A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆
For the -Painlevé equation with affine Weyl group symmetry of type ⁽¹⁾₆, a 2 × 2 matrix Lax form and a second-order scalar lax form were known. We give a new 3 × 3 matrix Lax form and a third order scalar equation related to it. Continuous limit is also discussed.
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| citation_txt | A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆. Kanam Park. SIGMA 19 (2023), 094, 17 pages |
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| description | For the -Painlevé equation with affine Weyl group symmetry of type ⁽¹⁾₆, a 2 × 2 matrix Lax form and a second-order scalar lax form were known. We give a new 3 × 3 matrix Lax form and a third order scalar equation related to it. Continuous limit is also discussed.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 094, 17 pages
A 3 × 3 Lax Form for the q-Painlevé Equation
of Type E6
Kanam PARK
National Institute of Technology, Toba College, 1-1, Ikegami-cho, Toba-shi, Mie, Japan
E-mail: paku-k@toba-cmt.ac.jp
Received December 01, 2022, in final form November 05, 2023; Published online November 18, 2023
https://doi.org/10.3842/SIGMA.2023.094
Abstract. For the q-Painlevé equation with affine Weyl group symmetry of type E
(1)
6 ,
a 2×2 matrix Lax form and a second order scalar lax form were known. We give a new 3×3
matrix Lax form and a third order scalar equation related to it. Continuous limit is also
discussed.
Key words: Lax formalism; q-Painlevé equation
2020 Mathematics Subject Classification: 14H70; 34M56; 39A13
1 Introduction
The q-Painlevé equation with affine Weyl group symmetry of type E
(1)
6 was first discovered in [9].
The well-known form of it is as follows
T : (a1, a2, a3, a4, a5, a6, a7, a8; f, g) →
(
a1/q, a2/q, a3/q, a4/q, a5, a6, a7, a8; f, g
)
,(
fg − 1
)
(fg − 1)
ff
=
(g − 1/a5)(g − 1/a6)(g − 1/a7)(g − 1/a8)
(g − a3)(g − a4)
,(
fg − 1
)(
fg − 1
)
gg
=
(
f − a5
)(
f − a6
)(
f − a7
)(
f − a8
)(
f − a1/q
)(
f − a2/q
) ,
q =
a1a2
a3a4a5a6a7a8
,
where f and g are dependent variables, a1, a2, . . . , a8 are parameters, q is a constant and the
overline symbol “¯” denotes the discrete time evolution.
In previous works, the following Lax forms for the q-Painlevé equation for type E
(1)
6 has been
obtained. In [12], a 2 × 2 matrix Lax pair was first derived as a reduction of the q-Garnier
system. The other approach gives a 2× 2 matrix Lax pair [2, 7]. In [16], a second order scalar
Lax form was obtained as a reduction from the q-Painlevé equation of type E8. The relation
between these Lax forms was given in [15].
In this article, we give a new Lax form with 3 × 3 matrix Lax pair. We derive such a Lax
form as a special case of the system investigated in our previous work [8]. As a result, we derive
an equation which is equivalent to the q-Painlevé equation of type E
(1)
6 [4, 10].
In the previous work [8], we defined a nonlinear q-difference system as a connection preserving
deformation of the following linear equation
Ψ(qz) = Ψ(z)A(z), A(z) = DXε1
1 (z)Xε2
2 (z) · · ·XεM
M (z),
D = diag[d1, d2, . . . , dN ],
mailto:paku-k@toba-cmt.ac.jp
https://doi.org/10.3842/SIGMA.2023.094
2 K. Park
Xi(z) = diag[u1,i, u2,i, . . . , uN,i] + Λ, Λ =
0 1 O
. . .
. . .
1
z 0
, (1.1)
where the exponents are εi = ±1 (1 ≤ i ≤ M), uj,i (1 ≤ j ≤ N) are dependent variables
and ci, dj are parameters which satisfy
N∏
j=1
uj,i = ci.
Since one can exchange the order of matrices X±1
i by suitable rational transformations of vari-
ables uj,i, the equation (1.1) essentially depends on M+, M−, where M± = #{εi|εi = ±1}.
The contents of this paper is as follows. In Section 2, we set up a linear q-difference equa-
tion (2.1), which is a case of (M+,M−, N) = (3, 0, 3) for the equation (1.1). And we discuss
about its two deformations. One deformation gives rise to a well known form of the q-Painlevé
equation of type E
(1)
6 , and the other deformation gives an equation for a non-standard direction.
Namely, it does not give a q-shift deformation for parameters. In Section 3, we derive a scalar
equation from the 3 × 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]. In Appendix A, we give deformations considered in Section 2 on root variables. In
Appendix B, we give explicit forms of coefficients of a single linear q-difference equation derived
in Section 3.
Remark 1.1. The equation (1.1) 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 [11].
And in case of (M+,M−, N) = (2, 0, 2n + 2), the equation (1.1) is also equivalent to the linear
q-difference equation related to the 2n-dimensional system q-P(n+1,n+1) [13]. In both cases, when
n = 1, they give rise to the q-PVI equation [5]. We note that the equation (2.1) is coincide to
the q-difference linear equation of the Lax form in [14] in case of (m, n)=(3, 1).
2 A 3 × 3 matrix Lax form
In this section, we consider two types of deformations for the linear q-difference equation (1.1)
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 Ψ(z) = [Ψ1(z),Ψ2(z),Ψ3(z)]:
Ψ(qz) = Ψ(z)A(z) = Ψ(z)A(z, t),
A(z) = DX1(z)X2(z)X3(z) =
b1d1 ∗ ∗
0 b2d2 ∗
0 0 b3d3
+
d1 0 0
∗ d2 0
∗ ∗ d3
z, (2.1)
where the matrices D and Xi(z) (1 ≤ i ≤ 3) stand for
D = diag[d1, d2, d3], Xi(z) = diag[u1,i, u2,i, u3,i] + Λ, Λ =
0 1 0
0 0 1
z 0 0
,
and uj,i (1 ≤ i, j ≤ 3) are dependent variables and ci, dj are parameters which satisfy
3∏
i=1
uj,i = bj , 1 ≤ j ≤ 3,
3∏
j=1
uj,i = ci, 1 ≤ i ≤ 3, b1b2b3 = c1c2c3. (2.2)
A 3× 3 Lax Form for the q-Painlevé Equation of Type E6 3
The first equation in (2.2) is equivalent to that the characteristic exponents at z = 0 of the
equation (2.1) are bjdj . The second equation in (2.2) is equivalent to the following condition:
|A(z)| = d1d2d3(z + c1)(z + c2)(z + c3). (2.3)
Through a gauge transformation by a 3×3 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 (2.3), two
of the remaining four components are determined by other components and parameters bj , ci, dj .
In this article, we will consider two deformations T1 and T2 for the equation (2.1) which act
on parameters bj , ci, dj as
T1 : (b1, b2, b3, c1, c2, c3, d1, d2, d3) →
(
b1, qb2,
b3
q
, c1, c2, c3, qd1, d2, qd3
)
,
T2 : (b1, b2, b3, c1, c2, c3, d1, d2, d3) →
(
b1d1
d2
,
b2d2
d3
, b3, c1, c2, c3q, d2, d3,
d1
q
)
.
2.1 The deformation T1
We will show that the deformation T1 gives rise to the standard form of the q-E
(1)
6 . We consider
the following q-difference linear equation:
Ψ(qz) = Ψ(z)A(z), A(z) =
b1d1 1 v1
0 b2d2 1
0 0 b3d3
+
d1 0 0
v2 d2 0
v3 v4 d3
z, (2.4)
|A(z)| = d1d2d3(z + c1)(z + c2)(z + c3). (2.5)
Although there are many ways of gauge fixings, fixing as the equation (2.4) makes relatively
easier to find a new change of variables from indeterminate points of time evolution equations
vi (∗ = T1(∗)).
As a connection preserving deformation for the equation (2.4), (2.5) we take the following
deformation for parameters
T1 : (b1, b2, b3, c1, c2, c3, d1, d2, d3) →
(
b1, qb2,
b3
q
, c1, c2, c3, qd1, d2, qd3
)
. (2.6)
Then, there is a matrix B(z) which satisfies the following deformation equation:
T1Ψ(z) = Ψ(z)B(z).
We derive and show the matrix B(z). First, the matrix B(z) is a rational function for z [5].
In fact, by the argument [5] to determine a coefficient matrix of a deformation equation, the
matrix B(z) is of degree one through the following:
(i) The parameters ci which satisfy |A(ci)| = 0 are constant by the deformation T1 (2.6).
Therefore, the matrix B(z) does not have poles at z = −qkci (k ∈ Z). Namely, the
matrix B(z) has poles only at z = 0 or z = ∞.
(ii) The deformation T1 (2.6) shifts characteristic exponents of the equation (2.4) at z = 0 as
follows:
T1 : (b1d1, b2d2, b3d3) → (qb1d1, qb2d2, b3d3).
Therefore, the matrix B(z) behaves nearby z = 0 as
B(z) = B0 +B1z +O
(
z2
)
.
4 K. Park
(iii) The deformation T1 (2.6) shifts characteristic exponents of the equation (2.4) at z = ∞ as
follows:
T1 : (d1, d2, d3) → (qd1, d2, qd3).
Therefore, the matrix B(z) behaves nearby z = ∞ as
B(z) = B1z +B0 +O
(
1
z
)
.
From the above (i)–(iii), the matrix B(z) is a polynomial in z of degree one. At last, we derive
an explicit form of the matrix B(z). We express the matrix B(z) as follows:
B(z) = B0 +B1z, Bi =
[
βi
j,k
]
1≤j,k≤3
, i = 1, 2. (2.7)
Comparing coefficients of z for a compatibility condition equation
B(z)A(z) = A(z)B(qz), ∗ = T1(∗),
for the matrices A(z) (2.4) and B(z) (2.7), we have the following three equations:
B0A0 = A0B0, B1A0 +B0A1 = qA0B1 +A1B0, B1A1 = qA1B1, (2.8)
where matrices Ai denote coefficients of zi for the matrix A(z) (2.4). Solving the first and the
third equations of (2.8), forms of the matrices B0 and B1 are as follows:
B0 =
0 0
β0
2,3(b2d2v1−b3d3v1−1)
b1d1−b3d3
0 0 β0
2,3
0 0 −β0
2,3 (b2d2 − b3d3)
,
B1 =
(d1−d2)β1
2,1
v2
0 0
β1
2,1 0 0
d1qv3v4β1
2,1+d2(v2v3β1
3,2−qv3v4β1
2,1)+v2(qv4v4β1
2,1−β1
3,2(d3qv3+v2v4))
(d1−d3)qv2v4
β1
3,2 −β1
3,2(d2−d3q)
v4
.
From the second equation of (2.8), we obtain explicit forms of time evolutions vi (1 ≤ i ≤ 4), the
components β0
2,3 and β1
2,1 of the matrix B(z) (2.7). Explicit forms of the remain components β0
2,3
and β1
2,1 are as follows:
β0
2,3 = β1
3,2q(b1d1 − b3d3)(d1 − d2 − v1v2)
(
−b22d
2
2qv1v2(d3q − d1 + v1v2)− b3d3d2qv2
− 2b23d
2
3d2qv1v2 + b3d3d2qv1v3 + b2d2
(
b3d3(d1q(−d3(q − 1)− 2v1v2)
+ qv1v2(d3q + 2v1v2) + d2(d3(q − 1)q + (2q − 1)v1v2)) + q
(
d3qv2 + d1(v1v3 − v2)
+ v1
(
−d2v3 + v22 − v1v3v2
)))
+ b3d3d2qv4 − b23d
2
3qv
2
1v
2
2 − b3d3qv1v
2
2 − b3d
2
3qv2
+ b3d1d3qv2 + b23d1d
2
3qv1v2 − b3d1d3qv1v3 + b3d3qv
2
1v2v3 − b3d1d3qv4
+ b3d3qv1v2v4 + b1d1
(
b3d3
(
d2qv1v2 − d3qv1v2 + d22(q − 1)− d1d2(q − 1)
)
+ b2d2q(d2(d3(−q) + d3 − v1v2) + d1d3(q − 1) + d3v1v2) + qv4(d1 − d2 − v1v2)
)
− b23d
2
3d
2
2q + b23d1d
2
3d2q + b3d3d2v2 + b23d
2
3d2v1v2 + b23d
2
3d
2
2
− b23d1d
2
3d2 + d2qv3 − d1qv3 + qv1v2v3
)−1
,
β1
2,1 = β1
3,2
qv2(b1d1v1 − b2d2v1 + 1)
v2(1− b2d2v1) + b3d3(−d1 + d2 + v1v2) + b1d1(d1 − d2)
.
A 3× 3 Lax Form for the q-Painlevé Equation of Type E6 5
We remark that time evolutions vi are independent for the component β1
3,2. Therefore, we
set β1
3,2 = 1.
From the above, a deformation equation T1Ψ(z) = Ψ(z)B(z) is expressed as follows:
T1Ψ(z) = Ψ(z)B(z), B(z) =
0 0 w1
0 0 w2
0 0 w3
+
w4 0 0
w5 0 0
w6 1 w7
z, (2.9)
|B(z)| = (w1w5 − w2w4)z
2,
T1 : (b1, b2, b3, c1, c2, c3, d1, d2, d3; v1, v4)
→
(
b1, qb2,
b3
q
, c1, c2, c3, qd1, d2, qd3; v1, v4
)
. (2.10)
Theorem 2.1. Through a compatibility condition of the equations (2.4), (2.9), (2.10),
B(z)A(z) = A(z)B(qz), (2.11)
we obtain the following equations:
(fg − 1)
(
fg − 1
)
ff
=
(g − 1/c1)(g − 1/c2)(g − 1/c3)(g − d2/b3d3)
(g − 1/b2)(g − d2/b1d1)
,(
fg − 1
)(
fg − 1
)
gg
=
(
f − c1
)(
f − c2
)(
f − c3
)(
f − b3d3/d2
)(
f − b3d3/d1
)(
f − b3/q
) , (2.12)
where
f =
b3d3
d3 − v1v4
, g =
d2v1
b2d2v1 − 1
, (2.13)
and ∗ stands for T1(∗).
The equation (2.12) is the well known form of the q-Painlevé equation of type E6 [4, 10] (see
also [6]).
Proof. The result is obtained by a direct computation of the compatibility condition of (2.11).
Since the computation is rather heavy, we will give a comment how to do it efficiently. Though
the 2 variables v2, v3 can be represented by the rational functions of the remaining two variables
v1, v4 by the relation (2.5), it is more efficient to do this elimination after the calculation of the
compatibility condition (2.11) in 4 variables, and then reduce it to 2 variables. In this way, we
get the following time evolutions for v1, v4 as rational functions of v1, v4:
v1 =
C1(v1, v4)C2(v1, v4)
D1(v1, v4)D2(v1, v4)
, v4 =
C3(v1, v4)
D3(v1, v4)D4(v1, v4)
, (2.14)
where
C3(v1, v4), D2(v1, v4) : polynomials in v1, v4 of degree (4, 3),
C2(v1, v4) : a polynomial in v1, v4 of degree (3, 1),
C1(v1, v4), D3(v1, v4), D4(v1, v4) : polynomials in v1, v4 of degree (2, 1),
D1(v1, v4) : a polynomial in v1, v4 of degree (1, 0).
The remaining task is to rewrite the equation (2.14) to (2.12). A useful way to solve it is to
look at the singularities of the equations [3]. Namely, we investigate the points at which the
6 K. Park
right hand side of the equations (2.14) are indeterminate. We focus on the equation D3(v1, v4)
of them
D3(v1, v4) = d3(−b2d2v1 + b3d2v1 + 1) + v1v4(b2d2v1 − 1). (2.15)
Investigating common zero points of the equation (2.15) and the other polynomials Ck(v1, v4),
Dl(v1, v4), we find 4 indeterminate points as follows:
(v1, v4) =
(
− 1
(u−1 − b2)d2
,−
(
u−1 − b3
)(
u−1 − b2
)
d3d2
u−1
)
,
u = c−1
1 , c−1
2 , c−1
3 , d2/b3d3. (2.16)
The other 4 points are the following:
(v1, v1v4) = (∞,∞),
(
1
b2d2 − b1d1
,∞
)
,
(
1
b2d2
, d3 − d1
)
,
(
1
b2d2
, 0
)
.
In view of the form of the points in (2.16), we define the variables f , g as follows:
(v1, v1v4) =
(
− 1(
g−1 − b2
)
d2
,
(f − b3)d3
f
)
. (2.17)
By the transformation (2.17), the equation D3(v1, v4) = 0 is transformed to an equation fg = 1.
Through the correspondence (2.17) and the time evolution equations for the variables v1, v4
(2.14), we have time evolution equations for f , g (2.12). ■
2.2 The deformation T2
In this subsection, we take a deformation equation which is one of that considered in the previous
work [8]. It corresponded to the permutations of the matricesXi(z)
±1. We consider the following
Lax pair:
Ψ(qz) = Ψ(z)A(z),
A(z) = DX1(z)X2(z)X3(z) =
b1d1 1 ∗
0 b2d2 1
0 0 b3d3
+
d1 0 0
∗ d2 0
∗ ∗ d3
z, (2.18)
T2Ψ(z) = Ψ(z)B(z), B(z) = X3(z/q)
−1, |B(z)| = q
z + qc3
,
T2 : (b1, b2, b3, c1, c2, c3, d1, d2, d3;x, y)
→
(
b1d1
d2
,
b2d2
d3
, b3, c1, c2, c3q, d2, d3,
d1
q
;x, y
)
, (2.19)
where we define variables x, y and gauge freedom w1, w2 with the variables uj,i in (2.18) as
follows:
x =
u1,1u1,2u2,1u2,2
u1,1 + u2,2
, y =
1
u1,1u1,2 (u2,1 + u3,2)
, w1 = u1,1, w2 = u1,3. (2.20)
Theorem 2.2. Through a compatibility condition of the equations (2.18), (2.19), (2.20),
B(z)A(z) = A(z)B(qz), (2.21)
A 3× 3 Lax Form for the q-Painlevé Equation of Type E6 7
we obtain the following equations:
x+ c1
x+ c2
=
E1E2
F1F2
,
c1y + 1
c2y + 1
=
E1E3
F1F3
, (2.22)
where
E1 = b1d1(b2d2(1− xy) + c1d3y(c2 + x)) + c1d2d3x(c2y + 1),
E2 = b1d1(b2d2(1− xy) + c3d3qy(c1 + x)) + c3d2d3qx(c1y + 1),
E3 = b1d1(b2d2(1− xy) + c2d3y(c1 + x)) + c3d2d3qx(c1y + 1),
F1 = b1d1(b2d2(1− xy) + c2d3y(c1 + x)) + c2d2d3x(c1y + 1),
F2 = b1d1(b2d2(1− xy) + c3d3qy(c2 + x)) + c3d2d3qx(c2y + 1),
F3 = b1d1(b2d2(1− xy) + c1d3y(c2 + x)) + c3d2d3qx(c2y + 1), (2.23)
and ∗ stands for T2(∗).
Proof. Solving a compatibility condition (2.21) with (2.20) for the variables x, y, we obtain
the following equations
x =
(xy − 1)G1(x, y)
H1(x, y)
, y =
xG2(x, y)
H2(x, y)
,
where Gk(x, y), Hk(x, y) (k = 1, 2) are polynomials in variables x, y. The polynomial G1(x, y) is
of degree (1, 1), G2(x, y) is of degree (1, 2) and Hk(x, y) are of degree (2, 2). Using a method [6]
for finding point configuration, a configuration of points for the equation (2.22), (2.23) is as
follows:
(x, y) =
(
−c1,−
1
c1
)
,
(
−c2,−
1
c2
)
,
(
−b1d1
d2
,− d2
b1d1
)
,
(
−b1b2
c3
,− c3
b1b2
)
,
(−b2, 0),
(
−b1b2d1
c3d3q
, 0
)
,
(
0,− 1
b1
)
,
(
0,− b2d2
c1c2d3
)
. (2.24)
Calculating x+c1
x+c2
and c1y+1
c2y+1 , respectively, we have equations (2.22), (2.23). ■
Remark 2.3. The time evolution equations uj,i are derived by solving the compatibility condi-
tion (2.21). For another derivation using the transformations which correspond to permutations
of the matrices Xεi
i (z), see [8, Proposition 2.1].
Before ending this subsection, we show a relation between pairs of the variables (f, g) in (2.12)
and (x, y) in (2.22).
Proposition 2.4. The equations (2.4), (2.13) and (2.18), (2.20) are equivalent with each other
if the variables f , g and x, y are related as
f =
c1c2c3(b2 + x)(xy − 1)
b1b2y(c1 + x)(c2 + x) + c3x(b2(1− xy) + c2xy + c1y(c2 + x) + x)
,
g = − y(b1b2 + c3x)
c3(b2(1− xy) + x) + b1b2xy
. (2.25)
Proof. Comparing the coefficient matrix A(z) of the equation (2.4), (2.13) to the coefficient
matrix A(z) of the equation (2.18), (2.20), we can solve for the variables x, y and gauge freedom
w1, w2 in terms of variables f , g and parameters bj , ci, dj (1 ≤ i, j ≤ 3). Then we have the
desired relation between pairs of variables (f, g) and (x, y) (2.25). ■
In Appendix A, we give pictures of a point configuration of the equation (2.12) and the
configuration (2.24), root basis associated with them, and three deformations T1, T2 and T 3
2 on
root variables attached with root basis.
8 K. Park
3 A scalar equation related to the q-E
(1)
6
In this section, we derive a scalar q-difference equation from the matrix q-difference equation
(2.18), (2.20) for an unknown function Ψ(z) = [Ψ1(z),Ψ2(z),Ψ3(z)] and its properties.
Before deriving, we state about a characterization of a linear q-difference equation. Linear
differential equations are characterized by its singular points and characteristic exponents. Simi-
larly, linear q-difference equations are also characterized by its singular points and characteristic
exponents. We consider the following n-th order q-difference equation
Pn(z)Φ(q
nz) + Pn−1(z)Φ
(
qn−1z
)
+ · · ·+ P0(z)Φ(z) = 0,
Pk(z) = pk,0 + pk,1z + · · ·+ pk,n+l−kz
n+l−k, 0 ≤ k ≤ n, l ∈ Z≥0,
p0,0, p0,n+l, pn,0, pn,l ̸= 0. (3.1)
In the q-difference equation (3.1), singular points are at z = 0 and z = ∞. And characteristic
exponents of solutions at z = 0 and z = ∞ are given as solutions of the following characteristic
equations respectively
Pn(0)λ
n + Pn−1(0)λ
n−1 + · · ·+ P0(0) = 0,
Pn(∞)qn−1µn + Pn−1(∞)qn−2µn−1 + · · ·+ P1(∞)µ+ P0(∞) = 0. (3.2)
To characterize the q-difference equation (3.1) is namely to determine coefficients pk,m. The
number of coefficients pk,m in the equation (3.1) is (n+1)(n+2l+2)
2 . Through the following, total
3n+ 2l − 1 coefficients pk,m are determined:
(i) We express parameters ai and bj as zeroes of the coefficients Pn(z) and P0(z):
Pn(z) = pn,l(z − a1)(z − a2) · · · (z − al),
P0(z) = p0,n+l(z − b1)(z − b2) · · · (z − bn+l). (3.3)
The parameters ai and bj indicate poles of solutions of the equation (3.1). By the expres-
sions (3.3), (2l + n) coefficients pn,m, p0,m′ (m ̸= l, m′ ̸= n+ l) are determined.
(ii) We put parameters ci and dj as characteristic exponents at z = 0 and z = ∞ respectively
z = 0: c1, c2, . . . , cn,
z = ∞ : d1, d2, . . . , dn. (3.4)
There is the following relation between parameters ai, bj , ck and dl (q-Fuchs’ relation)
(−1)nq
n(n−1)
2
n∏
j=1
dj
n+l∏
i=1
bi =
l∏
k=1
ak
n∏
j=1
cj . (3.5)
By the conditions (3.4) and (3.5), solving relations between roots and coefficients of char-
acteristic equations (3.2) at z = 0 and z = ∞ for the equation (3.1), (2n− 1) coefficients
pk,n+l−k pk′,0 (k ̸= n, k′ ̸= 0) are determined.
Example 3.1. In case (n, l) = (2, 0), the equation (3.1) has 6 coefficients pk,m. From the
conditions (i) and (ii), the equation (3.1) is characterized uniquely up to normalization. This
equation is equivalent to the q-hypergeometric equation via a gauge transformation.
A 3× 3 Lax Form for the q-Painlevé Equation of Type E6 9
From the above (i) and (ii), the number of remain coefficients pk,m is 1
2(n−1)(n+2l−2), which is
the number of accessary parameters. If z = a1 is an apparent singularity for the equation (3.1),
namely all solutions of the equation (3.1) are regular at z = a1, we have the following relations:
P0(a1/q) = 0, and f :=
P0(a1)
P1(a1/q)
=
P1(a1)
P2(a1/q)
= · · · = Pn−1(a1)
Pn(a1/q)
, (3.6)
where f is a parameter. The above equations (3.6) correspond to a non-logarithmic condition
via a Laplace transformation z ↔ Tz. The relations (3.6) determine n coefficients pk,m.
From now on, we derive a scalar q-difference equation from the matrix q-difference equation
(2.18), (2.20) for an unknown function Ψ(z) = [Ψ1(z),Ψ2(z),Ψ3(z)] and its properties. Elimi-
nating functions Ψ2(z) and Ψ3(z) in the equation (2.18), (2.20), we obtain the following third
linear q-difference equation for Φ(z) := Ψ1(z):
L(z) := P3(z)Φ
(
q3z
)
+ P2(z)Φ
(
q2z
)
+ P1(z)Φ(qz) + P0(z)Φ(z) = 0, (3.7)
where
P3(z) = p31(z − u),
P2(z) = p22z
2 + p21z + p20,
P1(z) = p13z
3 + p12z
2 + p11z + p10,
P0(z) = −P3(qz)d1d2d3(z + c1)(z + c2)(z + c3). (3.8)
Here, the coefficients pk,l (1 ≤ k ≤ 3, 0 ≤ l ≤ 3) in the polynomials Pk(z) (3.8) depend on
parameters bj , ci, dj , and a variable u defined as the zero of P3(z). The variable u is expressed
in terms of x, y as follows
u =
I1(x, y)I2(x, y)
J1(x, y)J2(x, y)
, (3.9)
where
I1(x, y) = y(b1(c2xy + c1y(c2 + x) + x) + c1c2(1− xy)),
I2(x, y) = b2d3I1(x, y)− b2d2J1(x, y) + c3d3x(c1y + 1)(c2y + 1),
J1(x, y) = b2(b1y + 1)(xy − 1),
J2(x, y) = d2(J1(x, y)− x(c1y + 1)(c2y + 1))− d3I1(x, y).
Explicit forms of the polynomials Pj(z) (0 ≤ j ≤ 3) (3.8) are given in appendix.
Then we have
Lemma 3.2. The equation L(z) = 0 (3.7) has the following properties:
(i) it is a linear four term equation between Φ
(
qjz
)
(0 ≤ j ≤ 3) and its coefficients Pj(z) are
polynomials for z of degree 4− j,
(ii) a polynomial P0(z) has four zero points at z = −ci (1 ≤ i ≤ 3), u/q,
(iii) the exponents of solutions Φ(z) are b1d1, qb2d2, qb3d3 (at z = 0) and d1, d2, d3 (at z = ∞),
(iv) a point z = u such that P3(z) = 0 is an apparent singularity, namely we have
v :=
P0(u)
P1(u/q)
=
P1(u)
P2(u/q)
=
P2(u)
P3(u/q)
. (3.10)
Conversely, the equation L(z) = 0 (3.7) is uniquely characterized by these properties (i)–(iv) up
to normalization.
10 K. Park
Proof. The properties (i)–(iv) follows by computation through eliminating Ψ2(z), Ψ3(z) in
(2.18), (2.20). The converse can be confirmed that coefficients Pj(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′(z) = P ′
3(z)Φ
(
q3z
)
+ P ′
2(z)Φ
(
q2z
)
+ P ′
1(z)Φ(qz) + P ′
0(z)Φ(z) = 0, (3.11)
P ′
3(z) = p′31(z − u),
P ′
2(z) = p′22z
2 + p′21z + p′20,
P ′
1(z) = p′13z
3 + p′12z
2 + p′11z + p′10,
P ′
0(z) = p′04(z − u/q)(z + c1)(z + c2)(z + c3).
From the property (iii), the condition of the exponents of solutions Φ(z) at z = 0 determines the
coefficients p′31, p
′
20, p
′
10 and the condition of the exponents of solutions Φ(z) at z = ∞ deter-
mines the coefficients p′22, p
′
13. The remaining coefficients except for p′04 are determined by the
property (iv). If we put the normalization factor p′04 as q
5uvc1c2d1d2d3, the function L′(z) (3.11)
equals to the function L(z) (3.7). ■
In the following, viewing Φ
(
qiz
)
(0 ≤ i ≤ 3) as parameters, we regard the scalar q-difference
equation L(z) = 0 (3.7) as an algebraic curve in variables (u, v) ∈ P1 × P1. We represent as the
curve as P (u, v) = 0. The features of the curve are the following.
Lemma 3.3. The algebraic curve P (u, v) = 0 has the following properties:
(i) The polynomial P (u, v) has the following form:
P (u, v) =
∑
0≤j≤3
0≤i≤4−j
ci,ju
ivj , c0,0 := c0z
2Φ(qz).
The coefficients ci,j depend on bi, ci, di, q, z, Φ
(
qiz
)
.
(ii) It passes the following 8 points:
(u, v) = (0, qb1d1),
(
0, q2b2d2
)
,
(
0, q2b3d3
)
, (qz,∞),
(z, 0), (−c1, 0), (−c2, 0), (−c3, 0),
and 3 points in the coordinate (r, s) = (u, v/u)
(r, s) =
(
∞, q2d1
)
,
(
∞, q2d2
)
,
(
∞, q2d3
)
.
(iii) At u = z the equation P (u, v) = 0 has the following property:( ∑
0≤i≤3−j
ci,j+1z
i
)∣∣∣
z→qz,Φ(qkz)→Φ(qk+1z)
=
∑
0≤i≤4−j
ci,j(qz)
i, j = 0, 1, 2. (3.12)
Conversely, the equation P (u, v) = 0 is uniquely characterized by these properties (i)–(iii) up to
normalization factor c0.
Proof. The properties (i)–(iii) follow for the polynomial P (u, v). Conversely, we consider a poly-
nomial
P ′(u, v) :=
∑
0≤j≤3
0≤i≤4−j
c′i,ju
ivj , c′0,0(z) = c′0Φ(qz).
A 3× 3 Lax Form for the q-Painlevé Equation of Type E6 11
The polynomial P ′(u, v) has 14 coefficients. From the property (ii), 10 coefficients are described
in terms of parameters bj , cj , dj and the coefficient c′0 as follows:
P ′(u, v) = c′0zΦ(qz)
(u− qz)
b1b2b3d1d2d3q6
v3
+
(
c′1,2u− c′0
zΦ(qz)
b1b2b3d1d2d3q4
(
(d1 + d2 + d3)u
2(b2d2q + b3d3q + b1d1)z
))
v2
+
(
c′2,1u
2 + c′1,1u+ c′0zΦ(qz)
((
1
d1
+
1
d2
+
1
d3
)
u3
q2b1b2b3
− z
(
1
b1d1q
+
1
b2d2q2
+
1
b3d3q2
)))
v
+ c′0zΦ(qz)
(c1 + u)(c2 + u)(c3 + u)(z − u)
b1b2b3
.
The remaining 3 parameters c′1,1, c
′
1,2, c
′
2,1 are determined in terms of parameters bi, ci, di, q, z,
Φ
(
qiz
)
, c′0 by the property (iii). Namely, the property (iii) gives 3 linear inhomogeneous equa-
tions among c′1,1, c
′
1,2, c
′
2,1, c
′
1,1
∣∣
z→qz
, c′1,2
∣∣
z→qz
, c′2,1
∣∣
z→qz
. Though these relations are apparently
q-difference equations, we can solve them algebraically. For example, in the equation (3.12), we
solve c′12 when j = 2. Then when j = 0, we solve c′21|z→qz,Φ(qkz)→Φ(qk+1z). And finally solving c′11
when j = 1, they algebraically can be solved. ■
Explicit forms of the coefficients ci,j(z) of the polynomial P (u, v) are in Appendix B.
3.1 Relations among pairs of variables (f, g), (x, y) and (u, v)
Proposition 3.4. Under the relations among variables (f, g), (x, y) and (u, v):
f =
c1c2c3(b2 + x)(xy − 1)
b1b2y(c1 + x)(c2 + x) + c3x(b2(1− xy) + c2xy + c1y(c2 + x) + x)
,
g = − y(b1b2 + c3x)
c3(b2(1− xy) + x) + b1b2xy
, (3.13)
x =
K1(u, v)K2(u, v)
L1(u, v)
, y =
K3(u, v)
L2(u, v)
, (3.14)
K1(u, v) = b2v
(
v − q2(b2d2 + d3u)
)
,
K2(u, v) = b1b2
(
q2
(
d3u(d2q
2(c3 + u)− v)− d2v(b2 + u)
)
+ v2
)
+ c1c2d3q
2
(
b2d2q
2(c3 + u)− c3v
)
,
K3(u, v) = b2d2q
2u
(
d3q
2(c3 + u)− v
)
,
L1(u, v) = q2
(
b22d2
(
d2q
2
(
d3q
2(c1 + u)(c3 + u)− uv
)(
d3q
2(c2 + u)(c3 + u)− uv
)
+ v
(
d3q
2v(b1(2c3 + u) + u(c3 + 2u))− d23q
4u(b1 + u)(c3 + u)− uv2
))
− b32d
2
2q
2v(d3q
2(b1 + u)(c3 + u)− uv)− b2c3d3v
(
b1v
(
v − d3q
2u
)
+ d2q
2
(
d3q
2(c2u+ c1(2c2 + u))(c3 + u)− (c1 + c2)uv
))
+ c1c2c
2
3d
2
3q
2v2
)
,
L2(u, v) = c1c2d3q
2
(
b2d2q
2(c3 + u)− c3v
)
+ b1b2v
(
v − q2(b2d2 + d3u)
)
,
the corresponding Lax equations (2.4) with (2.17), (2.18) with (2.20) and (2.18) with (3.9),
(3.10) are equivalent. Conversely, such relations among (f, g), (x, y) and (u, v) are uniquely
determined as (3.13), (3.14).
Proof. Using the relation (3.13), we can check that the equation (2.4) with (2.17) is equivalent
to (2.18) with (2.20). Similarly, using the relation (3.14), we can check that the equation (2.18)
12 K. Park
with (2.20) is equivalent to (2.18) with (3.9), (3.10). 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 E6 was described. The linear differential equation of the Lax pair is as follows
d
dz
Ψ(z) = Ψ(z)
(
Ab
1
z
+
Ab
2
z − 1
)
, Ab
3 := −
(
Ab
1 +Ab
2
)
, (4.1)
where the matrices Ab
i (1 ≤ i ≤ 3) are 3× 3 matrices with different eigenvalues.
We show that the linear q-difference equation (2.1) reduces to the equation (4.1) via a continu-
ous limit q → 1. The equation (2.1) takes the following form after a scale transformation z → −z
and gauge transformations
(1− z)Ψ(qz) = Ψ(z)A(z), A(z) =
b1d1 k1 v1
0 b2d2 k2
0 0 b3d3
−
d1 0 0
v2 d2 0
v3 v4 d3
z,
|A(z)| = d1d2d3(z − c1)(z − c2)(z − c3), (4.2)
where kj (j = 1, 2) are constants. We put q = eh and consider the limit h → 0. We set
bi = qβi , ci = qγi , di = qδi , 1 ≤ i ≤ 3,
kj = hlj , j = 1, 2, vm = hum, 1 ≤ m ≤ 4, (4.3)
where lj are constants. By using Taylor’s expansion for (4.2), (4.3)
(l.h.s.) = (1− z)Ψ(z) + h(1− z)z
d
dz
Ψ(z) +O
(
h2
)
,
(r.h.s.) = (1− z)Ψ(z)
+ hΨ(z)
β1 − δ1(z − 1) l1 u1
−u2z β2 − δ2(z − 1) l2
−u3z −u4z β3 − δ3(z − 1)
+O
(
h2
)
,
we find the following limit as h → 0:
d
dz
Ψ(z) = Ψ(z)
(
A1
z
+
A2
z − 1
)
,
A1 =
β1 + δ1 l1 u1
0 β2 + δ2 l2
0 0 β3 + δ3
, A2 =
−β1 −l1 −u1
u2 −β2 −l2
u3 u4 −β3
,
A3 := −(A1 +A2) =
δ1 0 0
u2 δ2 0
u3 u4 δ3
, (4.4)
where eigenvalues of the matrix A2 are γi (1 ≤ i ≤ 3) by the condition of the determinant of the
matrix A(z) (4.2). Therefore, the linear q-difference equation (2.1) reduces to the equation (4.1)
via a continuous limit q → 1.
In the following, we consider a continuous limit q → 1 of the result in Section 2.1. In
Section 2.1, through a compatibility condition of the equations (2.11), we derived a standard
A 3× 3 Lax Form for the q-Painlevé Equation of Type E6 13
q-Painlevé equation of type E6. We take the following equation as a deformation equation for
the differential equation (4.4) which is rewritten version of (2.10):
TΨ(z) = B(z)Ψ(z), B(z) =
w1 w2 w3
0 0 0
0 0 0
+
w4 0 0
1 0 0
w5 w6 w7
z,
|B(z)| = (w3w6 − w2w7)z
2,
T : (β1, β2, β3, γ1, γ2, γ3, δ1, δ2, δ3) → (β1 − 1, β2 + 1, β3, γ1, γ2, γ3, δ1 + 1, δ2, δ3 + 1). (4.5)
Solving a compatibility condition for the equation (4.4), (4.5), we obtain the following additional-
difference Painlevé equation of type E6 [6, 10]:
(f + g)
(
f + g
)
=
(g + γ1)(g + γ2)(g + γ3)(g + β1 + δ1 − δ2)
(g + β2 + 1)(g + β3 − δ2 + δ3 + 1)
,
(
f + g
)(
f + g
)
=
(
f − γ1
)(
f − γ2
)(
f − γ3
)(
f − β1 − δ1 + δ2
)(
f − β1 + β2 − 1
)(
f − β1 − δ1 + δ3 + 1
) ,
where
f = β3 +
u1u4
l1
, g =
l1l2
u1
− β2,
and ∗ stands for T (∗). From the above, we derive the additional-difference Painlevé equation of
type E6 solving a compatibility condition of the Lax pair via a continuous limit q → 1.
A Deformations T1, T2 and T 3
2 on root variables
In this appendix, we show that how the deformations T1, T2 and T 3
2 act on root variables.
We consider a pair of root basis of {αi} (i = 0, 1, . . . , 6) and {δj} (j = 0, 1, 2) as symmetry
type E
(1)
6 and surface type A
(1)
2 , respectively.
α3α2α1 α4 α5
α6
α0
δ1 δ2
δ0
Figure 1. The Dynkin diagram of E
(1)
6 and A
(1)
2 .
Pictures of a point configuration of the equation (2.12) and the configuration (2.24) are
presented in Figures 2 and 3, respectively. From these pictures, we take a pair of root basis {αi}
and {δj} as symmetry type E
(1)
6 and surface type A
(1)
2 as follows:
α0 = E7 − E8, α1 = E6 − E5, α2 = H2 − E1 − E6, α3 = E1 − E2,
α4 = E2 − E3, α5 = E3 − E4, α6 = H1 − E1 − E7,
δ0 = H1 +H2 − E1 − E2 − E3 − E4, δ1 = H1 − E5 − E6, δ2 = H2 − E7 − E8,
δ = α0 + α1 + 2α2 + 3α3 + 2α4 + α5 + 2α6 = δ0 + δ1 + δ2, (A.1)
where we stand for δ as a null root. The choice of the above root basis is the same as in [6].
14 K. Park
g = 0
f = 0
fg = 1
p4 (c3,
1
c3
)
p3 (c2,
1
c2
)
p2 (c1,
1
c1
)
p1 ( b3d3d2
, d2
b3d3
)(0, 1
b2
) p6
(0, d2
b1d1
) p5
p7
(b3, 0)
p8
( qb3d3d1
, 0)
Figure 2. A point configuration of the equation (2.12).
y = 0
x = 0
xy = 1
q4 (−c2,− 1
c2
)
q3 (−c1,− 1
c1
)
q2 (− b1d1
d2
,− d2
b1d1
)
q1 (− b1b2
c3
,− c3
b1b2
)
(0,− 1
b1
) q6
(0,− b2d2
c1c2d3
) q5
q7
(−b2, 0)
q8
(− b1b2d1
qc3d3
, 0)
Figure 3. A point configuration (2.24).
A.1 A deformation T1 on root variables
We take variables ai (i = 0, 1, . . . , 6) as root variables attached to the root αi (A.1) associated
with a point configuration in coordinate (f, g) (see Figure 2)
a0 =
qd3
d1
, a1 =
b1d1
b2d2
, a2 =
b2d2
b3d3
, a3 =
b3d3
c1d2
,
a4 =
c1
c2
, a5 =
c2
c3
, a6 =
d2
d3
, (A.2)
which satisfy q = a0a1a
2
2a
3
3a
2
4a5a
2
6. Then we have the following statement.
Proposition A.1. The action T1 (2.10) on the root variables ai in (A.2) is given by the trans-
lation
T1(a0, a1, a2, a3, a4, a5, a6) = (a0, a1, qa2, a3, a4, a5, a6/q). (A.3)
Proof. Applying (2.10), we obtain the desired result (A.3). ■
A 3× 3 Lax Form for the q-Painlevé Equation of Type E6 15
A.2 Deformations T2 and T 3
2 on root variables
The deformation T2 (2.19) is not a translation on parameters bi, ci, dj but a deformation T 3
2
gives a translation on them
T 3
2 : (b1, b2, b3, c1, c2, c3, d1, d2, d3) →
(
qb1, qb2, b3, c1, c2, q
3c3,
d1
q
,
d2
q
,
d3
q
)
. (A.4)
We show that how the deformations T2 (2.19) and T 3
2 (A.4) on root variables.
We take variables a′i (i = 0, 1, . . . , 6) as root variables attached to the root αi (A.1) associated
with a point configuration in coordinate (x, y) (see Figure 3)
a′0 =
qc3d3
b1d1
, a′1 =
b1b2d2
c1c2d3
, a′2 =
b2
c3
, a′3 =
c3d1
b2d2
,
a′4 =
c1d2
b1d1
, a′5 =
c2
c1
, a′6 =
b1
c3
, (A.5)
which satisfy q = a′0a
′
1a
′
2
2a′3
3a′4
2a′5a
′
6
2. Then we have the following statement.
Proposition A.2. The actions T2 (2.19) and T 3
2 on the root variables a′i in (A.5) are given as
follows:
T2(a
′
0, a
′
1, a
′
2, a
′
3, a
′
4, a
′
5, a
′
6) =
(
q
a′6
, a′0
2
a′1a
′
2a
′
3a
′
6
2
,
1
a′0a
′
3a
′
6
,
q
a′2
,
a′0a
′
2a
′
3a
′
4a
′
6
q
, a′5,
a′2a
′
3a
′
6
q
)
,
T 3
2 (a
′
0, a
′
1, a
′
2, a
′
3, a
′
4, a
′
5, a
′
6) =
(
a′0q
2, a′1q
3,
a′2
q2
, a′3q
2,
a′4
q
, a′5,
a′6
q2
)
. (A.6)
Proof. Applying (2.19) and (A.4), we obtain the desired results (A.6). ■
B Explicit forms of coefficients in Section 3
In this appendix, we give explicit forms of Pj(z) (3.8) and the coefficients cij (0 ≤ j ≤ 3,
0 ≤ i ≤ 4− j) of the polynomial P (u, v) in variables u and v.
Explicit forms of Pj(z) (3.8) are as follows:
P3(z) = p31(z − u),
P2(z) = p22z
2 + p21z + p20,
P1(z) = p13z
3 + p12z
2 + p11z + p10,
P0(z) = −P3(qz)d1d2d3(z + c1)(z + c2)(z + c3),
where the coefficients pj,k (1 ≤ j ≤ 3, 0 ≤ k ≤ 3) are
p31 = −q2uvc1c2,
p22 = q4uvc1c2(d1 + d2 + d3),
p21 = −quvc1c2
(
ud3q
3 + (qu− b2)d2q
2 − b3d3q
2 − vq + v +
(
q3u− qb1
)
d1
)
,
p20 = −q2u2vc1c2(b1d1 + qb2d2 + qb3d3),
p13 = −q5uvc1c2(d1d2 + d2d3 + d3d1),
p12 = −q
(
ub1b2b3(u+ c2)d1d2d3q
5 + uc21c2(u+ c2)d1d2d3q
5
+ c1
(
u2c22d1d2d3q
5 + ub1b2b3d1d2d3q
5
− c2
(
qd1
(
qu
(
qud2
(
− ud3q
2 + vq + v
)
+ v(q(q + 1)ud3 − v)
)
16 K. Park
− b1
(
v − q2b2d2
)(
v − q2b3d3
))
+ v
(
d2
(
u(q(q + 1)ud3 − v)
+ b2
(
q2b3d3 − v
))
q2 + v
(
v − q2(u+ b3)d3
)))))
,
p11 = u
(
ub1b2b3(u+ c2)d1d2d3q
6 + uc21c2(u+ c2)d1d2d3q
6
+ c1
(
u2c22d1d2d3q
6 + ub1b2b3d1d2d3q
6
− c2
(
qd1
(
b1
(
b2d2
(
− b3d3q
3 + vq + v
)
q2 + v
(
q2(q + 1)b3d3 − v
))
− qu
(
v − q2ud2
)(
v − q2ud3
))
+ v
(
d2
(
u
(
q2ud3 − v
)
+ b2
(
q2(q + 1)b3d3 − v
))
q2 + v
(
v − q2(u+ b3)d3
)))))
,
p10 = q3u2vc1c2(qb2b3d2d3 + b1b2d1d2 + b1b3d1d3).
We give also explicit forms of the coefficients cij (0 ≤ j ≤ 3, 0 ≤ i + j ≤ 4) of the polynomial
P (u, v) in variables u and v:
c01 = −c0
z2
q2
(
q
b1d1
+
1
b2d2
+
1
b3d3
)
Φ(qz),
c02 = c0
z2(b1d1 + qb2d2 + qb3d3)Φ(qz)
q4b1b2b3d1d2d3
, c03 = −c0
z2Φ(qz)
q5b1b2b3d1d2d3
,
c10 = c0
(
z2
c1
+
z2
c2
+
z2
c3
− z
)
Φ(qz),
c11 = −c0z
(
(c1 + z)(c2 + z)(c3 + z)Φ(z)
b1b2b3q
−
(
1
b1d1q
+
1
b3d3q2
+
1
b2d2q2
+
1
b1d1q2
+
1
b3d3q3
+
1
b2d2q3
+
z2
b1b2b3d3q
+
z2
b1b2b3d2q
+
z2
b1b2b3d1q
)
Φ(qz)
+
(
d1 + d2 + d3
q2b1b2b3d1d2d3
z − b1d1 + qb2d2 + b3d3
q4b1b2b3d1d2d3
)
Φ
(
q2z
)
− 1
b1b2b3d1d2d3q4
Φ
(
q3z
))
,
c12 = c0z
((
d1 + d2 + d3
q3b1b2b3d1d2d3
z − b1d1 + qb2d2 + qb3d3
q5b1b2b3d1d2d3
)
Φ(qz) +
(−1 + q)
q5b1b2b3d1d2d3
Φ
(
q2z
))
,
c13 = c0z
Φ(qz)
q6b1b2b3d1d2d3
,
c20 = c0z
((
1
c1c2
+
1
c2c3
+
1
c3c1
)
z −
(
1
c1
+
1
c2
+
1
c3
))
Φ(qz),
c21 = c0
(
(c1 + z)(c2 + z)(c3 + z)
b1b2b3q2
Φ(z)
−
(
(1 + q)(d1d2 + d2d3 + d3d1)
q2b1b2b3d1d2d3
+
1
q3b3d3
+
1
q3b2d2
+
1
q2b1d1
)
Φ(qz)
+
(
d1 + d2 + d3
q2b1b2b3d1d2d3
z +
b1d1 + qb2d2 + qb3d3
q4b1b2b3d1d2d3
)
Φ
(
q2z
))
,
c22 = −c0
(d1 + d2 + d3)Φ(qz)
q4b1b2b3d1d2d3
zΦ(qz),
c30 = c0
(
z2
c1c2c3
−
(
1
c1c2
+
1
c2c3
+
1
c3c1
)
z
)
Φ(qz),
c31 =
(d1d2 + d2d3 + d3d1)z
b1b2b3d1d2d3q2
Φ(qz), c40 = −c0
zΦ(qz)
b1b2b3
.
A 3× 3 Lax Form for the q-Painlevé Equation of Type E6 17
Acknowledgements
The author would like to express her gratitude to Professor Yasuhiko Yamada for valuable sug-
gestions and encouragement. And the author is grateful to referees for giving helpful comments
to improve the manuscript. She also thanks supports from JSPS KAKENHI Grant Numbers
17H06127 and 26287018 for the travel expenses in accomplishing this study.
References
[1] Boalch P., Quivers and difference Painlevé equations, in Groups and Symmetries,CRM Proc. Lect. Notes,
Vol. 47, American Mathematical Society, Providence, RI, 2009, 25–51, arXiv:0706.2634.
[2] Dzhamay A., Knizel A., q-Racah ensemble and q-P(E
(1)
7 /A
(1)
1 ) discrete Painlevé equation, Int. Math. Res.
Not. 2020 (2020), 9797–9843, arXiv:1903.06159.
[3] Dzhamay A., Takenawa T., On some applications of Sakai’s geometric theory of discrete Painlevé equations,
SIGMA 14 (2018), 075, 20 pages, arXiv:1804.10341.
[4] Grammaticos B., Ramani A., The hunting for the discrete Painlevé equations, Regul. Chaotic Dyn. 5 (2000),
53–66.
[5] Jimbo M., Sakai H., A q-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145–154,
arXiv:chao-dyn/9507010.
[6] Kajiwara K., Noumi M., Yamada Y., Geometric aspects of Painlevé equations, J. Phys. A 50 (2017), 073001,
164 pages, arXiv:1509.08186.
[7] Knizel A., Moduli spaces of q-connections and gap probabilities, Int. Math. Res. Not. 2016 (2016), 6921–
6954, arXiv:1506.06718.
[8] Park K., A certain generalization of q-hypergeometric functions and their related connection preserving
deformation II, Funkcial. Ekvac. 65 (2022), 311–328, arXiv:2005.04992.
[9] Ramani A., Grammaticos B., Hietarinta J., Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67
(1991), 1829–1832.
[10] Ramani A., Grammaticos B., Tamizhmani T., Tamizhmani K.M., Special function solutions of the discrete
Painlevé equations, Comput. Math. Appl. 42 (2001), 603–614.
[11] Sakai H., A q-analog of the Garnier system, Funkcial. Ekvac. 48 (2005), 273–297.
[12] Sakai H., Lax form of the q-Painlevé equation associated with the A
(1)
2 surface, J. Phys. A 39 (2006),
12203–12210.
[13] Suzuki T., A q-analogue of the Drinfeld–Sokolov hierarchy of type A and q-Painlevé system, in Algebraic
and Analytic Aspects of Integrable Systems and Painlevé Equations, Contemp. Math., Vol. 651, American
Mathematical Society, Providence, RI, 2015, 25–38, arXiv:1105.4240.
[14] Suzuki T., A Lax formulation of a generalized q-Garnier system, Math. Phys. Anal. Geom. 24 (2021), 38,
12 pages, arXiv:2103.15336.
[15] Witte N.S., Ormerod C.M., Construction of a Lax pair for the E
(1)
6 q-Painlevé system, SIGMA 8 (2012),
097, 27 pages, arXiv:1207.0041.
[16] Yamada Y., Lax formalism for q-Painlevé equations with affine Weyl group symmetry of type E
(1)
n , Int.
Math. Res. Not. 2011 (2011), 3823–3838, arXiv:1004.1687.
https://doi.org/10.1090/crmp/047/04
https://arxiv.org/abs/0706.2634
https://doi.org/10.1093/imrn/rnz211
https://doi.org/10.1093/imrn/rnz211
https://arxiv.org/abs/1903.06159
https://doi.org/10.3842/SIGMA.2018.075
https://arxiv.org/abs/1804.10341
https://doi.org/10.1007/BF00398316
https://arxiv.org/abs/chao-dyn/9507010
https://doi.org/10.1088/1751-8121/50/7/073001
https://arxiv.org/abs/1509.08186
https://doi.org/10.1093/imrn/rnv366
https://arxiv.org/abs/1506.06718
https://doi.org/10.1619/fesi.65.311
https://arxiv.org/abs/2005.04992
https://doi.org/10.1103/PhysRevLett.67.1829
https://doi.org/10.1016/S0898-1221(01)00180-8
https://doi.org/10.1619/fesi.48.273
https://doi.org/10.1088/0305-4470/39/39/S13
https://doi.org/10.1090/conm/651/13037
https://doi.org/10.1090/conm/651/13037
https://arxiv.org/abs/1105.4240
https://doi.org/10.1007/s11040-021-09412-3
https://arxiv.org/abs/2103.15336
https://doi.org/10.3842/SIGMA.2012.097
https://arxiv.org/abs/1207.0041
https://doi.org/10.1093/imrn/rnq232
https://doi.org/10.1093/imrn/rnq232
https://arxiv.org/abs/1004.1687
1 Introduction
2 A 3 times 3 matrix Lax for
2.1 The deformation T_1
2.2 The deformation T_2
3 A scalar equation related to the q-E_6^(1)
3.1 Relations among pairs of variables (f, g), (x, y) and (u, v)
4 Continuous limit
A Deformations T_1, T_2 and T_2^3 on root variables
A.1 A deformation T_1 on root variables
A.2 Deformations T_2 and T_2^3 on root variables
B Explicit forms of coefficients in Section 3
References
|
| id | nasplib_isofts_kiev_ua-123456789-212037 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T14:41:36Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Park, Kanam 2026-01-23T10:10:27Z 2023 A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆. Kanam Park. SIGMA 19 (2023), 094, 17 pages 1815-0659 2020 Mathematics Subject Classification: 14H70; 34M56; 39A13 arXiv:2211.16706 https://nasplib.isofts.kiev.ua/handle/123456789/212037 https://doi.org/10.3842/SIGMA.2023.094 For the -Painlevé equation with affine Weyl group symmetry of type ⁽¹⁾₆, a 2 × 2 matrix Lax form and a second-order scalar lax form were known. We give a new 3 × 3 matrix Lax form and a third order scalar equation related to it. Continuous limit is also discussed. The author would like to express her gratitude to Professor Yasuhiko Yamada for valuable suggestions and encouragement. And the author is grateful to referees for giving helpful comments to improve the manuscript. She also thanks supports from JSPS KAKENHI Grant Numbers 17H06127 and 26287018 for the travel expenses in accomplishing this study. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆ Article published earlier |
| spellingShingle | A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆ Park, Kanam |
| title | A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆ |
| title_full | A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆ |
| title_fullStr | A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆ |
| title_full_unstemmed | A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆ |
| title_short | A 3 × 3 Lax Form for the -Painlevé Equation of Type ₆ |
| title_sort | 3 × 3 lax form for the -painlevé equation of type ₆ |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212037 |
| work_keys_str_mv | AT parkkanam a33laxformforthepainleveequationoftype6 AT parkkanam 33laxformforthepainleveequationoftype6 |