-Middle Convolution and -Painlevé Equation
A -deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear -difference equation associated with the -Painlevé VI equation. Then we obtain integral transformations. We investigate the -middle convolution in terms of the affine Weyl group symmetry of the -P...
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| description | A -deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear -difference equation associated with the -Painlevé VI equation. Then we obtain integral transformations. We investigate the -middle convolution in terms of the affine Weyl group symmetry of the -Painlevé VI equation. We deduce an integral transformation on the -Heun equation.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 056, 21 pages
q-Middle Convolution and q-Painlevé Equation
Shoko SASAKI a, Shun TAKAGI a and Kouichi TAKEMURA b
a) Department of Mathematics, Faculty of Science and Engineering, Chuo University,
1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
b) Department of Mathematics, Ochanomizu University,
2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan
E-mail: takemura.kouichi@ocha.ac.jp
Received January 31, 2022, in final form July 08, 2022; Published online July 20, 2022
https://doi.org/10.3842/SIGMA.2022.056
Abstract. A q-deformation of the middle convolution was introduced by Sakai and Yama-
guchi. We apply it to a linear q-difference equation associated with the q-Painlevé VI
equation. Then we obtain integral transformations. We investigate the q-middle convolution
in terms of the affine Weyl group symmetry of the q-Painlevé VI equation. We deduce an
integral transformation on the q-Heun equation.
Key words: q-Painlevé equation; q-Heun equation; middle convolution; integral transforma-
tion
2020 Mathematics Subject Classification: 33E10; 34M55; 39A13
1 Introduction
The middle convolution was introduced by Katz [8] for local systems on a punctured Riemann
sphere, and Dettweiler and Reiter [3, 4] reformulated it for the Fuchsian system of differential
equations. Here the Fuchsian system of differential equations is the system of linear differential
equations written as
dY
dx
=
(
A1
x− t1
+
A2
x− t2
+ · · ·+ Ar
x− tr
)
Y, (1.1)
where Y is a column vector with n entries and A1, A2, . . . , Ar are constant matrices of size n×n.
We review briefly the definition of the middle convolution for equation (1.1) (or the tuple of the
matrices (A1, . . . , Ar)). Let λ ∈ C and Fi, i = 1, . . . , r, be the matrix of size nr×nr of the form
Fi =
O · · · O · · · O
...
...
...
A1 · · · Ai + λIn · · · Ar
...
...
...
O · · · O · · · O
(i), (1.2)
where In is the identity matrix of size n. Then the correspondence of the tuple of matrices
(A1, . . . , Ar) 7→ (F1, . . . , Fr) (or the correspondence of the associated Fuchsian system) is called
the convolution. The convolution does not preserve the irreducibility in general. It is shown
that the following subspaces K,L of Cnr are preserved by the action of Fi, i = 1, . . . , r,
K =
kerA1
...
kerAr
, L = ker(F1 + F2 + · · ·+ Fr).
mailto:takemura.kouichi@ocha.ac.jp
https://doi.org/10.3842/SIGMA.2022.056
2 S. Sasaki, S. Takagi and K. Takemura
We denote the linear transformation induced from the action of Fi on the quotient space
Cnr/(K + L) by F i. The correspondence of the tuple of matrices (A1, . . . , Ar) 7→
(
F 1, . . . , F r
)
(or the correspondence of the associated Fuchsian system) is called the middle convolution.
It was shown in [3] that the convolution is related with Euler’s integral transformation. Let Y (x)
be a solution of equation (1.1). Set
Wj(x) =
Y (x)
x− tj
, W (x) =
W1(x)
...
Wr(x)
.
Then W (x) is a column vector with nr entries. We apply Euler’s integral transformation for
each entry of W (x), i.e., we set
G(x) =
∫
∆
W (s)(x− s)λ ds,
where ∆ is an appropriate cycle in C with the variable s. Then the function G(x) satisfies the
following Fuchsian system of differential equation
dY
dx
=
(
F1
x− t1
+
F2
x− t2
+ · · ·+ Fr
x− tr
)
Y,
where F1, . . . , Fr were defined in equation (1.2).
Sakai and Yamaguchi [10] constructed a theory of a q-deformation of the middle convolution
for systems of q-difference equations. Here the system is described as
Y (qx) =
(
B∞ +
B1
1− x/t1
+ · · ·+ Br
1− x/tr
)
Y (x),
where Y (x) is a column vector with n entries and B∞, B1, . . . , Br are constant matrices of size
n×n. The construction of the q-middle convolution is similar to the case of the Fuchsian system
of differential equations. For details, see Section 2.
In this paper we apply the q-middle convolution to linear q-difference equations which are
related to the q-Painlevé VI equation
yy
a3a4
=
(z − tb1)(z − tb2)
(z − b3)(z − b4)
,
zz
b3b4
=
(y − ta1)(y − ta2)
(y − a3)(y − a4)
, (1.3)
with the constraint b1b2a3a4 = qa1a2b3b4. Here y and z denotes the time evolution t 7→ qt
of y and z, and the parameters a1, . . . , a4, b1, . . . , b4 are time-independent. The q-Painlevé VI
equation was introduced by Jimbo and Sakai [6] as a q-deformation of the Painlevé VI equa-
tion. They obtained equation (1.3) by introducing a q-analogue of the monodromy preserving
deformation, and it is related to the linear q-difference equations
Y (qx) = A(x)Y (x), (1.4)
where A(x) is a 2× 2 matrix with polynomial entries (see equation (3.2) for details). The linear
q-difference equation which we apply the q-middle convolution is not equation (1.4) but the
transformed equation
Y (qx) = B(x)Y (x),
B(x) =
A(x)
c0(x− ta1)(x− ta2)
= B∞ +
B1
1− x/(ta1)
+
B2
1− x/(ta2)
(1.5)
q-Middle Convolution and q-Painlevé Equation 3
for some constant c0. By applying the q-middle convolution with the parameter λ, we obtain
2× 2 matrices, if we choose the constants c0 and λ suitably. Then we obtain a correspondence
of the parameters, and we may regard it as a correspondence of the q-Painlevé VI equations.
On the other hand, it is known that the q-Painlevé VI equation has a symmetry of the affine
Weyl group of type D
(1)
5 , and a realization of the symmetry was described in the review [7]
of Kajiwara, Noumi and Yamada. In this paper, we express the symmetry by the q-middle
convolution in terms of the generators of the affine Weyl group.
In [11], a relationship between the q-Painlevé VI equation and the q-Heun equation(
x− h1q
1/2
)(
x− h2q
1/2
)
g(x/q) + l3l4
(
x− l1q
−1/2
)(
x− l2q
−1/2
)
g(qx)
−
{
(l3 + l4)x
2 + Ex+ (l1l2l3l4h1h2)
1/2
(
h
1/2
3 + h
−1/2
3
)}
g(x) = 0
was studied from a viewpoint of the initial value space. In particular, the q-Heun equation
was obtained from the linear q-difference equation associated to the q-Painlevé VI equation by
specializing the parameters. On the other hand, the q-middle convolution induces an integral
transformation of the linear q-difference equation. By considering a particular specialization,
the linear q-difference equation turns out to be the q-Heun equation and we obtain an integral
transformation of q-Heun equation.
This paper is organized as follows. In Section 2, we review a part of the theory of the
q-middle convolution established by Sakai and Yamaguchi [10]. In Section 3, we recall the
linear q-difference equation associated to the q-Painlevé VI equation and calculate the q-middle
convolution for it. In Section 4, we investigate the symmetry by the q-middle convolution in
terms of the Weyl group symmetry of the q-Painlevé VI equation. For this purpose, we clarify
a relationship between equation (1.5) and the Lax pair in [7]. In Section 5, we obtain an integral
transformation on the q-Heun equation. In Section 6, we give concluding remarks.
2 q-middle convolution
We recall the q-convolution and the q-middle convolution introduced by Sakai and Yamagu-
chi [10].
Let B = (B∞;B1, . . . , BN ) be the tuple of the square matrices of the same size and b =
(b1, b2, . . . , bN ) be the tuple of the non-zero complex numbers which are different from one
another. We denote by EB,b the linear q-difference equations
Y (qx) = B(x)Y (x), B(x) = B∞ +
N∑
i=1
Bi
1− x/bi
.
Definition 2.1 (q-convolution, [10]). Let B = (B∞;B1, . . . , BN ) be the tuple of m×m matrices
and (b1, b2, . . . , bN ) be the tuple of the non-zero complex numbers which are different one another.
Set B0 = Im − B∞ − B1 − · · · − BN , We define the q-convolution cλ : (B∞;B1, . . . , BN ) 7→
(F∞;F1, . . . , FN ) as follows:
F = (F∞;F1, . . . , FN ) is a tuple of (N + 1)m× (N + 1)m matrices,
Fi =
O
B0 · · · Bi −
(
1− qλ
)
Im · · · BN
O
(i+1), 1 ≤ i ≤ N,
F∞ = I(N+1)m − F̂ , F̂ =
B0 · · · BN
...
. . .
...
B0 · · · BN
.
4 S. Sasaki, S. Takagi and K. Takemura
Let ξ ∈ C \ {0}. The q-convolution induces the q-analogue of the Euler’s integral transfor-
mation in terms of the Jackson integral∫ ξ∞
0
f(x) dqx = (1− q)
∞∑
n=−∞
qnξf(qnξ)
for the solutions of the q-difference equations. Note that the value of the Jackson integral may
depend on the value ξ.
Theorem 2.2 ([10, Theorem 2.1]). Let Y (x) be a solution of EB,b. Set b0 = 0 and
Pλ(x, s) =
(
qλ+1s/x; q
)
∞
(qs/x; q)∞
=
∞∏
i=0
x− qi+λ+1s
x− qi+1s
.
Define the function Ŷ (x) by
Ŷi(x) =
∫ ξ∞
0
Pλ(x, s)
s− bi
Y (s) dqs, i = 0, . . . , N, Ŷ (x) =
Ŷ0(x)
...
ŶN (x)
.
Then the function Ŷ (x) satisfies the equation EF,b, i.e.,
Ŷ (qx) =
(
F∞ +
N∑
i=1
Fi
1− x/bi
)
Ŷ (x).
Although the original theorem by Sakai and Yamaguchi was restricted to the case ξ = 1 in
the Jackson integral, we may just extend it to the case ξ ∈ C \ {0}, which was motivated by the
theory of the Jackson integral due to Aomoto [1]. The convergence of the Jackson integrals Ŷi(x),
i = 0, . . . , N , would not be considered in [10]. In this paper, we discuss the Jackson integrals
formally and we do not consider the convergence in details. Namely, we discuss the Jackson
integrals under the assumption that the integrals converge absolutely and Theorem 2.2 holds
true with the convergence. Thus we use the phrasing “formally” in theorems which are related
to the Jackson integral. An aspect of convergence on Theorem 2.2 will be discussed in [2].
The q-middle convolution is defined by considering an appropriate quotient space.
Definition 2.3 (q-middle convolution, [10]). We define the F -invariant subspaces K and L
of (Cm)N+1 as follows
K = KV =
kerB0
...
kerBN
, L = LV(λ) = ker
(
F̂ −
(
1− qλ
)
I(N+1)m
)
.
We denote the action of Fk on the quotient space (Cm)N+1/(K + L) by F k, k = ∞, 1, . . . , N .
Then the q-middle convolution mcλ is defined by the correspondence EB,b 7→ EF,b, where
F =
(
F∞;F 1, . . . , FN
)
.
The q-middle convolution mcλ would induce the integral transformation of the solutions by
applying the integral transformation on the q-convolution, although it would be necessary to
consider the subspace K + L ⊂ (Cm)N+1.
q-Middle Convolution and q-Painlevé Equation 5
3 Linear q-difference equation associated
to q-Painlevé VI equation and q-middle convolution
We recall the linear q-difference equation
Y (qx) = A(x)Y (x), (3.1)
which was discussed by Jimbo and Sakai [6] to obtain the q-Painlevé VI equation by the con-
nection preserving deformation.
We take the 2× 2 matrix A(x) in equation (3.1) to be of the form
A(x) = A0(t) +A1(t)x+A2x
2, A2 =
(
χ1 0
0 χ2
)
,
A0(t) has eigenvalues tθ1, tθ2,
detA(x) = χ1χ2(x− ta1)(x− ta2)(x− a3)(x− a4). (3.2)
Then we have the following relation
χ1χ2a1a2a3a4 = θ1θ2. (3.3)
Note that the relations to the parameter of the q-Painlevé VI equation in equation (1.3) are
given by b1 = a1a2/θ1, b2 = a1a2/θ2, b3 = 1/(qχ1), b4 = 1/χ2.
We need accessory parameters to determine uniquely the elements of the matrix A(x). Write
A(x) =
(
a11(x) a12(x)
a21(x) a22(x)
)
.
Then a12(x) is a linear polynomial. We introduce the parameters w, y, z and impose the
condition
a12(x) = χ2w(x− y), a11(x)|x=y = (y − ta1)(y − ta2)/(qz). (3.4)
Then the elements of A(x) are determined as
A(x) =
(
χ1((x− y)(x− α) + z1) χ2w(x− y)
χ1w
−1(γx+ δ) χ2((x− y)(x− β) + z2)
)
, (3.5)
where
α =
1
χ1 − χ2
[
y−1((θ1 + θ2)t− χ1z1 − χ2z2)− χ2((a1 + a2)t+ a3 + a4 − 2y)
]
,
β =
1
χ1 − χ2
[
−y−1((θ1 + θ2)t− χ1z1 − χ2z2) + χ1((a1 + a2)t+ a3 + a4 − 2y)
]
,
γ = z1 + z2 + (y + α)(y + β) + (α+ β)y − a1a2t
2 − (a1 + a2)(a3 + a4)t− a3a4,
δ = y−1
(
a1a2a3a4t
2 − (αy + z1)(βy + z2)
)
and
z1 =
(y − ta1)(y − ta2)
qχ1z
, z2 = qχ1(y − a3)(y − a4)z.
We consider the q-middle convolution for the q-difference equation
Y (qx) = B(x)Y (x), B(x) =
A(x)
c0(x− ta1)(x− ta2)
, (3.6)
6 S. Sasaki, S. Takagi and K. Takemura
where c0 is a constant which will be fixed later. Note that, if Ỹ (x) is a solution of equation (3.1)
and the parameter µ satisfies qµ = 1/
(
c0a1a2t
2
)
, then the function
Y (x) = xµ(x/(ta1); q)∞(x/(ta2); q)∞Ỹ (x)
satisfies equation (3.6). Write
B(x) = B∞ +
B1
1− x/(ta1)
+
B2
1− x/(ta2)
=
(
b11(x) b12(x)
b21(x) b22(x)
)
. (3.7)
Then
B∞ =
1
c0
(
χ1 0
0 χ2
)
, B2 = B1|a1↔a2 , (3.8)
B1 =
a2
tθ1(a1 − a2)
(
(y − ta1)b
[1]
1 /(qyz(χ1 − χ2)) wχ2(y − ta1)
−b
[1]
1 b
[1]
2 /
(
q2wy2z2(χ1 − χ2)
2
)
−χ2b
[1]
2 /(qyz(χ1 − χ2))
)
,
where
b
[1]
1 = q2χ2
1χ2(y − a3)(y − a4)z
2 + (y − ta2)(χ2y − χ1ta1)
− qχ1
{
2χ2y
2 − (χ1ta1 + χ2ta2 + χ2a3 + χ2a4)y + t(θ1 + θ2)
}
z,
b
[1]
2 = q2χ1(y − a3)(y − a4)(χ1y − χ2ta1)z
2 + (y − ta1)
2(y − ta2)
− q(y − ta1)
{
2χ1y
2 − (χ2ta1 + χ1ta2 + χ1a3 + χ1a4)y + t(θ1 + θ2)
}
z.
It is shown directly that detB1 = 0 and detB2 = 0. Set B0 = I2 −B∞ −B1 −B2(= I2 −B(0)).
Then the condition detB0 = 0 is equivalent to c0 = θ1/(ta1a2) or c0 = θ2/(ta1a2). We now
impose the condition detB0 = 0. For this purpose, we restrict to the case c0 = θ1/(ta1a2).
Note that the case c0 = θ2/(ta1a2) can be discussed by replacing the parameters as θ1 ↔ θ2.
We eliminate the parameter θ2 by equation (3.3). It follows from detB0 = 0, detB1 = 0 and
detB2 = 0 that there exists non-zero vectors ( v01v02 ), (
v11
v12 ), (
v21
v22 ) such that
B0
(
v01
v02
)
=
(
0
0
)
, B1
(
v11
v12
)
=
(
0
0
)
, B2
(
v21
v22
)
=
(
0
0
)
. (3.9)
We normalize the vectors by setting
v01 = qwyzθ1(χ1 − χ2), v11 = qwyzθ1χ2(χ1 − χ2), v21 = qwyzθ1χ2(χ1 − χ2). (3.10)
We now apply Definition 2.1. Namely we set
F1 =
O O O
B0 B1 −
(
1− qλ
)
I2 B2
O O O
, F2 =
O O O
O O O
B0 B1 B2 −
(
1− qλ
)
I2
,
F̂ =
B0 B1 B2
B0 B1 B2
B0 B1 B2
, F∞ = I6 − F̂ . (3.11)
The invariant subspaces K and L are described as
K = KV =
kerB0
kerB1
kerB2
, L = LV(λ) = ker
(
F̂ −
(
1− qλ
)
I6
)
. (3.12)
q-Middle Convolution and q-Painlevé Equation 7
Hence a basis of the space K is
v01
v02
0
0
0
0
,
0
0
v11
v12
0
0
,
0
0
0
0
v21
v22
.
If qλ = χ2ta1a2/θ1 (resp. qλ = χ1ta1a2/θ1) then dim(L) = 1 and the vector t(0, 1, 0, 1, 0, 1)
(resp. t(1, 0, 1, 0, 1, 0)) is a basis of the space L. Here we continue the discussion by setting qλ =
χ2ta1a2/θ1.
We introduce the matrix P by
P =
0 0 0 v01 0 0
g1 g3 1 v02 0 0
0 0 0 0 v11 0
g2 g4 1 0 v12 0
0 0 0 0 0 v21
0 0 1 0 0 v22
,
g1 = qzθ1 + ya2 − qyzχ1a2 − ta1a2,
g2 = y(qzχ1 − 1)(a1 − a2),
g3 = −a2(y − ta1),
g4 = y(a1 − a2).
(3.13)
Then detP = −q4w3y4z4θ31χ
2
2(χ1 − χ2)
3(a1 − a2)(χ1ta1a2 − θ1), and the matrix P is invertible
if detP ̸= 0. Set
F̃1 = P−1F1P, F̃2 = P−1F2P, F̃∞ = P−1F∞P.
Then it follows from the invariance of the space K + L that the i.j elements of these matrices
for i ∈ {1, 2} and j ∈ {3, 4, 5, 6} are equal to zero. Thus, they admit the following expression:
F̃1 =
(
F 1 O
∗
)
, F̃2 =
(
F 2 O
∗ ∗
)
, F̃∞ =
(
F∞ O
∗
)
,
where F 1, F 2, F∞ are 2×2 matrices. The matrix F∞ is diagonal, which follows from the choice
of the parameters g1, . . . , g4. We can restrict the q-difference equation
Y̌ (qx) =
(
F̃∞ +
F̃1
1− x/(ta1)
+
F̃2
1− x/(ta2)
)
Y̌ (x) (3.14)
of size 6 to that of size 2 by choosing the first two components and we write
Y (qx) = F (x)Y (x), F (x) = F∞ +
F 1
1− x/(ta1)
+
F 2
1− x/(ta2)
. (3.15)
Then
F∞ =
(
χ1ta1a2/θ1 0
0 1
)
, F 2 = F 1|a1↔a2 ,
F 1 =
a1
qyzθ1(a1 − a2)(θ1 − χ1ta1a2)
(
−(y − ta1)f
[1]
1 a22 −f
[1]
2 (y − ta1)a2
f
[1]
1 f
[1]
3 a2 f
[1]
2 f
[1]
3
)
, (3.16)
8 S. Sasaki, S. Takagi and K. Takemura
where
f
[1]
1 = q2χ2
1χ2(y − a3)(y − a4)z
2 + (y − ta2)(yχ2 − χ1ta1)
− qχ1{2χ2y
2 − (χ1ta1 + χ2ta2 + χ2a3 + χ2a4)y + t(θ1 + θ2)}z,
f
[1]
2 = qχ1χ2a2(y − a3)(y − a4)z − (y − ta2)(a2χ2y − θ1),
f
[1]
3 = q(ya2χ1 − θ1)z − a2(y − ta1).
It is expected that the q-middle convolution induces an integral transformation. Let Y (x) be
a solution to Y (qx) = B(x)Y (x) in equation (3.6) and write
Y (x) =
(
y1(x)
y2(x)
)
.
It follows from Theorem 2.2 that the q-difference equation
Ŷ (qx) = F (x)Ŷ (x), F (x) = F∞ +
F1
1− x/(ta1)
+
F2
1− x/(ta2)
has a solution written as
Ŷ (x) =
Ŷ0(x)
Ŷ1(x)
Ŷ2(x)
=
∫ ξ∞
0 s−1Pλ(x, s)y1(s) dqs∫ ξ∞
0 s−1Pλ(x, s)y2(s) dqs∫ ξ∞
0 (s− ta1)
−1Pλ(x, s)y1(s) dqs∫ ξ∞
0 (s− ta1)
−1Pλ(x, s)y2(s) dqs∫ ξ∞
0 (s− ta2)
−1Pλ(x, s)y1(s) dqs∫ ξ∞
0 (s− ta2)
−1Pλ(x, s)y2(s) dqs
. (3.17)
Set Y̌ (x) = P−1Ŷ (x). Then it satisfies equation (3.14). Write
Y̌ (x) =
y̌1(x)
y̌2(x)
...
y̌6(x)
, P−1 =
p11 p12 · · · p16
p21 p22 · · · p26
...
...
. . .
...
p61 p62 · · · p66
.
Then the function Y (x) =
(
y̌1(x)
y̌2(x)
)
satisfies equation (3.15). On the other hand, it follows from
equation (3.17) that
y̌1(x) =
∫ ξ∞
0
{(
p11
s
+
p13
s− ta1
+
p15
s− ta2
)
y1(s) +
(
p12
s
+
p14
s− ta1
+
p16
s− ta2
)
y2(s)
}
× Pλ(x, s) dqs.
By a straightforward calculation, we have
p12
s
+
p14
s− ta1
+
p16
s− ta2
=
tθ1
qwyzχ2(χ1ta1a2 − θ1)s
b12(s),
p11
s
+
p13
s− ta1
+
p15
s− ta2
=
tθ1
qwyzχ2(χ1ta1a2 − θ1)s
(b11(s)− 1),
q-Middle Convolution and q-Painlevé Equation 9
where b12(s) and b11(s) are elements of the matrix B(s) in equation (3.7). Therefore(
p11
s
+
p13
s− ta1
+
p15
s− ta2
)
y1(s) +
(
p12
s
+
p14
s− ta1
+
p16
s− ta2
)
y2(s)
=
tθ1
qwyzχ2(χ1ta1a2 − θ1)s
{−y1(s) + b11(s)y1(s) + b12(s)y2(s)}.
Hence it follows from y1(qs) = b11(s)y1(s) + b12(s)y2(s) that
y̌1(x) =
tθ1
qwyzχ2(χ1ta1a2 − θ1)
∫ ξ∞
0
y1(qs)− y1(s)
s
Pλ(x, s) dqs.
We are going to obtain the integral representation without using y1(qs). It follows from the
definitions of the Jackson integral and the function Pλ(x, s) that∫ ξ∞
0
y1(qs)Pλ(x, s)
dqs
s
=
∫ ξ∞
0
y1(s)Pλ(x, s/q)
dqs
s
=
∫ ξ∞
0
y1(s)
x− qλs
x− s
Pλ(x, s)
dqs
s
.
See [12] for details. Hence∫ ξ∞
0
y1(qs)− y1(s)
s
Pλ(x, s) dqs =
∫ ξ∞
0
1
s
(
x− qλs
x− s
− 1
)
y1(s)Pλ(x, s) dqs
= (1− qλ)
∫ ξ∞
0
y1(s)
x− s
Pλ(x, s) dqs.
We recall that qλ = χ2ta1a2/θ1. Thus, we obtain
y̌1(x) =
t(χ2ta1a2 − θ1)
qwyzχ2(χ1ta1a2 − θ1)
∫ ξ∞
0
y1(s)
s− x
Pλ(x, s) dqs. (3.18)
We also calculate y̌2(x). It follows from equation (3.17) that
y̌2(x) =
∫ ξ∞
0
{(
p21
s
+
p23
s− ta1
+
p25
s− ta2
)
y1(s) +
(
p22
s
+
p24
s− ta1
+
p26
s− ta2
)
y2(s)
}
× Pλ(x, s) dqs.
By a straightforward calculation, we have
p22
s
+
p24
s− ta1
+
p26
s− ta2
=
θ1{(ta1a2 − qzθ1)s+ yta1a2(qzχ1 − 1)}
qwyzχ2a1a2(χ1ta1a2 − θ1)(s− y)s
b12(s),
p21
s
+
p23
s− ta1
+
p25
s− ta2
=
θ1{(ta1a2 − qzθ1)s+ yta1a2(qzχ1 − 1)}
qwyzχ2a1a2(χ1ta1a2 − θ1)(s− y)s
b11(s)
− t{(ta1a2 − qzθ1)χ1s+ yθ1(qzχ1 − 1)}
qwyzχ2(χ1ta1a2 − θ1)(s− y)s
.
Therefore it follows from y1(qs) = b11(s)y1(s) + b12(s)y2(s) that(
p21
s
+
p23
s− ta1
+
p25
s− ta2
)
y1(s) +
(
p22
s
+
p24
s− ta1
+
p26
s− ta2
)
y2(s)
=
θ1{(ta1a2 − qzθ1)s+ yta1a2(qzχ1 − 1)}
qwyzχ2a1a2(χ1ta1a2 − θ1)(s− y)s
y1(qs)
− t{(ta1a2 − qzθ1)χ1s+ yθ1(qzχ1 − 1)}
qwyzχ2(χ1ta1a2 − θ1)(s− y)s
y1(s).
10 S. Sasaki, S. Takagi and K. Takemura
We obtain that∫ ξ∞
0
(ta1a2 − qzθ1)s+ ta1a2y(qzχ1 − 1)
s− y
y1(qs)Pλ(x, s)
dqs
s
=
∫ ξ∞
0
(ta1a2 − qzθ1)s+ qta1a2y(qzχ1 − 1)
s− qy
y1(s)Pλ(x, s/q)
dqs
s
=
∫ ξ∞
0
(ta1a2 − qzθ1)s+ qta1a2y(qzχ1 − 1)
s− qy
(
1 +
(1− qλ)s
x− s
)
y1(s)Pλ(x, s)
dqs
s
.
Hence
y̌2(x) =
1
qwyzχ2a1a2
∫ ξ∞
0
{
qzθ1
s− qy
− ta1a2
s− y
+
(
qzθ1 − ta1a2
χ1ta1a2 − θ1
− q2yz
s− qy
)
χ2ta1a2 − θ1
x− s
}
× y1(s)Pλ(x, s) dqs. (3.19)
In summary, we obtain the following theorem by the q-middle convolution.
Theorem 3.1. Let Y (x) be a solution to
Y (qx) =
(
B∞ +
B1
1− x/(ta1)
+
B2
1− x/(ta2)
)
Y (x), Y (x) =
(
y1(x)
y2(x)
)
, (3.20)
where B∞, B1 and B2 are defined in equation (3.8). The function
Y (x) =
(
y̌1(x)
y̌2(x)
)
defined by equations (3.18) and (3.19) formally satisfies
Y (qx) =
(
F∞ +
F 1
1− x/(ta1)
+
F 2
1− x/(ta2)
)
Y (x), (3.21)
where F∞, F 1 and F 2 are defined in equation (3.16).
Thus, we obtain the correspondence of the systems of linear q-difference equations associated
with the q-Painlevé VI equation by the q-middle convolution. To give the correspondence of the
parameters by the q-middle convolution in the form of the equation Y (qx) =
{
A0(t) + A1(t)x
+A2x
2
}
Y (x) in equation (3.2), we need to transform equation (3.21).
Let c̃ be a non-zero constant which will be fixed later. Set
Ã(x) = c̃(x− ta1)(x− ta2)
(
F∞ +
F 1
1− x/(ta1)
+
F 2
1− x/(ta2)
)
(3.22)
and write Ã(x) = Ã(x, t) = Ã0(t) + Ã1(t)x+ Ã2x
2. Then we have
Ã2 =
(
c̃χ1ta1a2/θ1 0
0 c̃
)
,
Ã0(t) has the eigenvalues
c̃χ2t
3a21a
2
2
θ1
and
c̃t2a1a2θ2
θ1
,
det Ã(x, t) =
c̃2χ1ta1a2
θ1
(x− ta1)(x− ta2)
(
x− χ2ta1a2a3
θ1
)(
x− χ2ta1a2a4
θ1
)
.
q-Middle Convolution and q-Painlevé Equation 11
Hence the action of the q-middle convolution to the parameters is described as
χ1 →
c̃χ1ta1a2
θ1
, χ2 → c̃, {tθ1, tθ2} →
{
c̃t2a1a2θ2
θ1
,
c̃χ2t
3a21a
2
2
θ1
}
,
{a1, a2} → {a1, a2}, {a3, a4} →
{
χ2ta1a2a3
θ1
,
χ2ta1a2a4
θ1
}
. (3.23)
We investigate the action to the parameters y and z. Recall that y and z are determined by
equation (3.4). We denote the images of y and z by ỹ and z̃. Let ã11(x) (resp. ã12(x)) be the
upper left entry (resp. the upper right entry) of the matrix Ã(x). Then the value ỹ is the zero
of the linear function ã12(x), and we have
ỹ =
χ2ta1a2{qχ1(y − a3)(y − a4)z − (y − ta1)(y − ta2)}
qχ1χ2ta1a2(y − a3)(y − a4)z − θ1(y − ta1)(y − ta2)
y
=
qz
(y − a3)(y − a4)
(y − ta1)(y − ta2)
− 1
χ1
qz
(y − a3)(y − a4)
(y − ta1)(y − ta2)
− θ1
χ1χ2ta1a2
y. (3.24)
The value z̃ satisfies ã11(x)|x=ỹ = (ỹ − ta1)(ỹ − ta2)/(qz̃). Since
ã11(x)|x=ỹ =
c̃
qχ2z
(y − ta1)(y − ta2)
(y − a3)(y − a4)
(
ỹ − χ2ta1a2a3
θ1
)(
ỹ − χ2ta1a2a4
θ1
)
,
we have
z̃ = z
χ2
c̃
(y − a3)(y − a4)
(y − ta1)(y − ta2)
(ỹ − ta1)(ỹ − ta2)
(ỹ − χ2ta1a2a3/θ1)(ỹ − χ2ta1a2a3/θ1)
. (3.25)
4 q-middle convolution and Weyl group symmetry
of q-Painlevé VI equation
We investigate the transformation of the parameters induced by the q-middle convolution in
terms of the Weyl group symmetry associated with the q-Painlevé VI equation. Kajiwara,
Noumi and Yamada gave a survey on discrete Painlevé equations in [7]. They presented a list of
the Weyl group symmetry and a Lax pair for each discrete Painlevé equation. The q-Painlevé VI
equation by Jimbo and Sakai [6] corresponds to the equation q-P
(
D
(1)
5
)
in [7]. In this section,
we make a correspondence between the parameters of the q-Painlevé VI equation and those of
the equation q-P
(
D
(1)
5
)
.
The equation q-P
(
D
(1)
5
)
was obtained by the compatibility condition of the Lax pair L1
and L2 in [7]. The operator L1 is defined by
L1y(x) =
{
x(gν1 − 1)(gν2 − 1)
qg
− ν1ν2ν3ν4(g − ν5/κ2)(g − ν6/κ2)
fg
}
y(x)
+
ν1ν2(x− qν3)(x− qν4)
q(qf − x)
(gy(x)− y(x/q))
+
(x− κ1/ν7)(x− κ1/ν8)
q(f − x)
(
1
g
y(x)− y(qx)
)
, (4.1)
which is independent from the time evolution. Here the parameters are constrained by the
relation
κ21κ
2
2 = qν1ν2ν3ν4ν5ν6ν7ν8. (4.2)
In this paper, we do not use the operator L2, which contains the operation of the time evolution.
12 S. Sasaki, S. Takagi and K. Takemura
The correspondence between the parameters of the q-Painlevé VI equation and those of the
equation q-P
(
D
(1)
5
)
was made by considering the linear q-difference equation Y (qx) = A(x)Y (x)
in equation (3.1) and L1y(x) = 0 in equation (4.1).
For the system of q-difference equation
Y (qx) = A(x)Y (x), A(x) =
(
a11(x) a12(x)
a21(x) a22(x)
)
, Y (x) =
(
y1(x)
y2(x)
)
,
we calculate the q-difference equation for y1(x) by eliminating y2. The system of the q-difference
equation is written as
y1(qx) = a11(x)y1(x) + a12(x)y2(x),
y2(qx) = a21(x)y1(x) + a22(x)y2(x).
We substitute y2(x)= a21(x/q)y1(x/q)+a22(x/q)y2(x/q) into y1(qx)= a11(x)y1(x)+a12(x)y2(x).
Then we have
y1(qx) = a11(x)y1(x) + a12(x)a21(x/q)y1(x/q) + a12(x)a22(x/q)y2(x/q).
We elimilate y2(x/q) by the relation y2(x/q) = {y1(x)− a11(x/q)y1(x/q)}/a12(x/q). Thus,
y1(qx)
a12(x)
−
{
a11(x)
a12(x)
+
a22(x/q)
a12(x/q)
}
y1(x) +
a11(x/q)a22(x/q)− a12(x/q)a21(x/q)
a12(x/q)
y1(x/q) = 0.
We restrict it to the case that the matrix elements are fixed to equation (3.5). Then it follows
from equations (3.2) and (3.5) that
y1(qx)
x− y
−
{
a11(x)
x− y
+
a22(x/q)
x/q − y
}
y1(x)
+
χ1χ2(x/q − ta1)(x/q − ta2)(x/q − a3)(x/q − a4)
x/q − y
y1(x/q) = 0. (4.3)
Let ϕ(x) be the function such that ϕ(qx) = d0(x − ta1)(x − ta2)ϕ(x). Set u(x) = y1(x)/ϕ(x).
Then it follows from equation (4.3) that the function u(x) satisfies
(x− ta1)(x− ta2)
x− y
d0u(qx)−
{
a11(x)
x− y
+
a22(x/q)
x/q − y
}
u(x)
+
χ1χ2(x/q − a3)(x/q − a4)
x/q − y
1
d0
u(x/q) = 0.
We rewrite it by using equation (3.5) as
(x− ta1)(x− ta2)
x− y
{
d0u(qx)−
u(x)
qz
}
+
χ1χ2(x/q − a3)(x/q − a4)
x/q − y
{
1
d0
u(x/q)− qzu(x)
}
−
{
χ1(x− α) +
χ1z1
x− y
+ χ2(x/q − β) +
χ2z2
x/q − y
− (x− ta1)(x− ta2)
qz(x− y)
− qzχ1χ2(x/q − a3)(x/q − a4)
x/q − y
}
u(x) = 0. (4.4)
q-Middle Convolution and q-Painlevé Equation 13
It is seen that the poles x = y and x = qy are cancelled on the coefficient of u(x). By a straight-
forward calculation (see [12] for details), equation (4.4) is written as{
x(qzχ1 − 1)(zχ2 − 1)
q2z
− χ1χ2a3a4
(qz − ta1a2/θ1)(qz − ta1a2/θ2)
q2yz
}
u(x)
+
χ1χ2(a3 − x/q)(a4 − x/q)
qy − x
{
qzu(x)− 1
d0
u(x/q)
}
+
(x− ta1)(x− ta2)
q(y − x)
{
u(x)
qz
− d0u(qx)
}
= 0. (4.5)
We compare equation (4.5) for the case d0 = 1 with equation (4.1). Then we obtain the following
correspondence:
qz = g, y = f, χ1 = ν1, χ2 = qν2,
ta1a2
θ1
=
ν5
κ2
,
ta1a2
θ2
=
ν6
κ2
,
a3 = ν3, a4 = ν4, ta1 =
κ1
ν7
, ta2 =
κ1
ν8
. (4.6)
On the relation of the parameters, the condition χ1χ2a1a2a3a4 = θ1θ2 is equivalent to equa-
tion (4.2). Thus, we obtained a correspondence between the parameters of the q-Painlevé VI
equation in [6] and those of the equation q-P
(
D
(1)
5
)
in [7].
Sakai [9] established that each discrete Painlevé equation has the symmetry in terms of the
affine Weyl group, and the description of the symmetry was reviewed explicitly by Kajiwara,
Noumi and Yamada in [7]. The q-Painlevé VI equation has the symmetry of the affine Weyl
group of the type D
(1)
5 . We describe the action of the operators s0, . . . , s5 for the parameters
(κ1, κ2, ν1, . . . , ν8) ∈ (C×)10 and (f, g) ∈ P1 × P1 as follows
s0 : ν7 ↔ ν8, s1 : ν3 ↔ ν4, s4 : ν1 ↔ ν2, s5 : ν5 ↔ ν6,
s2 : ν3 →
k1
ν7
, ν7 →
k1
ν3
, k2 →
k1k2
ν3ν7
, g → g
f − ν3
f − k1/ν7
,
s3 : ν1 →
k2
ν5
, ν5 →
k2
ν1
, k1 →
k1k2
ν1ν5
, f → f
g − 1/ν1
g − ν5/k2
. (4.7)
The omitted variables are invariant by the action, i.e., s2(f) = f . Then we can confirm that
these operations satisfy the relations of the Weyl group W
(
D
(1)
5
)
whose Dynkin diagram is as
follows i
i
i
i
i i
0
1
4
5
2 3
On the other hand, the q-middle convolution induces the transformation of the parameters
of the q-Painlevé VI equation given in equation (3.23), although there was an arbitrary param-
eter c̃. We describe it in terms of the Weyl group action by specializing the parameter c̃ in
equation (3.22).
Proposition 4.1. We specify the parameter c̃ in equation (3.22) by setting c̃ = χ2. Then the
transformation of the parameters of the q-Painlevé VI equation which is induced by the q-middle
convolution coincides with the action
s5s2s1s0s2s3s2s0s1s2 (4.8)
14 S. Sasaki, S. Takagi and K. Takemura
by the generators of W
(
D
(1)
5
)
, and it is written as
ν1 → q
ν1ν2ν5
κ2
, ν2 → ν2,
κ1
ν7
→ κ1
ν7
,
κ1
ν8
→ κ1
ν8
,
ν3 → q
ν2ν3ν5
κ2
, ν4 → q
ν2ν4ν5
κ2
,
κ2
ν5
→ q
ν2ν5
ν6
,
κ2
ν6
→ q2
ν22ν5
κ2
,
f → f̃ =
(f − ν3)(f − ν4)
(f − κ1/ν7)(f − κ1/ν8)
g − 1
ν1
(f − ν3)(f − ν4)
(f − κ1/ν7)(f − κ1/ν8)
g − κ2
qν1ν2ν5
f,
g → g̃ =
(f − ν3)(f − ν4)
(f − κ1/ν7)(f − κ1/ν8)
(f̃ − κ1/ν7)(f̃ − κ1/ν8)
(f̃ − qν2ν3ν5/κ2)(f̃ − qν2ν4ν5/κ2)
g. (4.9)
Proof. Set c̃ = χ2. Then it follows from equations (3.23), (3.24) and (3.25) that we may write
the transformation of the parameters induced by the q-middle convolution as
χ1 →
χ1χ2ta1a2
θ1
, χ2 → χ2, a1 → a1, a2 → a2,
a3 →
χ2ta1a2a3
θ1
, a4 →
χ2ta1a2a4
θ1
, tθ1 →
χ2t
2a1a2θ2
θ1
, tθ2 →
χ2
2t
3a21a
2
2
θ1
,
y → ỹ =
qz
(y − a3)(y − a4)
(y − ta1)(y − ta2)
− 1
χ1
qz
(y − a3)(y − a4)
(y − ta1)(y − ta2)
− θ1
χ1χ2ta1a2
y,
z → z̃ =
(y − a3)(y − a4)
(y − ta1)(y − ta2)
(ỹ − ta1)(ỹ − ta2)
(ỹ − χ2ta1a2a3/θ1)(ỹ − χ2ta1a2a3/θ1)
z. (4.10)
By the correspondence in equation (4.6), it is rewritten as equation (4.9).
We show that the transformation of the parameters given in equation (4.9) coincides with
the consequence of the action given in equation (4.8). We apply the operation s2s0s1s2 to f
and g. Then
s2s0s1s2(f) = f, s2s0s1s2(g) =
(f − ν3)(f − ν4)
(f − κ1/ν7)(f − κ1/ν8)
g.
Note that we define the composition of the transformations as automorphisms of the algebra
(symbolical composition in [7, Remark 2.1]). Since s3(f) = f(g − 1/ν1)/(g − ν5/k2), we have
s2s0s1s2s3(f) =
(f − ν3)(f − ν4)
(f − κ1/ν7)(f − κ1/ν8)
g − 1
ν1
(f − ν3)(f − ν4)
(f − κ1/ν7)(f − κ1/ν8)
g − κ2
qν1ν2ν6
f.
Here we used equation (4.2). By comparing it with equation (4.9), we obtain
s5s2s0s1s2s3(f) = f̃ .
We consider the operation to g. Since g is invariant under the actions of s3 and s5, we have
s5s2s0s1s2s3(g) = s2s0s1s2(g) =
(f − ν3)(f − ν4)
(f − κ1/ν7)(f − κ1/ν8)
g.
q-Middle Convolution and q-Painlevé Equation 15
We set s = s5s2s0s1s2s3. Then
s5s2s1s0s2s3s2s0s1s2(g) =
s(f)− s(ν3)
s(f)− s(κ1/ν7)
s(f)− s(ν4)
s(f)− s(κ1/ν8)
s(g)
=
f̃ − s(ν3)
f̃ − sκ1/ν7)
f̃ − s(ν4)
f̃ − s(κ1/ν8)
f − ν3
f − κ1/ν7
f − ν4
f − κ1/ν8
g.
It follows from equation (4.2) that
s
(
κ1
ν7
)
=
qν2ν4ν5
κ2
, s
(
κ1
ν8
)
=
qν2ν3ν5
κ2
, s(ν3) =
κ1
ν8
, s(ν4) =
κ1
ν7
.
Hence
s5s2s1s0s2s3s2s0s1s2(g) = g̃,
s5s2s1s0s2s3s2s0s1s2(f) = s5s2s0s1s2s3(f) = f̃ .
The action of the operation s5s2s1s0s2s3s2s0s1s2 to the other parameters is given by
κ1 →
qκ1ν2ν5
κ2
, κ2 →
q2ν22ν
2
5
κ2
,
ν1 →
qν1ν2ν5
κ2
, ν2 → ν2, ν3 →
qν2ν3ν5
κ2
, ν4 →
qν2ν4ν5
κ2
,
ν5 →
qν2ν5ν6
κ2
, ν6 → ν5, ν7 →
qν2ν5ν7
κ2
, ν8 →
qν2ν5ν8
κ2
,
and it recovers equation (4.9). ■
5 Integral transformation on q-Heun equation
We interpret the integral transformation in Theorem 3.1 as the one for solutions of the single
second-order linear q-difference equations.
Proposition 5.1.
(i) The function y1(x) in equation (3.20) satisfies{
x(qχ1z − 1)(χ2z − 1)
z
− χ1χ2a3a4
(qz − ta1a2/θ1)(qz − ta1a2/θ2)
yz
}
y1(x)
+
χ1χ2(x− qa3)(x− qa4)
qy − x
{
qzy1(x)−
ta1a2
θ1
y1(x/q)
}
+
q(x− ta1)(x− ta2)
y − x
{
y1(x)
qz
− θ1
ta1a2
y1(qx)
}
= 0. (5.1)
(ii) The function y̌1(x) in equation (3.21) satisfies{
x(qtθ2z̃ − a3a4)(χ2z̃ − 1)
a3a4z̃
−
t2a1a2(qχ2θ2z̃ − θ1)
(
qχ2
2ta1a2z̃ − θ1
)
θ21ỹz̃
}
y̌1(x)
+
χ2tθ2(x− qtθ2/(χ1a4))(x− qtθ2/(χ1a3))
a3a4(qỹ − x)
{
qz̃y̌1(x)−
1
χ2
y̌1(x/q)
}
+
q(x− ta1)(x− ta2)
ỹ − x
{
y̌1(x)
qz̃
− χ2y̌1(qx)
}
= 0, (5.2)
where ỹ and z̃ are determined by equation (4.10).
16 S. Sasaki, S. Takagi and K. Takemura
Proof. It follows from equation (3.6) that the function y1(x) in equation (3.20) satisfies equa-
tion (4.5) with the condition d0 = c0 = θ1/(ta1a2). Then we obtain (i).
Equation (3.21) is obtained as the form of equation (3.20) by replacing the parameters as
(y, z) → (ỹ, z̃) and equation (3.23) up to the ambiguity of the parameter w, and the q-difference
equation for y̌1(x) does not depend on the parameter w. Hence (ii) follows from equation (3.22)
with the condition c̃ = χ2. ■
Theorem 5.2. Assume that y1(x) is a solution to equation (5.1) and λ satisfies qλ = χ2a1a2t/θ1.
Then the function
y̌1(x) =
∫ ξ∞
0
y1(s)
s− x
Pλ(x, s) dqs (5.3)
formally satisfies equation (5.2).
Proof. Let bij(x) (i, j ∈ {1, 2}) be the elements of the matrix B(x) in equation (3.7). The
function y1(x) is given as a solution to equation (5.1). Define the function y2(x) by y1(qx) =
b11(x)y1(x) + b12(x)y2(x). Since equation (5.1) is written as
y1(qx)
b12(x)
−
{
b11(x)
b12(x)
+
b22(x/q)
b12(x/q)
}
y1(x) +
b11(x/q)b22(x/q)− b12(x/q)b21(x/q)
b12(x/q)
y1(x/q) = 0,
we obtain the equality y2(x) = b21(x/q)y1(x/q) + b22(x/q)y2(x/q). Then the function Y (x) =
t(y1(x), y2(x)) satisfies equation (3.20). Hence it follows from Theorem 2.2 that the function
Y̌ (x) = t(y̌1(x), y̌2(x)) satisfies equation (3.21). Therefore the function y̌1(x) in equation (5.3)
satisfies equation (5.2) by Proposition 5.1(ii). ■
Corollary 5.3. Let µ, µ′ and λ be the constants such that qµ = ν5/κ2, qµ
′
= qν2 and
qλ = qν2ν5/κ2. Let y(x) be a solution to the equation L1y(x) = 0 which was described in
equation (4.1). Then the function
y̌(x) = xµ
′
∫ ξ∞
0
y(s)
s− x
sµPλ(x, s) dqs (5.4)
formally satisfies the equation L̃1y̌(x) = 0, where the operator L̃1 is obtained from L1 by replacing
the parameters in accordance with the action of s5s2s1s0s2s3s2s0s1s2 (see equation (4.9)).
Proof. Let y(x) be a solution to the equation L1y(x) = 0 (see equation (4.1)). Set y1(x) =
xµy(x). Then function y1(x) satisfies{
x(gν1 − 1)(gν2 − 1)
qg
− ν1ν2ν3ν4(g − ν5/κ2)(g − ν6/κ2)
fg
}
y1(x)
+
ν1ν2(x− qν3)(x− qν4)
q(qf − x)
(
gy1(x)−
ν5
κ2
y1(x/q)
)
+
(x− κ1/ν7)(x− κ1/ν8)
q(f − x)
(
1
g
y1(x)−
κ2
ν5
y1(qx)
)
= 0,
and it is written as equation (5.1) by the correspondence in equation (4.6). It follows from
Theorem 5.2 that the function
y̌1(x) =
∫ ξ∞
0
y(s)
s− x
sµPλ(x, s) dqs
satisfies equation (5.2). We apply the correspondence in equation (4.6) for the parameters and
set y̌(x) = xµ
′
y̌1(x). Then it is seen that the function y̌(x) is written as equation (5.4) and
satifies the equation L̃1y̌(x) = 0. ■
q-Middle Convolution and q-Painlevé Equation 17
We specialize the parameters y and z in Theorem 5.2 as
y = a3, z =
(ta1 − a3)(ta2 − a3)
qt(θ1 + θ2) + a23(qχ1 + χ2) + Eqθ1a3/(ta1a2)
. (5.5)
Then equation (5.1) is written as
(x− ta1)(x− ta2)y1(qx) +
χ1χ2t
2a21a
2
2(x− a3)(x− qa4)
qθ21
y1(x/q)
−
{
ta1a2(χ2 + qχ1)
qθ1
x2 + Ex+ t2a1a2
(
1 +
θ2
θ1
)}
y1(x) = 0, (5.6)
and equation (5.2) is written as
(x− ta1)(x− ta2)y̌1(qx) +
χ1ta1a2
qθ1
(
x− tθ2
χ1a4
)(
x− qtθ2
χ1a3
)
y̌1(x/q)
−
{
qχ1ta1a2 + θ1
qθ1
x2 + Ex+ t2a1a2
χ2ta1a2 + θ2
θ1
}
y̌1(x) = 0. (5.7)
Note that these equations are the q-Heun equation, and the specialization of the parameters is
related to the initial value space of q-P
(
D
(1)
5
)
(see [11]). By replacing the parameters, we obtain
the following theorem on the q-Heun equation.
Theorem 5.4. Assume that l1l2l3l4 = h1h2h3q
2 and g(x) satisfies the q-Heun equation writ-
ten as(
x− h1q
1/2
)(
x− h2q
1/2
)
g(x/q) + l3l4
(
x− l1q
−1/2
)(
x− l2q
−1/2
)
g(qx)
−
{
(l3 + l4)x
2 + l3l4Ex+ (l1l2l3l4h1h2)
1/2
(
h
1/2
3 + h
−1/2
3
)}
g(x) = 0. (5.8)
Let λ be the value satisfying qλ = q/l4. Then the function
ǧ(x) =
∫ ξ∞
0
g(s)
s− x
Pλ(x, s) dqs
formally satisfies(
x− h′1q
1/2
)(
x− h′2q
1/2
)
ǧ(x/q) + l′3l
′
4
(
x− l′1q
−1/2
)(
x− l′2q
−1/2
)
ǧ(qx)
−
{
(l′3 + l′4)x
2 + l′3l
′
4Ex+ (l′1l
′
2l
′
3l
′
4h
′
1h
′
2)
1/2
(
(h′3)
1/2 + (h′3)
−1/2
)}
ǧ(x) = 0, (5.9)
where
l′1 = l1, l′2 = l2, l′3 = l3, l′4 = q, h′1 = qh1/l4, h′2 = qh2/l4, h′3 = l4/(qh3).
Proof. Let l1, l2, l3, l4, h1, h2 and h3 be the values such that l1l2l3l4 = h1h2h3q
2. We evaluate
the values a1, a2, a3 and a4 as a3 = h1q
1/2, a4 = h2q
−1/2, ta1 = l1q
−1/2 and ta2 = l2q
−1/2, and
fix the ratios θ1/χ1 and θ1/χ2 by l3 = θ1/(χ1ta1a2) and l4 = qθ1/(χ2ta1a2). It follows from
the relation χ1χ2a1a2a3a4 = θ1θ2 in equation (3.3) that h3 = l1l2l3l4/
(
h1h2q
2
)
= θ1/θ2. Then
equation (5.8) is written as equation (5.6) by setting g(x) = y1(x). We apply Theorem 5.2, where
the parameters y and z are specialized as equation (5.5). Then we obtain equation (5.7) by the
q-integral transformation. By rewriting the parameters a1, a2, a3, a4, θ1/χ1, θ1/χ2 and θ1/θ2,
we obtain equation (5.9). ■
18 S. Sasaki, S. Takagi and K. Takemura
6 Concluding remarks
In this paper, we applied the q-middle convolution to a linear q-difference equation associated
with the q-Painlevé VI equation, and we obtain integral transformations as a consequence.
We investigated the symmetry by the q-middle convolution in terms of the affine Weyl group
symmetry of the q-Painlevé VI equation. As an application, we obtained an integral transfor-
mation on the q-Heun equation. Note that our result is a q-analogue of the results in [5, 13, 14]
on the Painlevé VI equation, the middle convolution and Heun’s differential equation.
In [10], the q-convolution was connected to the q-integral transformation by the Jackson
integral. On the other hand, the convolution for the system of Fuchsian differential equations
was connected to the Euler’s integral transformation, and there are several choice of the cycles
on the integration by the Pochhammer contour. Thus, there is a problem to find more cycles on
the q-integral transformation associated with the q-convolution. Other problems related with
the general q-middle convolution may be found from our explicit application of the specified
q-middle convolution.
The q-Painlevé VI equation was denoted by q-P
(
D
(1)
5
)
in [7], and the Weyl group symmetry
and the Lax pair of discrete Painlevé equations including q-P
(
D
(1)
5
)
were reviewed in [7]. The
equations q-P
(
E
(1)
6
)
and q-P
(
E
(1)
7
)
are also q-analogue of the Painlevé VI equation. We hope to
extend the symmetry of integral transformations to the cases q-P
(
E
(1)
6
)
and q-P
(
E
(1)
7
)
by using
the q-middle convolution, which might be related with the variants of q-Heun equation [15, 16].
There is also a problem to connect the degenerated q-Painlevé equations
(
e.g., q-P
(
A
(1)
4
))
with
the q-middle convolution. In this direction, the theory of the q-middle convolution for the
non-Fuchsian q-difference equation is anticipated.
A Middle convolution for other parameters
In Section 3, we discussed the middle convolution which is related to the q-Painlevé VI equation.
The space L was defined in equation (3.12) and we imposed the condition qλ = χ2a1a2t/θ1 in
Section 3, which induces dim(L) = 1. In the appendix, we discuss the case qλ = χ1a1a2t/θ1,
which also induces dim(L) = 1.
We continue the argument in the case qλ = χ1a1a2t/θ1 by replacing the matrix P in equa-
tion (3.13) with
P =
0 0 1 v01 0 0
g1 g3 0 v02 0 0
0 0 1 0 v11 0
g2 g4 0 0 v12 0
0 0 1 0 0 v21
0 0 0 0 0 v22
,
g3 = −q(χ1a2y − θ1)z + a2(y − ta1),
g4 = y(a1 − a2)(qχ1z − 1),
g1 = −χ2a2
{
q2χ1θ1(y − a3)(y − a4)(χ1y − χ2ta1)z
2 + θ1(y − ta1)
2(y − ta2)
− q(y− ta1)
(
2θ1χ1y
2− θ1(χ2ta1+ χ1ta2+ χ1a3+ χ1a4)y+ t
(
θ21+ χ1χ2a1a2a3a4
))
z
}
,
g2 = (a1 − a2)y
{
q2χ2
1χ2θ1(y − a3)(y − a4)z
2 + θ1χ2(y − ta1)(y − ta2)
− qχ1
(
2χ2θ1y
2 − χ2θ1(ta1 + ta2 + a3 + a4)y + t
(
θ21 + χ2
2a1a2a3a4
))
z
}
.
Recall that v01, . . . , v22 were defined by equations (3.9) and (3.10). Then detP = q3w2y3z3 ×
χ2(χ1 − χ2)
2(a1 − a2)θ
2
1(χ2ta1a2 − θ1)v
2
22, and the matrix P is invertible if detP ̸= 0. Set
F̃1 = P−1F1P, F̃2 = P−1F2P, F̃∞ = P−1F∞P,
q-Middle Convolution and q-Painlevé Equation 19
(see equation (3.11)). Then they admit the following expression:
F̃1 =
(
F 1 O
∗ ∗
)
, F̃2 =
(
F 2 O
∗ ∗
)
, F̃∞ =
(
F∞ O
∗ ∗
)
,
where F 1, F 2, F∞ are 2× 2 matrices given by
F∞ =
(
1 0
0 χ2ta1a2/θ1
)
, F 2 = F 1|a1↔a2 ,
F 1 =
a1
qyzθ21(a1 − a2)(χ2ta1a2 − θ1)
(
−θ1f
[1]
1 f
[1]
2 −a2f
[1]
2
χ2a2θ1f
[1]
1 f
[1]
3 χ2a
2
2f
[1]
3
)
,
where
f
[1]
1 = qχ1χ2a2(y − a3)(y − a4)z − (χ2a2y − θ1)(y − ta2),
f
[1]
2 = q(χ1a2y − θ1)z − a2(y − ta1),
f
[1]
3 = q2χ1θ1(y − a3)(y − a4)(χ1y − χ2ta1)z
2 + θ1(y − ta1)
2(y − ta2)
− q(y− ta1)
(
2χ1θ1y
2− θ1(χ2ta1+ χ1ta2+ χ1a3+ χ1a4)y+ t
(
θ21+ χ1χ2a1a2a3a4
))
z.
Write
Y (qx) = F (x)Y (x), F (x) = F∞ +
F 1
1− x/(ta1)
+
F 2
1− x/(ta2)
. (A.1)
Note that the upper right entry of F (x) is written as
ta1a2((ta1a2 − qθ1z)x+ ta1a2y(qχ1z − 1))
qyzθ21(x− ta1)(x− ta2)(χ2ta1a2 − θ1)
.
As discussed in Section 3, equation (A.1) is related to an integral transformation. Let Y (x)
be a solution to Y (qx) = B(x)Y (x) in equation (3.6) and write Y (x) =
(
y1(x)
y2(x)
)
. By applying
Theorem 2.2, it is shown that the function Y (x) =
(
y̌1(x)
y̌2(x)
)
defined by
y̌j(x) =
∫ ξ∞
0
{(
pj1
s
+
pj3
s− ta1
+
pj5
s− ta2
)
y1(s) +
(
pj2
s
+
pj4
s− ta1
+
pj6
s− ta2
)
y2(s)
}
× Pλ(x, s) dqs, j = 1, 2,
formally satisfies equation (A.1), where pjk is the (j, k)-entry of the matrix P−1.
By a straightforward calculation, it is shown that
p21
s
+
p23
s− ta1
+
p25
s− ta2
=
−tθ21(χ1 − χ2)
c(χ2ta1a2 − θ1)s
b21(s),
p22
s
+
p24
s− ta1
+
p26
s− ta2
=
−tθ21(χ1 − χ2)
c(χ2ta1a2 − θ1)s
(b22(s)− 1),
p12
s
+
p14
s− ta1
+
p16
s− ta2
=
θ1((ta1a2 − qzθ1)s+ yta1a2(qzχ1 − 1))
qcwyzχ2a1a2(χ2ta1a2 − θ1)(s− y)s
b12(s),
p11
s
+
p13
s− ta1
+
p15
s− ta2
=
θ1((ta1a2 − qzθ1)s+ yta1a2(qzχ1 − 1))
qcwyzχ2a1a2(χ2ta1a2 − θ1)(s− y)s
b11(s)
− t(χ1(ta1a2 − qzθ1)s+ yθ1(qzχ1 − 1))
qcwyzχ2(χ2ta1a2 − θ1)(s− y)s
,
20 S. Sasaki, S. Takagi and K. Takemura
c = q2χ2
1χ2θ1(y − a3)(y − a4)z
2 + θ1(yχ2 − χ1ta2)(y − ta1)
− qχ1
(
2χ2θ1y
2 − θ1(χ2ta1 + χ1ta2 + χ2a3 + χ2a4)y + t
(
χ1χ2a1a2a3a4 + θ21
))
z,
where bjk(s) are elements of the matrix B(s) in equation (3.7). It follows from y1(qs) =
b11(s)y1(s) + b12(s)y2(s) that
y̌1(x) =
(ta1a2 − qzθ1)
qcwyzχ2a1a2(χ2ta1a2 − θ1)
×
∫ ξ∞
0
{
−(χ1ta1a2−θ1)y1(s)+θ1
(
1+
yta1a2(qzχ1−1)
(ta1a2−qzθ1)s
)
(y1(qs)−y(s))
}
Pλ(x, s)
s− y
dqs.
Hence, the integral representation of y̌1(x) in the case qλ = χ1ta1a2/θ1 is more complicated than
that in the case qλ = χ2ta1a2/θ1. On the other hand, it follows from y2(qs) = b21(s)y1(s) +
b22(s)y2(s) that
y̌2(x) =
−tθ21(χ1 − χ2)
c(χ2ta1a2 − θ1)
∫ ξ∞
0
y2(qs)− y2(s)
s
Pλ(x, s) dqs.
To give the correspondence of the parameters by the q-middle convolution in the form of
the equation Y (qx) =
{
A0(t) + A1(t)x + A2x
2
}
Y (x) in equation (3.2), we need to transform
equation (3.21).
Let c̃ and d̃ be a non-zero constant which will be fixed later. Set x = d̃x̃,
Ã(x̃) = c̃(x− ta1)(x− ta2)
(
F∞ +
F 1
1− x/(ta1)
+
F 2
1− x/(ta2)
)
and write Ã(x̃) = Ã(x̃, t) = Ã0(t) + Ã1(t)x̃+ Ã2x̃
2. Then we have
Ã2 =
(
c̃d̃2θ1/(ta1a2) 0
0 c̃d̃2χ2
)
,
Ã0(t) has the eigenvalues c̃χ1t
2a1a2 and c̃tθ2,
det Ã(x̃, t) =
c̃2d̃4χ2θ1
ta1a2
(
x̃− ta1
d̃
)(
x̃− ta2
d̃
)(
x̃− χ1ta1a2a3
d̃θ1
)(
x̃− χ1ta1a2a4
d̃θ1
)
.
Hence the action of the q-middle convolution to the parameters in the case qλ = χ1ta1a2/θ1 is
described as
χ1 →
c̃d̃2θ1
ta1a2
, χ2 → c̃d̃2χ2, {a1, a2} →
{
a1
d̃
,
a2
d̃
}
,
{a3, a4} →
{
χ1ta1a2a3
d̃θ1
,
χ1ta1a2a4
d̃θ1
}
, {tθ1, tθ2} →
{
c̃χ1t
2a1a2, c̃tθ2
}
.
We investigate the action to the parameters y and z. We denote the images of y and z by ỹ
and z̃. Let ã11(x̃) (resp. ã12(x̃)) be the upper left entry (resp. the upper right entry) of the
matrix Ã(x̃). Then the value ỹ is the zero of the linear function ã12(x̃), and we have
ỹ =
ta1a2(qχ1z − 1)
d̃(qθ1z − ta1a2)
y =
χ1ta1a2(qz − 1/χ1)
d̃θ1(qz − ta1a2/θ1)
y.
The value z̃ satisfies ã11(x̃)|x̃=ỹ =
(
ỹ − ta1/d̃
)(
ỹ − ta2/d̃
)
/(qz̃), and we obtain
z̃ =
z
c̃d̃2
.
q-Middle Convolution and q-Painlevé Equation 21
Set d̃ = χ1ta1a2/θ1 and c̃d̃2 = 1. By applying the correspondence in equation (4.6), the trans-
formation of the parameters is written as
ν1 →
κ2
ν5
, ν2 → ν2,
{
κ1
ν7
,
κ1
ν8
}
→
{
κ1κ2
ν7ν1ν5
,
κ1κ2
ν8ν1ν5
}
, {ν3, ν4} → {ν3, ν4},{
κ2
ν5
,
κ2
ν6
}
→
{
ν1,
κ2
ν6
}
, f → g − 1/ν1
g − ν5/κ2
f, g → g.
Hence we obtain the following proposition.
Proposition A.1. We specify the parameters c̃ and d̃ by setting d̃ = χ1ta1a2/θ1 and c̃d̃2 = 1.
Then the transformation of the parameters induced by the q-middle convolution in the case
qλ = χ1ta1a2/θ1 is realized by the action of s3 given in equation (4.7).
Acknowledgements
The authors are grateful to the referees for careful reading of the manuscript and valuable
comments. The third author was supported by JSPS KAKENHI Grant Number JP18K03378.
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1 Introduction
2 q-middle convolution
3 Linear q-difference equation associated to q-Painleve VI equation and q-middle convolution
4 q-middle convolution and Weyl group symmetry of q-Painleve VI equation
5 Integral transformation on q-Heun equation
6 Concluding remarks
A Middle convolution for other parameters
References
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| id | nasplib_isofts_kiev_ua-123456789-211731 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T06:43:27Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sasaki, Shoko Takagi, Shun Takemura, Kouichi 2026-01-09T12:54:09Z 2022 -Middle Convolution and -Painlevé Equation. Shoko Sasaki, Shun Takagi and Kouichi Takemura. SIGMA 18 (2022), 056, 21 pages 1815-0659 2020 Mathematics Subject Classification: 33E10; 34M55; 39A13 arXiv:2201.03960 https://nasplib.isofts.kiev.ua/handle/123456789/211731 https://doi.org/10.3842/SIGMA.2022.056 A -deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear -difference equation associated with the -Painlevé VI equation. Then we obtain integral transformations. We investigate the -middle convolution in terms of the affine Weyl group symmetry of the -Painlevé VI equation. We deduce an integral transformation on the -Heun equation. The authors are grateful to the referees for careful reading of the manuscript and valuable comments. The third author was supported by JSPS KAKENHI Grant Number JP18K03378. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications -Middle Convolution and -Painlevé Equation Article published earlier |
| spellingShingle | -Middle Convolution and -Painlevé Equation Sasaki, Shoko Takagi, Shun Takemura, Kouichi |
| title | -Middle Convolution and -Painlevé Equation |
| title_full | -Middle Convolution and -Painlevé Equation |
| title_fullStr | -Middle Convolution and -Painlevé Equation |
| title_full_unstemmed | -Middle Convolution and -Painlevé Equation |
| title_short | -Middle Convolution and -Painlevé Equation |
| title_sort | -middle convolution and -painlevé equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211731 |
| work_keys_str_mv | AT sasakishoko middleconvolutionandpainleveequation AT takagishun middleconvolutionandpainleveequation AT takemurakouichi middleconvolutionandpainleveequation |