are $\gamma_{i}$ ($1\leq i\leq 3$) by
the condition of the determinant of the matrix $\mathcal{A}(z)$ (4.2).
Therefore the linear $q$-difference equation (2.1) reduces to the equation
(4.1) via a continuous limit $q\to 1$.

In the following, we consider a continuous limit $q\to 1$ of the result in
§2.1. In the subsection §2.1, through a compatibility condition of the
equations (2.10), we derived a standard $q$-Painlevé equation of type $E_{6}$.
We take the following equation as a deformation equation for the differential
equation (4.5) which is rewritten version of (2.9)

$\begin{array}[]{c}T\Psi(z)=B(z)\Psi(z),\quad
B(z)=\begin{bmatrix}w_{1}&w_{2}&w_{3}\\\ 0&0&0\\\
0&0&0\end{bmatrix}+\begin{bmatrix}w_{4}&0&0\\\ 1&0&0\\\
w_{5}&w_{6}&w_{7}\end{bmatrix}z,\vspace{6pt}\\\
|B(z)|=(w_{3}w_{6}-w_{2}w_{7})z^{2},\vspace{6pt}\\\
T:(\beta_{1},\beta_{2},\beta_{3},\gamma_{1},\gamma_{2},\gamma_{3},\delta_{1},\delta_{2},\delta_{3})\to(\beta_{1}-1,\beta_{2}+1,\beta_{3},\gamma_{1},\gamma_{2},\gamma_{3},\delta_{1}+1,\delta_{2},\delta_{3}+1).\end{array}$
( 4.6)

Solving a compatibility condition for the equation (4.5), (4.6), we obtain the
following additional-difference Painlevé equation of type $E_{6}$ [5], [9]

$\begin{array}[]{c}(f+g)(\overline{f}+g)=\cfrac{\left(g+\gamma_{1}\right)\left(g+\gamma_{2}\right)\left(g+\gamma_{3}\right)\left(g+\beta_{1}+\delta_{1}-\delta_{2}\right)}{\left(g+\beta_{2}+1\right)\left(g+\beta_{3}-\delta_{2}+\delta_{3}+1\right)},\vspace{6pt}\\\
(\overline{f}+\overline{g})(\overline{f}+g)=\cfrac{(\overline{f}-\gamma_{1})(\overline{f}-\gamma_{2})(\overline{f}-\gamma_{3})(\overline{f}-\beta_{1}-\delta_{1}+\delta_{2})}{(\overline{f}-\beta_{1}+\beta_{2}-1)(\overline{f}-\beta_{1}-\delta_{1}+\delta_{3}+1)}.\end{array}$
( 4.7)

where,

$f=\beta_{3}+\frac{u_{1}u_{4}}{l_{1}},\quad
g=\frac{l_{1}l_{2}}{u_{1}}-\beta_{2},$ ( 4.8)

and $\overline{*}$ stands for $T(*)$. From the above, we derive the
additional-difference Painlevé equation of type $E_{6}$ solving a
compatibility condition of the Lax pair via a continuous limit $q\to 1$.

## Acknowledgement

The author would like to express her gratitude to Professor Yasuhiko Yamada
for valuable suggestions and encouragement. She also thanks supports from JSPS
KAKENHI Grant Numbers 17H06127 and 26287018 for the travel expenses in
accomplishing this study.

## Appendix

In the appendix, we give an explicit forms of $P_{j}(z)$ (3.2)

$\begin{array}[]{l}P_{3}(z)=p_{31}(z-u),\\\
P_{2}(z)=p_{22}z^{2}+p_{21}z+p_{20},\\\
P_{1}(z)=p_{13}z^{3}+p_{12}z^{2}+p_{11}z+p_{10},\\\
P_{0}(z)=-P_{3}(qz)d_{1}d_{2}d_{3}(z+c_{1})(z+c_{2})(z+c_{3}),\end{array}$ (
4.9)

where the coefficients $p_{j,k}$ $(1\leq j\leq 3,0\leq k\leq 3)$ are

$\begin{array}[]{l}p_{31}=-q^{2}uvc_{1}c_{2},\vspace{6pt}\\\
p_{22}=q^{4}uvc_{1}c_{2}\left(d_{1}+d_{2}+d_{3}\right),\vspace{6pt}\\\
p_{21}=-quvc_{1}c_{2}\left(ud_{3}q^{3}+\left(qu-
b_{2}\right)d_{2}q^{2}-b_{3}d_{3}q^{2}-vq+v+\left(q^{3}u-qb_{1}\right)d_{1}\right),\vspace{6pt}\\\
p_{20}=-q^{2}u^{2}vc_{1}c_{2}\left(b_{1}d_{1}+qb_{2}d_{2}+qb_{3}d_{3}\right),\vspace{6pt}\\\
p_{13}=-q^{5}uvc_{1}c_{2}\left(d_{1}d_{2}+d_{2}d_{3}+d_{3}d_{1}\right),\vspace{6pt}\\\
p_{12}=-q(ub_{1}b_{2}b_{3}\left(u+c_{2}\right)d_{1}d_{2}d_{3}q^{5}+uc_{1}^{2}c_{2}\left(u+c_{2}\right)d_{1}d_{2}d_{3}q^{5}\vspace{6pt}\\\
\qquad+c_{1}(u^{2}c_{2}^{2}d_{1}d_{2}d_{3}q^{5}+ub_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{5}\vspace{6pt}\\\
\qquad-
c_{2}(qd_{1}(qu\left(qud_{2}\left(-ud_{3}q^{2}+vq+v\right)+v\left(q(q+1)ud_{3}-v\right)\right)\vspace{6pt}\\\
\qquad-
b_{1}\left(v-q^{2}b_{2}d_{2}\right)\left(v-q^{2}b_{3}d_{3}\right))+v(d_{2}(u\left(q(q+1)ud_{3}-v\right)\vspace{6pt}\\\
\qquad+b_{2}\left(q^{2}b_{3}d_{3}-v\right))q^{2}+v\left(v-q^{2}\left(u+b_{3}\right)d_{3}\right))))),\vspace{6pt}\\\
p_{11}=u(ub_{1}b_{2}b_{3}\left(u+c_{2}\right)d_{1}d_{2}d_{3}q^{6}+uc_{1}^{2}c_{2}\left(u+c_{2}\right)d_{1}d_{2}d_{3}q^{6}\vspace{6pt}\\\
\qquad+c_{1}(u^{2}c_{2}^{2}d_{1}d_{2}d_{3}q^{6}+ub_{1}b_{2}b_{3}d_{1}d_{2}d_{3}q^{6}\vspace{6pt}\\\
\qquad-
c_{2}(qd_{1}(b_{1}\left(b_{2}d_{2}\left(-b_{3}d_{3}q^{3}+vq+v\right)q^{2}+v\left(q^{2}(q+1)b_{3}d_{3}-v\right)\right)\vspace{6pt}\\\
\qquad-
qu\left(v-q^{2}ud_{2}\right)\left(v-q^{2}ud_{3}\right))+v(d_{2}(u\left(q^{2}ud_{3}-v\right)\vspace{6pt}\\\
\qquad+b_{2}\left(q^{2}(q+1)b_{3}d_{3}-v\right))q^{2}+v\left(v-q^{2}\left(u+b_{3}\right)d_{3}\right))))),\vspace{6pt}\\\
p_{10}=q^{3}u^{2}vc_{1}c_{2}\left(qb_{2}b_{3}d_{2}d_{3}+b_{1}d_{1}\left(b_{2}d_{2}+b_{3}d_{3}\right)\right).\end{array}$
( 4.10)

We give also an explicit forms of the coefficients $c_{ij}$ $(0\leq j\leq
3,0\leq i+j\leq 4)$ of the polynomial $P(u,v)$ in variables $u$, $v$:

$\begin{array}[]{l}c_{01}=c_{0}\cfrac{z^{2}}{q^{2}}\left(-\cfrac{q}{b_{1}d_{1}}-\cfrac{1}{b_{2}d_{2}}-\cfrac{1}{b_{3}d_{3}}\right)\Phi(qz),\vspace{6pt}\\\
c_{02}=c_{0}\cfrac{z^{2}\left(b_{1}d_{1}+qb_{2}d_{2}+qb_{3}d_{3}\right)\Phi(qz)}{q^{4}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{03}=-c_{0}\cfrac{z^{2}\Phi(qz)}{q^{5}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{10}=c_{0}\left(\cfrac{c_{1}c_{2}z^{2}}{b_{1}b_{2}b_{3}}+\cfrac{z^{2}}{c_{1}}+\cfrac{z^{2}}{c_{2}}-z\right)\Phi(qz),\vspace{6pt}\\\
c_{11}=-c_{0}\cfrac{z}{q^{4}b_{1}b_{2}b_{3}c_{1}c_{2}d_{1}d_{2}d_{3}}(b_{1}d_{1}(qb_{2}d_{2}\left(q^{2}b_{3}\left(z+c_{1}\right)\left(z+c_{2}\right)d_{3}\Phi(z)-(q+1)c_{1}c_{2}\Phi(qz)\right)\vspace{6pt}\\\
\quad\qquad+c_{1}c_{2}\left(\Phi\left(q^{2}z\right)-q(q+1)b_{3}d_{3}\Phi(qz)\right))\vspace{6pt}\\\
\quad\qquad+c_{1}c_{2}(zd_{3}\Phi\left(q^{2}z\right)q^{2}+zd_{1}\left(-qzd_{3}\Phi(qz)+qd_{2}\left(\left(z+c_{1}\right)\left(z+c_{2}\right)d_{3}\Phi(z)-z\Phi(qz)\right)+\Phi\left(q^{2}z\right)\right)q^{2}\vspace{6pt}\\\
\quad\qquad+b_{3}d_{3}\Phi\left(q^{2}z\right)q-d_{2}\left(q\left(qz^{2}+(q+1)b_{2}b_{3}\right)d_{3}\Phi(qz)-\left(qz+b_{2}\right)\Phi\left(q^{2}z\right)\right)q-\Phi\left(q^{3}z\right))),\vspace{6pt}\\\
c_{12}=c_{0}\cfrac{z}{q^{5}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}}(zd_{3}\Phi(qz)q^{2}+\left(qz-
b_{2}\right)d_{2}\Phi(qz)q-b_{3}d_{3}\Phi(qz)q-\Phi\left(q^{2}z\right)q\vspace{6pt}\\\
\qquad\qquad+\left(q^{2}z-b_{1}\right)d_{1}\Phi(qz)+\Phi\left(q^{2}z\right)),\vspace{6pt}\\\
c_{13}=c_{0}\cfrac{z\Phi(qz)}{q^{6}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{20}=c_{0}\left(\cfrac{c_{1}z\left(z-c_{2}\right)z+z^{2}c_{2}}{b_{1}b_{2}b_{3}}-\cfrac{-z^{2}+c_{1}z+c_{2}z}{c_{1}c_{2}}\right)\Phi(qz),\vspace{6pt}\\\
c_{21}=c_{0}\cfrac{b_{1}d_{1}}{q^{4}b_{1}b_{2}b_{3}c_{1}c_{2}d_{1}d_{2}d_{3}}(qb_{2}d_{2}\left(qb_{3}\left(z+c_{1}\right)\left(z+c_{2}\right)d_{3}\Phi(z)-c_{1}c_{2}\Phi(qz)\right)\vspace{6pt}\\\
\qquad\quad+c_{1}c_{2}\left(\Phi\left(q^{2}z\right)-qb_{3}d_{3}\Phi(qz)\right))-c_{0}c_{1}c_{2}(zd_{1}(-\left(z+c_{1}\right)\left(z+c_{2}\right)d_{2}d_{3}\Phi(z)+(q+1)zd_{2}\Phi(qz)\vspace{6pt}\\\
\qquad\quad+(q+1)zd_{3}\Phi(qz)-\Phi\left(q^{2}z\right))q^{2}-zd_{3}\Phi\left(q^{2}z\right)q^{2}-b_{3}d_{3}\Phi\left(q^{2}z\right)q+d_{2}(q\left((q+1)z^{2}+b_{2}b_{3}\right)d_{3}\Phi(qz)\vspace{6pt}\\\
\qquad\quad-\left(qz+b_{2}\right)\Phi\left(q^{2}z\right))q+\Phi\left(q^{3}z\right)),\vspace{6pt}\\\
c_{22}=-c_{0}\cfrac{z\left(d_{1}+d_{2}+d_{3}\right)\Phi(qz)}{q^{4}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{30}=c_{0}\left(-\cfrac{-z^{2}+c_{1}z+c_{2}z}{b_{1}b_{2}b_{3}}-\cfrac{1}{c_{1}c_{2}}\right)\Phi(qz),\vspace{6pt}\\\
c_{31}=c_{0}\cfrac{\left(d_{1}d_{2}+d_{2}d_{3}+d_{3}d_{1}\right)z\Phi(qz)}{q^{2}b_{1}b_{2}b_{3}d_{1}d_{2}d_{3}},\vspace{6pt}\\\
c_{40}=-c_{0}\cfrac{z\Phi(qz)}{b_{1}b_{2}b_{3}}.\end{array}$ ( 4.11)

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