Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾
We present the τ-functions for the hypergeometric solutions to the q-Painlevé system of type E₇⁽¹⁾ in a determinant formula whose entries are given by the basic hypergeometric function ₈W₇. By using the W(D₅) symmetry of the function ₈W₇, we construct a set of twelve solutions and describe the actio...
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| Дата: | 2009 |
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| Мова: | English |
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Інститут математики НАН України
2009
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| Цитувати: | Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ / T. Masuda // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 18 назв. — англ. |
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Masuda, T. 2019-02-19T17:41:58Z 2019-02-19T17:41:58Z 2009 Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ / T. Masuda // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 18 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33D15; 33D05; 33D60; 33E17 https://nasplib.isofts.kiev.ua/handle/123456789/149150 We present the τ-functions for the hypergeometric solutions to the q-Painlevé system of type E₇⁽¹⁾ in a determinant formula whose entries are given by the basic hypergeometric function ₈W₇. By using the W(D₅) symmetry of the function ₈W₇, we construct a set of twelve solutions and describe the action of ~W(D₆⁽¹⁾) on the set. This paper is a contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions” (July 21–25, 2008, MPIM, Bonn, Germany). The author would like to express his sincere thanks to Professors M. Noumi and Y. Yamada for valuable discussions and comments. Especially, he owes initial steps of this work, including the formulation by means of the lattice τ -functions and the bilinear equations, to discussions with them. The author would also thank Professors K. Kajiwara and Y. Ohta for stimulating discussions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ Article published earlier |
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Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ |
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Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ Masuda, T. |
| title_short |
Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ |
| title_full |
Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ |
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Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ |
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Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ |
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hypergeometric τ-functions of the q-painlevé system of type e₇⁽¹⁾ |
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Masuda, T. |
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Masuda, T. |
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2009 |
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English |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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Article |
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We present the τ-functions for the hypergeometric solutions to the q-Painlevé system of type E₇⁽¹⁾ in a determinant formula whose entries are given by the basic hypergeometric function ₈W₇. By using the W(D₅) symmetry of the function ₈W₇, we construct a set of twelve solutions and describe the action of ~W(D₆⁽¹⁾) on the set.
|
| issn |
1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/149150 |
| citation_txt |
Hypergeometric τ-Functions of the q-Painlevé System of Type E₇⁽¹⁾ / T. Masuda // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 18 назв. — англ. |
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AT masudat hypergeometricτfunctionsoftheqpainlevesystemoftypee71 |
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2025-11-26T10:11:26Z |
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2025-11-26T10:11:26Z |
| _version_ |
1850618546189500416 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 035, 30 pages
Hypergeometric τ -Functions
of the q-Painlevé System of Type E
(1)
7
?
Tetsu MASUDA
Department of Physics and Mathematics, Aoyama Gakuin University,
5-10-1 Fuchinobe, Sagamihara, Kanagawa, 229-8558, Japan
E-mail: masuda@gem.aoyama.ac.jp
Received November 27, 2008, in final form March 10, 2009; Published online March 24, 2009
doi:10.3842/SIGMA.2009.035
Abstract. We present the τ -functions for the hypergeometric solutions to the q-Painlevé
system of type E
(1)
7 in a determinant formula whose entries are given by the basic hyper-
geometric function 8W7. By using the W (D5) symmetry of the function 8W7, we construct
a set of twelve solutions and describe the action of W̃ (D(1)
6 ) on the set.
Key words: q-Painlevé system; q-hypergeometric function; Weyl group; τ -function
2000 Mathematics Subject Classification: 33D15; 33D05; 33D60; 33E17
1 Introduction
A natural framework for discrete Painlevé equations by means of the geometry of rational
surfaces has been proposed by Sakai [17]. Each equation is defined by the group of Cremona
transformations on a family of surfaces obtained by blowing-up at nine points on P2. According
to the types of rational surfaces, the discrete Painlevé equations are classified in terms of affine
root systems. Also, their symmetries are described by means of affine Weyl groups, the lattice
part of which gives rise to difference equations. For instance, the q-Painlevé system of type E
(1)
7 ,
which is the main object of this paper, is a discrete dynamical system defined on a family of
rational surfaces parameterized by nine-point configurations on P2 such that six points among
them are on a conic and other three are on a line [17]. An explicit expression for the system of
q-difference equations is given by [15]
(fg − tt)(fg − t2)
(fg − 1)(fg − 1)
=
(f − b1t)(f − b2t)(f − b3t)(f − b4t)
(f − b5)(f − b6)(f − b7)(f − b8)
,
(fg − t2)(fg − tt)
(fg − 1)(fg − 1)
=
(
g − t
b1
)(
g − t
b2
)(
g − t
b3
)(
g − t
b4
)
(
g − 1
b5
)(
g − 1
b6
)(
g − 1
b7
)(
g − 1
b8
) , (1.1)
where t is the independent variable and the time evolution of the dependent variables is given
by g = g(qt) and f = f(t/q). The parameters bi (i = 1, 2, . . . , 8) satisfy b1b2b3b4 = q and
b5b6b7b8 = 1.
Similarly to the Painlevé differential equations, the discrete Painlevé equations admit par-
ticular solutions expressible in terms of various hypergeometric functions. Regarding the q-
difference Painlevé equations, the hypergeometric solutions to those equations have been con-
structed by means of a geometric approach and direct linearization of the q-difference Riccati
?This paper is a contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy
Problems, and Hypergeometric Functions” (July 21–25, 2008, MPIM, Bonn, Germany). The full collection is
available at http://www.emis.de/journals/SIGMA/Elliptic-Integrable-Systems.html
mailto:masuda@gem.aoyama.ac.jp
http://dx.doi.org/10.3842/SIGMA.2009.035
http://www.emis.de/journals/SIGMA/Elliptic-Integrable-Systems.html
2 T. Masuda
equations [9, 10]. In particular, the Riccati solution to the system of q-difference equations (1.1)
is expressed in terms of the q-hypergeometric series
8W7(a0; a1, . . . , a5; q, z) = 8ϕ7
(
a0, qa
1/2
0 ,−qa
1/2
0 , a1 · · · , a5
a
1/2
0 ,−a
1/2
0 , qa0/a1, · · · , qa0/a5
; q, z
)
=
∞∑
k=0
(1− a0q
2k)
(1− a0)
(a0; q)k
(q; q)k
5∏
i=1
(ai; q)k
(qa0/ai; q)k
zk, z =
q2a2
0
a1a2a3a4a5
, (1.2)
where (a; q)k =
k−1∏
i=0
(1− aqi).
The purposes of this paper are to propose a formulation for the q-Painlevé system of type E
(1)
7
by means of the lattice τ -functions and to completely determine the τ -functions for the hyper-
geometric solutions (hypergeometric τ -functions for short) of the system.
This paper is organized as follows. In Section 2, we give a formulation for the q-Painlvé
system of type E
(1)
7 in terms of the lattice τ -functions. Section 3 is devoted to a preparation for
constructing the hypergeometric τ -functions. We decompose the lattice, each of whose elements
indicates the τ -function, into a family of six-dimensional lattices.
In Sections 4–6, we construct the hypergeometric τ -functions. We find that a q-analogue of
the double gamma function appears as a normalization factor of the hypergeometric τ -functions
in Section 4. In Section 5, we find that a class of bilinear equations for the lattice τ -functions
yields the contiguity relations for the q-hypergeometric function 8W7. As is well-known, the q-
hypergeometric function 8W7 possesses the W (D5)-symmetry [13]. From that, we can construct
a set of twelve solutions corresponding to the coset W (D6)/W (D5), and describe the action of
W̃
(
D
(1)
6
)
on the set of solutions.
One of the important features of the hypergeometric solutions to the continuous and discrete
Painlevé equations is that they can be expressed in terms of Wronskians or Casorati deter-
minants [4, 12, 7, 3, 16]. In Section 6, we show that the hypergeometric τ -functions of the
q-Painlevé system of type E
(1)
7 are expressed by “two-directional Casorati determinants”. As
a consequence, we get an explicit expression for the hypergeometric solutions to the q-difference
Painlevé equation (1.1), which is proposed in Corollary 6.1.
2 The q-Painlevé system of type E
(1)
7
2.1 The discrete Painlevé system of type E
(1)
8
At first, we give a brief review of the formulation for the discrete Painlevé system of type E
(1)
8
in terms of the lattice τ -functions [8, 11].
Let L =
9
⊕
i=0
Zei be a lattice with a basis {e0, e1, . . . , e9}, and define a symmetric bilinear form
〈 , 〉 : L × L → Z by
〈e0, e0〉 = −1, 〈ei, ei〉 = 1 (i = 1, 2, . . . , 9), 〈ei, ej〉 = 0 (i, j = 0, 1, . . . , 9; i 6= j).
Consider the affine Weyl group W (E(1)
8 ) = 〈s0, s1, . . . , s8〉 associated with the Dynkin diagram
1 2 3 4 5 6 7 8
0
d d d d d d d d
d
The lattice L admits a natural linear action of W (E(1)
8 ) defined by si .Λ = Λ − 〈hi,Λ〉hi for
Λ ∈ L, where hi (i = 0, 1, . . . , 8) are the simple coroots defined by h0 = e0 − e1 − e2 − e3 and
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 3
hi = ei − ei+1 (i = 1, . . . , 8). The canonical central element c = 3e0 − e1 − · · · − e9 is orthogonal
to all the simple coroots hi, and hence W (E(1)
8 )-invariant.
The parameter space for the discrete Painlevé system of type E
(1)
8 is the ten-dimensional
vector space
9
⊕
i=0
Cei, whose coordinates are denoted by εi = 〈ei, ·〉 (i = 0, 1, . . . , 9). The root
lattice Q(E(1)
8 ) =
8
⊕
i=0
Zαi is generated by the simple roots α0 = ε0−ε1−ε2−ε3 and αi = εi−εi+1
(i = 1, . . . , 8). The affine Weyl group W (E(1)
8 ) acts on the coordinate function εi in a similar
way to on the basis ei. The W (E(1)
8 )-invariant element corresponding to c is given by δ = 〈c, ·〉 =
3ε0 − ε1 − · · · − ε9, which is called the null root and plays the role of the scaling constant for
difference equations in the context of the discrete Painlevé equations. For simplicity, we denote
the reflection sα with respect to the root α = εij = εi − εj or α = εijk = ε0 − εi − εj − εk for
i, j, k ∈ {1, 2, . . . , 9} by sij or sijk, respectively. Also, we often use the notation eij = ei− ej and
eijk = e0 − ei − ej − ek.
For each α ∈ Q
(
E
(1)
8
)
, the action of the translation operator Tα ∈ W
(
E
(1)
8
)
is given by [6]
Tα(Λ) = Λ + 〈c,Λ〉h−
(
1
2
〈h, h〉〈c,Λ〉+ 〈h, Λ〉
)
c (Λ ∈ L) (2.1)
by using the element h ∈ L such that α = 〈h, ·〉. Note that we have TαTβ = TβTα and
wTαw−1 = Tw.α for any w ∈ W
(
E
(1)
8
)
. When α = εij or εijk, we also denote the translation Tα
simply by Tij or Tijk, respectively. They can be expressed by
Tij = sil1l2sil3l4sl5l6l7sil3l4sil1l2sij , {i, j, l1, . . . , l7} = {1, 2, . . . , 9},
Tijk = sl1l2l3sl4l5l6sl1l2l3sijk, {i, j, k, l1, . . . , l6} = {1, 2, . . . , 9}.
Let us introduce a family of dependent variables τΛ = τΛ(ε), ε = (ε0, . . . , ε9), indexed by
Λ ∈ M , where M is the W (E(1)
8 )-orbit defined by
M = W
(
E
(1)
8
)
. e1 = {Λ ∈ L | 〈c,Λ〉 = −1, 〈Λ,Λ〉 = 1} ⊂ L.
The action of W
(
E
(1)
8
)
on the lattice τ -functions τΛ is defined by w(τΛ) = τw.Λ for any w ∈
W
(
E
(1)
8
)
. The discrete Painlevé system of type E
(1)
8 is equivalent to the overdetermined system
defined by the bilinear equations
[εjk][εjkl]τeiτe0−ei−el
+ [εki][εkil]τejτe0−ej−el
+ [εij ][εijl]τek
τe0−ek−el
= 0
for any mutually distinct indices i, j, k, l ∈ {1, 2, . . . , 9}, as well as their W
(
E
(1)
8
)
-transforms
[w(εjk)][w(εjkl)]τw.eiτw.(e0−ei−el) + (i, j, k)-cyclic = 0
for any w ∈ W
(
E
(1)
8
)
. Here, [x] is a nonzero odd holomorphic function on C satisfying the
Riemann relation
[x + y][x− y][u + v][u− v] = [x + u][x− u][y + v][y − v]− [x + v][x− v][y + u][y − u]
for any x, y, u, v ∈ C. There are three classes of such functions; elliptic, trigonometric and
rational. These three cases correspond to the three types of difference equations, namely, elliptic
difference, q-difference and ordinal difference, respectively. The lattice part of W
(
E
(1)
8
)
gives
rise to the difference Painlevé equation.
4 T. Masuda
2.2 The q-Painlevé system of type E
(1)
7
Let us propose a formulation for the q-Painlevé system of type E
(1)
7 by means of the lattice
τ -functions, using by the notation introduced in the previous subsection. A derivation of the
formulation is discussed in Appendices.
The q-Painlevé system of type E
(1)
7 is a discrete dynamical system defined on a family of
rational surfaces parameterized by nine-point configurations on P2 such that six points among
them are on a conic C and other three are on a line L [17]. Here, we set p1, p2, p3, p4, p5, p6 ∈ C
and p7, p8, p9 ∈ L. In what follows, the symbols C and L also mean the index sets C =
{1, 2, 3, 4, 5, 6} and L = {7, 8, 9}, respectively. And we often use i, j, k, . . . and r, s as the
elements of C and L, respectively. In this setting, the symmetric groups SC
6 = 〈s12, . . . , s56〉 and
SL
3 = 〈s78, s89〉 naturally act on the configuration space as the permutation of the points on C
and L, respectively. Also, the standard Cremona transformation with respect to (p1, p2, p7) is
well-defined as a birational action on the space. They generate the affine Weyl group W (E(1)
7 ) =
〈s12, s23, s34, s45, s56, s78, s89, s127〉. The associated Dynkin diagram and its automorphism are
realized by
c c c c c c cc
e89 e78 e127 e23 e34 e45 e56
e12
and π = s123s47s58s69, respectively. Thus we find that the extended affine Weyl group W̃
(
E
(1)
7
)
= 〈s12, s23, s34, s45, s56, s78, s89, s127, π〉 acts on the configuration space.
The lattice τ -functions τΛ = τΛ(ε) for the q-Painlevé system of type E
(1)
7 are indexed by
Λ ∈ ME7 = W̃
(
E
(1)
7
)
. e1 = MC
∐
ML,
where
MC = {Λ ∈ M | 〈e789,Λ〉 = 0} = W
(
E
(1)
7
)
. e1,
ML = {Λ ∈ M | 〈e789,Λ〉 = −1} = W
(
E
(1)
7
)
. e7.
The action of W̃ (E(1)
7 ) on the lattice τ -functions τΛ is defined by w(τΛ) = τw.Λ for any w ∈
W̃
(
E
(1)
7
)
. The q-Painlevé system of type E
(1)
7 is equivalent to the overdetermined system defined
by the bilinear equations
[εrs]τC
ej
τL
e0−ei−ej
= [εijs]τL
er
τC
e0−ei−er
− [εijr]τL
es
τC
e0−ei−es
,
[εjk]τL
er
τC
e0−ei−er
= [εikr]τC
ej
τL
e0−ei−ej
− [εijr]τC
ek
τL
e0−ei−ek
, (2.2)
[εij ][εijr]τC
ek
τC
e0−ek−er
+ (i, j, k)-cyclic = 0,
[εij ][εkl]τL
e0−ei−ej
τL
e0−ek−el
+ (i, j, k)-cyclic = 0, (2.3)
τC
ei
τC
e0−ei−e9 − τC
ej
τC
e0−ej−e9 + [εij ][εij9] dLτL
e7τ
L
e8 = 0,
τL
e0−e1−e4τ
L
e0−e2−e3 − τL
e0−e1−e3τ
L
e0−e2−e4 + [ε12][ε34] dCτC
e5τ
C
e6 = 0 (2.4)
for mutually distinct indices i, j, k, l ∈ C and r, s ∈ L, as well as their W̃ (E(1)
7 )-transforms. The
superscript C (resp. L) denotes that the τ -function is indexed by Λ ∈ MC (resp. Λ ∈ ML),
and we leave it out when it is unnecessarily. It is possible to fix the function [x] as [x] =
e(1
2x) − e(−1
2x), e(x) = eπ
√
−1x, without loss of generality. The factors dL and dC in (2.4)
correspond to the irreducible components of the anti-canonical devisor DL = e0 − e7 − e8 − e9
and DC = 2e0 − e1 − · · · − e6, respectively. These factors are W
(
E
(1)
7
)
-invariant and the action
of π is given by π : dL ↔ dC .
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 5
The translation operators with respect to the root vectors εij , εrs, εijr ∈ Q
(
E
(1)
7
)
are denoted
by Tij , Trs and Tijr, respectively. Also, there exist fifty six translation operators that move
a lattice point Λ ∈ ME7 to its nearest ones. Let us denote such operators by T̃ir, T̃ijk and T̃irs
according to the action on Q
(
E
(1)
7
)
;
T̃17 : ε78 7→ ε78 + δ, ε12 7→ ε12 − δ,
T̃123 : ε127 7→ ε127 − δ, ε34 7→ ε34 + δ,
for instance. We find that these operators can be realized as T̃ir = Tirs789, T̃ijk = s789Tijk and
T̃irs = Tirss789, respectively, in terms of the Weyl group W
(
E
(1)
8
)
. Then, from the formula (2.1),
the action on a lattice point can be calculated as T̃17(e9) = T17(e0 − e7 − e8) = e0 − e1 − e8, for
example. Note that we have the relations such as T̃19T̃178 = 1 and T̃123T̃456 = 1. The translation
operators with respect to the root vectors can be expressed by Tij = T̃irT̃
−1
jr , Trs = T̃−1
ir T̃is and
Tijr = T̃ijkT̃kr.
Proposition 2.1. If the lattice τ -functions τΛ (Λ ∈ ME7) satisfy the bilinear equations (2.2)
and their W̃
(
E
(1)
7
)
-transforms, then they also satisfy (2.3) and their W̃
(
E
(1)
7
)
-transforms.
This is easily verified by a direct calculation. From this proposition, we see that it is not
necessary to consider the bilinear equations (2.3) for constructing a solution to the q-Painlevé
system of type E
(1)
7 . However, as we will see Section 6, we use the bilinear equations of type (2.3)
in order to get a nicer determinant formula for the hypergeometric τ -functions. Then, we
will treat all types of bilinear equations below, although the discussion becomes technically
complicated as a consequence.
Let us introduce the dependent variables f and g by
f = e
(
1
8αl − 1
8αr + 1
4ε12 + 1
8δ
) e(1
4ε13)τe1τe0−e1−e2 − e(−1
4ε13)τe3τe0−e2−e3
e(1
4ε13)τe3τe0−e2−e3 − e(−1
4ε13)τe1τe0−e1−e2
,
g = e
(
1
8αr − 1
8αl + 1
4ε12 − 1
8δ
) e(1
4ε23)τe3τe0−e1−e3 − e(−1
4ε23)τe2τe0−e1−e2
e(1
4ε23)τe2τe0−e1−e2 − e(−1
4ε23)τe3τe0−e1−e3
with αl = 3ε127 + 2ε78 + ε89 and αr = 3ε34 + 2ε45 + ε56. Then, one get the explicit expression
for the q-difference equations (1.1), a derivation of which is discussed in Appendix C.
3 A family of six-dimensional lattices and the bilinear equations
As a preparation for constructing the hypergeometric τ -functions, we decompose the lattice ME7
into a family of six-dimensional lattices according to the value of the symmetric bilinear form
with the coroot vector e89 = e8 − e9;
ME7 =
∐
n∈Z
Mn, Mn =
{
Λ ∈ ME7 | 〈Λ, e89〉 = n
}
.
Parallel to this decomposition, let us consider the orthogonal complement of ε89 in the root
lattice Q
(
E
(1)
7
)
. Then we get the root lattice Q
(
D
(1)
6
)
corresponding to the Dynkin diagram
c c c c cc
cε12 ε23 ε34
ε45
ε56
ε127
δ − ε567
6 T. Masuda
Since we have ε127 + ε12 + 2ε23 + 2ε34 + 2ε45 + ε56 + (δ − ε567) = δ, the same δ denotes the
null root of Q
(
D
(1)
6
)
. The corresponding simple reflections generate the affine Weyl group
W
(
D
(1)
6
)
= 〈s127, s12, s23, s34, s45, s56, sδ−ε567〉. Note that the finite Weyl group W (D6) =
〈s127, s12, s23, s34, s45, s56〉 includes the symmetric group S6 = 〈s12, s23, s34, s45, s56〉 as a sub-
group. In this realization, an automorphism of the above Dynkin diagram can be expressed by
ρ = πs157s168s24s26s35s79 whose action on the simple roots of Q
(
D
(1)
6
)
is given by
ρ : ε12 ↔ δ − ε567, ε127 ↔ ε56, ε23 ↔ ε45.
The extended affine Weyl group W̃ (D(1)
6 ) = 〈s127, s12, s23, s34, s45, s56, sδ−ε567 , ρ〉 acts transi-
tively on each Mn. Regarding the translation operators, we have T̃i7, T̃ijk ∈ W̃
(
D
(1)
6
)
for
i, j, k ∈ C = {1, 2, . . . , 6}, which can be expressed in the form T̃α = ρw, w ∈ W
(
D
(1)
6
)
.
According to the location of the lattice τ -functions, one can classify the bilinear equa-
tions (2.2) into the following four types:
(A)n : Two on each of Mn−1, Mn and Mn+1, respectively.
(B)n : Four on Mn, and one on Mn+1 and Mn−1, respectively.
(C)n : Three on Mn+1 and Mn, respectively.
(D)n : Six on Mn.
The bilinear equations of type (C)n are further classified into two types. The first one is that
all of three τ -functions on Mn+1 belong to MC (or ML), which is denoted by (C)rn. The second
is that one of three τ -functions on Mn+1 belongs to MC (or ML), denoted by (C)in. Typical
bilinear equations are given by
(A)0 [ε89]τejτe0−ei−ej = [εij9]τe8τe0−ei−e8 − [εij8]τe9τe0−ei−e9 ,
(B)0 [ε78]τejτe0−ei−ej = [εij8]τe7τe0−ei−e7 − [εij7]τe8τe0−ei−e8 ,
(C)i0 [εjk]τe8τ2e0−ei−ej−ek−e8−e9 = [εik8]τe0−ek−e9τe0−ei−ej − [εij8]τe0−ej−e9τe0−ei−ek
,
(C)r0 [εij ]τe0−ei−ejτe0−ek−e9 + (i, j, k)-cyclic = 0,
(D)0 [εjk]τe7τe0−ei−e7 = [εik7]τejτe0−ei−ej − [εij7]τek
τe0−ei−ek
(3.1)
for mutually distinct indices i, j, k ∈ C. The bilinear equations (2.3) are also classified in a similar
way into four types, each of which we denote by (A)′n, (B)′n, (C)′n and (D)′n to distinguish them
from the bilinear equations (2.2). Typical equations are given by
(A)′0 [ε78][δ − ε569]τe9τ2e0−e1−e2−e3−e4−e9 + (7, 8, 9)-cyclic = 0,
(B)′0 [εij ][εkl]τe8τ2e0−ei−ej−ek−el−e8
= [εil8][εjk8]τe0−ei−ek
τe0−ej−el
− [εjl8][εik8]τe0−ej−ek
τe0−ei−el
,
(C)′0 [εij ][εij9]τek
τe0−ek−e9 + (i, j, k)-cyclic = 0,
(D)′0 [εij ][εkl]τijτkl + (i, j, k)-cyclic = 0, [εij ][εij7]τkτk7 + (i, j, k)-cyclic = 0 (3.2)
for mutually distinct indices i, j, k, l ∈ C. The bilinear equations (2.4) are also classified into
the type (A)dn, (B)dn, (C)dn and (D)dn. Typical equations are given by
(A)d0 τe8τ2e0−e1−e2−e3−e4−e8 − τe9τ2e0−e1−e2−e3−e4−e9 + [δ − ε567][ε89] dCτe5τe6 = 0,
(B)d0 τeiτe0−ei−e7 − τejτe0−ej−e7 + [εij ][εij7] dLτe8τe9 = 0,
τe8τ2e0−e1−e2−e3−e4−e8 − τe0−e1−e2τe0−e3−e4 − [ε128][ε348] dCτe5τe6 = 0,
(C)d0 τeiτe0−ei−e9 − τejτe0−ej−e9 + [εij ][εij9] dLτe7τe8 = 0,
(D)d0 τe0−e1−e4τe0−e2−e3 − τe0−e1−e3τe0−e2−e4 + [ε12][ε34] dCτe5τe6 = 0 (3.3)
for mutually distinct indices i, j ∈ C.
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 7
Lemma 3.1. Any bilinear equation of type (A)0 can be obtained by an action of W̃
(
D
(1)
6
)
on
the first equation of (3.1). Also, we have a similar situation regarding each case of type (B)0,
(C)r0, (C)i0 and (D)0, respectively.
Proof. Any lattice τ -function on M1 can be transformed to τe8 by an action of W̃
(
D
(1)
6
)
.
Searching for Λ ∈ M−1 such that 〈Λ + e8,Λ + e8〉 = 0, we find that the lattice τ -functions
on M−1 which can pair with τe8 are τe0−ei−e8 , τ2e0−ei−ej−ek−e7−e8 and τc+ei+e9−e7 for mutually
distinct indices i, j, k ∈ C. Any of them can be transformed to τe0−ei−e8 by an action of W (D6).
Since τe8 is invariant under the action of W (D6), we find that one of the pairs of the lattice
τ -functions in a bilinear equation of type (A)0 can be transformed to τe8τe0−ei−e8 by an action of
W̃
(
D
(1)
6
)
. Note that three pairs of the lattice τ -functions in a bilinear equation have a common
barycenter. Therefore, the bilinear equations of type (A)0 including the term τe8τe0−ei−e8 are
reduced to
[ε89]τejτe0−ei−ej = [εij9]τe8τe0−ei−e8 − [εij8]τe9τe0−ei−e9 ,
[ε89]τe7τe0−ei−e7 = [ε79]τe8τe0−ei−e8 − [ε78]τe9τe0−ei−e9 ,
which are transformed by the action of the Dynkin diagram automorphism ρ ∈ W̃
(
D
(1)
6
)
to each
other. The proof for the other types of bilinear equations is given in a similar way. �
From this lemma and similar consideration for the bilinear equations (3.2) and (3.3), we
immediately get the following proposition.
Proposition 3.1. Fix n ∈ Z.
1. All the bilinear equations of type (A)n can be transformed by the action of W̃
(
D
(1)
6
)
to one
another. Also, we have a similar situation regarding each case of type (B)n, (C)in, (C)rn
and (D)n, respectively.
2. All the bilinear equations of type (A)′n can be transformed by the action of W̃
(
D
(1)
6
)
to
one another. Also, we have a similar situation regarding each case of type (B)′n and (C)′n,
respectively. The set of all the bilinear equations of type (D)′n is decomposed to two orbits
by the action of W̃
(
D
(1)
6
)
.
3. All the bilinear equations of type (A)dn can be transformed by the action of W̃
(
D
(1)
6
)
to
one another. Also, we have a similar situation regarding each case of type (C)dn and (D)dn,
respectively. The set of all the bilinear equations of type (B)dn is decomposed to two orbits
by the action of W̃
(
D
(1)
6
)
.
Let us discuss the relationships among the above types of bilinear equations.
Proposition 3.2. If the lattice τ -functions satisfy all the bilinear equations of type (B)n, then
they also satisfy those of type (A)n; that is,
1. (B)n ⇒ (A)n.
Similarly, we have
2. (C)in ⇒ (C)rn.
Moreover, if τΛ 6= 0 for Λ ∈ Mn−1, we have the following:
3. (C)in−1 ⇒ (D)n.
4. (A)n, (C)in−1 ⇒ (C)in.
8 T. Masuda
Proof. It is sufficient to verify the statement for a certain n ∈ Z. The first and second
statements are easily verified for the case of n = 0.
3. (C)i0 ⇒ (D)1. Let us consider the following bilinear equation
[ε23]τe4τ2e0−e2−e3−e4−e5−e9 = [ε349]τe0−e2−e9τe0−e3−e5 − [ε249]τe0−e3−e9τe0−e2−e5
of type (C)i0. Multiplying this equation by τe0−e1−e9 and summing up its (1, 2, 3)-cyclic permu-
tations, we get a bilinear equation of type (D)1.
4. (A)0 and (C)i−1 ⇒ (C)i0. Let us consider the following bilinear equation of type (C)i−1
[εjk]τe9τ2e0−ei−ej−ek−e8−e9 = [εik9]τe0−ek−e8τe0−ei−ej − [εij9]τe0−ej−e8τe0−ei−ek
.
Multiplying both right and left-hand sides by τe8 and using the first equation of (3.1), we get
τe9 × [εjk]τe8τ2e0−ei−ej−ek−e8−e9
= τe0−ei−ej ×
(
[εik8]τe9τe0−ek−e9 + [ε89]τeiτe0−ei−ek
)
− {j ↔ k}
= τe9 ×
(
[εik8]τe0−ek−e9τe0−ei−ej − [εij8]τe0−ej−e9τe0−ei−ek
)
,
which is equivalent to the third equation of (3.1). �
Also, by not difficult but tedious procedure, we get the following propositions.
Proposition 3.3. If the lattice τ -functions satisfy all the bilinear equations of type (B)dn, then
they also satisfy those of type (A)dn; that is,
1. (B)dn ⇒ (A)dn.
Similarly, if τΛ 6= 0 for Λ ∈ Mn−1, we have the following:
2. (A)n, (C)dn−1 ⇒ (C)dn,
3. (C)in−1, (C)dn−1 ⇒ (D)dn,
4. (C)dn−1, (C)in−1, (B)n ⇒ (B)dn.
Proposition 3.4. If the lattice τ -functions satisfy all the bilinear equations of type (C)in, then
they also satisfy those of type (C)′n; that is,
1. (C)in ⇒ (C)′n.
Similarly, we have
2. (B)′n ⇒ (A)′n.
Moreover, if τΛ 6= 0 for Λ ∈ Mn−1, we have the following:
3. (C)′n−1, (D)n ⇒ (D)′n,
4. (B)′n, (C)in−1 ⇒ (B)n.
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 9
4 The construction of the τ -functions on M0
Hereafter, we construct the hypergeometric τ -functions for the q-Painlevé system of type E
(1)
7
by imposing the following boundary condition
τΛ−1 = 0 for any Λ−1 ∈ M−1 (4.1)
and τΛ0 6= 0 for any Λ0 ∈ M0. In this section, we discuss the construction of the τ -functions on
the lattice M0.
First, let us consider the following bilinear equations of type (A)0, (A)′0 and (A)d0
[ε89]τejτe0−ei−ej = [εij9]τe8τe0−ei−e8 − [εij8]τe9τe0−ei−e9 (i, j ∈ C),
[ε78][δ − ε569]τe9τ2e0−e1−e2−e3−e4−e9 + (7, 8, 9)-cyclic = 0,
[δ − ε567][ε89]dCτe5τe6 + τe8τ2e0−e1−e2−e3−e4−e8 − τe9τ2e0−e1−e2−e3−e4−e9 = 0. (4.2)
The boundary condition (4.1) leads us to
[ε89] = 0 ⇔ ε89 = ω ∈ Z. (4.3)
All the bilinear equations of type (A)0, (A)′0 and (A)d0 hold under the conditions (4.1) and (4.3),
since they can be obtained by the action of W̃
(
D
(1)
6
)
= 〈s127, s12, . . . , s56, sδ−ε567 , ρ〉 on (4.2) and
the coefficient [ε89] is W̃
(
D
(1)
6
)
-invariant.
Under the boundary condition (4.1), the bilinear equations of type (B)0, (B)′0 and (B)d0 are
expressed in terms of the lattice τ -functions on M0. Typical equations of these types are given by
[ε78]τejτe0−ei−ej = [εij8]τe7τe0−ei−e7 − [εij7]τe8τe0−ei−e8 ,
[εij ][εkl]τe8τ2e0−ei−ej−ek−el−e8 = [εil8][εjk8]τe0−ei−ek
τe0−ej−el
− [εjl8][εik8]τe0−ej−ek
τe0−ei−el
,
τeiτe0−ei−e7 − τejτe0−ej−e7 + [εij ][εij7] dLτe8τe9 = 0,
τe8τ2e0−e1−e2−e3−e4−e8 − τe0−e1−e2τe0−e3−e4 = [ε128][ε348] dCτe5τe6
for mutually distinct indices i, j, k, l ∈ C. These are reduced to
[ε79]τejτe0−ei−ej = [εij9]τe7τe0−ei−e7 , τeiτe0−ei−e7 = τejτe0−ej−e7 ,
τe0−e1−e2τe0−e3−e4 + [ε129][ε349] dCτe5τe6 = 0 (4.4)
and
[εil9][εjk9]τe0−ei−ek
τe0−ej−el
= [εjl9][εik9]τe0−ej−ek
τe0−ei−el
(4.5)
due to the conditions (4.1) and (4.3). Obviously, the equation (4.5) can be derived from the
third equation of (4.4) and its S6-transforms. Also, it is not difficult to see that all the bilinear
equations of type (D)0, (D)′0 and (D)d0 can be derived from the equations (4.4) and their W̃
(
D
(1)
6
)
-
transforms. Then, it is sufficient to consider the equations (4.4) and their W̃
(
D
(1)
6
)
-transforms
for constructing the hypergeometric τ -functions on M0.
Let us consider a pair of non-zero meromorphic functions (G(x), F (x)) satisfying the diffe-
rence equations G(x + δ) = ε [x]G(x) and F (x + δ) = G(x)F (x) with a constant ε ∈ C∗. When
Im δ > 0, a typical choice of such functions is given by
G(x) =
e
(
− δ
2
(
x/δ
2
))
(u; q)∞
, F (x) = e
(
− δ
2
(
x/δ
3
))
(u; q, q)∞,
10 T. Masuda
where u = e(x), q = e(δ), (u; q, q) =
∞∏
i,j=0
(1 − uqi+j) and ε = −1. For other choices of
(G(x), F (x), ε), see Appendix in [14]. In what follows, we fix a triplet (G+(x), F+(x), ε+) with
a constant factor ε+, namely we have
G+(x + δ) = ε+[x]G+(x), F+(x + δ) = G+(x)F+(x). (4.6)
Also, we introduce a pair of functions (G−(x), F−(x)) by the relations
F−(x) = F+(2δ + ω − x), G−(x)G+(δ + ω − x) = 1. (4.7)
Note that these functions satisfy the difference equations
G−(x + δ) = ε−[x]G−(x), F−(x + δ) = G−(x)F−(x) (4.8)
with ε− = (−1)ω+1ε+.
Moreover, we consider a triplet of functions (A+(x),B+(x), C+(x)) defined by the difference
equations
A+(x + δ)A+(x− δ)
A+(x)A+(x)
= e(αx + a),
B+(x + δ)B+(x− δ)
B+(x)B+(x)
= e(αx + b),
C+(x + δ)C+(x− δ)
C+(x)C+(x)
= e(−αx + c), (4.9)
where a, b, c and α are the complex constants satisfying e(2αδ + 4b + 2c) = (−1)ω+1 and
(−1)ω+1ε2+e(αω + 2a) + dC e(5b + 3c) = 0. A typical example of such functions is given by
A+(x) = e(δα
(
x/δ+1
3
)
+ a
(
x/δ
2
)
). Also, we introduce the functions A−(x), B−(x) and C−(x) by
the relations
A−(x) = A+(2δ + ω − x), B−(x) = B+(2δ − x), C−(x) = C+(2δ − x). (4.10)
Definition 4.1. For each Λ0 ∈ M0, we define the twelve functions τ
(a;±)
Λ0
(ε) (a ∈ C) by
τ
(a;±)
Λ0
(ε) = F±(ε79 + (〈e79,Λ0〉+ 1)δ)
∏
i,j∈C; i<j
F±κ
(a)
ij
(εij9 + (〈eij9,Λ0〉+ 1)δ)
×A±(ε79 + (〈e79,Λ0〉+ 1)δ)
∏
i,j∈C; i<j
A±κ
(a)
ij
(εij9 + (〈eij9,Λ0〉+ 1)δ)
×
∏
i∈Ca
B±(εia7 + (〈eia7,Λ0〉+ 1)δ)B±(εia + (〈eia,Λ0〉+ 1)δ)
× C±(εaa7 + (〈eaa7,Λ0〉+ 1)δ), (4.11)
where κ
(a)
ij is the sign factor defined by κ
(a)
ij = (−1)]({i,j}∩{a}) and Ca = C\{a}.
Theorem 4.1. The action of W̃
(
D
(1)
6
)
on the functions τ
(a;±)
Λ0
(ε) is described as follows:
1. For any translation operator T ∈ W̃
(
D
(1)
6
)
, we have τ
(a;±)
T.Λ0
(ε) = τ
(a;±)
Λ0
(T (ε)).
2. For any permutation σ ∈ S6, we have τ
(σ(a);±)
σ.Λ0
(ε) = τ
(a;±)
Λ0
(σ(ε)).
3. Take two mutually distinct indices i, j ∈ C.
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 11
(a) If a /∈ {i, j}, then τ
(a;±)
sij7.Λ0
(ε) = τ
(a;±)
Λ0
(sij7(ε)).
(b) If a ∈ {i, j}, then τ
(a;±)
sij7.Λ0
(ε) = τ
(b;∓)
Λ0
(sij7(ε)), where b is an index such that {a, b} =
{i, j}.
4. The action of the central element wc ∈ W (D6) is given by τ
(a;∓)
wc.Λ0
(ε) = τ
(a;±)
Λ0
(wc(ε)).
Proof. The first and second statements are obvious from the definition of τ
(a;±)
Λ0
(ε). The third
statement is guaranteed by the relations (4.7) and (4.10). Since we have
wc : ε79 7→ δ + ω − ε79, εij9 7→ δ + ω − εij9 (i, j ∈ C),
one can verify the fourth statement by using the relations (4.7) and (4.10). �
Remark 4.1. The central element wc ∈ W (D6) can be expressed by wc = s12s127s34s347s56s567.
It is easy to see that we have Twc = wcT
−1 for any translation operator T ∈ W̃ (D(1)
6 ).
Let S be a label set defined by S = {(a ; ε) | a ∈ C, ε = ±1}. By using the difference equa-
tions (4.6), (4.8) and (4.9), one can verify that the family of functions {τ (η)
Λ0
(ε)}Λ0∈M0 for each
label η ∈ S satisfies the bilinear equations (4.4). Also, the set of the functions {τ (η)
Λ0
(ε) | η ∈ S,
Λ0 ∈ M0} is consistent with respect to the action of W̃
(
D
(1)
6
)
in the sense of Theorem 4.1. Then,
we have the following theorem.
Theorem 4.2. For each label η ∈ S, the family of functions {τ (η)
Λ0
(ε)}Λ0∈M0 defined by (4.11)
satisfies all the bilinear equations of type (B)0, (B)′0, (B)d0, (D)0, (D)′0 and (D)d0 under the
conditions (4.1) and (4.3).
Before discussing the construction of the hypergeometric τ -functions on Mn for n ∈ Z≥1, we
mention those on Mn for n ∈ Z<0.
Lemma 4.1. For any fixed n ∈ Z<0, we have τΛn(ε) = 0 for any Λn ∈ Mn under the condi-
tions (4.1) and (4.3).
5 The construction of the τ -functions on M1
In this section, we construct the hypergeometric τ -functions on M1. We find that a class of bilin-
ear equations for the lattice τ -functions yields the contiguity relations for the q-hypergeometric
function 8W7 [5, 1]. As is well-known, the q-hypergeometric function 8W7 possesses the W (D5)-
symmetry [13]. From that, we can construct a set of twelve solutions corresponding to the coset
W (D6)/W (D5), and describe the action of W̃ (D(1)
6 ) on the set of solutions.
5.1 The q-hypergeometric function 8W7 and its transformation formula
Fix a complex number q with 0 < |q| < 1. Let us consider the basic hypergeometric function
8W7 = 8W7(a0; a1, . . . , a5; q, z) defined by (1.2). It is well-known that this function admits the
transformation formula [5, 1]
8W7(a0; a1, a2, a3, a4, a5; q, z) =
(
qa0,
qa0
a4a5
,
q2a2
0
a1a2a3a4
,
q2a2
0
a1a2a3a5
; q
)
∞(
qa0
a4
, qa0
a5
,
q2a2
0
a1a2a3
,
q2a2
0
a1a2a3a4a5
; q
)
∞
× 8W7
(
qa2
0
a1a2a3
; qa0
a2a3
, qa0
a1a3
, qa0
a1a2
, a4, a5; q, qa0
a4a5
)
,
12 T. Masuda
which can be expressed by the following identity
(q2a2
0/a1a2a3a4a5; q)∞
5∏
k=1
(qa0/ak; q)∞
(qa0; q)∞
8W7(a0; a1, . . . , a5; q, z)
=
(q2ã2
0/ã1ã2ã3ã4ã5; q)∞
5∏
k=1
(qã0/ãk; q)∞
(qã0; q)∞
8W7(ã0; ã1, . . . , ã5; q, z̃)
with respect to the coordinate transformation
ã0 = qa2
0/a1a2a3,
ã1 = qa0/a2a3, ã2 = qa0/a1a3, ã3 = qa0/a1a2,
ã4 = a4, ã5 = a5.
In this form, the function is manifestly invariant under the permutation of the parameters
a1, . . . , a5.
Assume that Im δ > 0. We relate the variables ai to εj by
a0 = e(δ − ε669), ai = e(δ − εi69) (i = 1, 2, . . . , 5), q = e(δ). (5.1)
Since the action of s457 ∈ W (D6) = 〈s12, s23, s34, s45, s56, s127〉 on the variables ai is given by
s457 : a0 7→ qa2
0/a1a2a3,
a1 7→ qa0/a2a3, a2 7→ qa0/a1a3, a3 7→ qa0/a1a2,
a4 7→ a4, a5 7→ a5,
we see that this action leads us to the above transformation formula for 8W7.
Let us introduce the function µ(6)(ε) that is invariant under the action of the symmetric
group S5 = 〈s12, s23, s34, s45〉 ⊂ S6 and satisfies
µ(6)(s457(ε))
µ(6)(ε)
=
g+(ε459)g+(2δ − ε669)
∏
i=4,5
g+(δ + ω − εi67)
g+(ε79)g+(2δ − ε669 − ε457)
∏
i=4,5
g+(δ + ω − εi6)
, (5.2)
where g+(x) is given by G+(x) =
g+(x)
(u; q)∞
with u = e(x) and q = e(δ). The relation (5.2) means
that the function
g+(2δ − ε669)
g+(ε79)
∏
i∈C6
1
g+(δ + ω − εi6)
µ(6)(ε) is invariant under the action of s457.
Then, we see that the function
µ(6)(ε)
G+(2δ − ε669)
G+(ε79)
∏
i∈C6
G−(εi6)Φ(6)(ε), (5.3)
where Φ(6)(ε) = 8W7(a0; a1, a2, a3, a4, a5; q, z), is invariant under the action of the finite Weyl
group W (D5) = 〈s12, s23, s34, s45, s127〉 ⊂ W (D6).
5.2 The contiguity relations for 8W7
It is also known that the q-hypergeometric function Φ(6) = 8W7 satisfies the following contiguity
relations [5, 1]
(a1 − a2)(1− z)Φ(6) = a1
5∏
i=3
(1− qa0/a1ai)
1− qa0/a1
Φ(6)|a1 7→a1/q
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 13
− a2
5∏
i=3
(1− qa0/a2ai)
1− qa0/a2
Φ(6)|a2 7→a2/q, (5.4)
(a2 − a1)(1− a0/a1a2)Φ(6) = (1− a1)(1− a0/a1)Φ(6)|a1 7→qa1
− (1− a2)(1− a0/a2)Φ(6)|a2 7→qa2 , (5.5)
(1− a0/a1)(1− z)Φ(6) =
5∏
i=1
(1− a0/ai)
(1− q−1a0)(1− a0)(1− q−1a1)
Φ(6)(−)
− q−1a1
5∏
i=2
(1− qa0/a1ai)
(1− q−1a1)(1− qa0/a1)
Φ(6)|a1 7→a1/q, (5.6)
Φ(6)|a1 7→qa1 − Φ(6) = q−1z
(1− qa0)(1− q2a0)
5∏
i=2
(1− ai)
(1− a0/a1)
5∏
i=1
(1− qa0/ai)
Φ(6)(+), (5.7)
where Φ(6)(±) = Φ(6)|a0 7→q±2a0,a1 7→q±1a1,...,a5 7→q±1a5
.
Noticing that the action of translation operators T̃i7 ∈ W̃
(
D
(1)
6
)
(i ∈ C) on the variables ai
(i = 0, 1, . . . , 5) is given by
T̃i7 : ai 7→ q−1ai, T̃67 : a0 7→ q−2a0, ai 7→ q−1ai (i ∈ C6),
we see that the contiguity relations (5.4) and (5.5) can be rewritten as
(−1)ω[εjk][ε79] Φ(6)(ε) =
∏
l∈C6\{j,k}
[εjl9]
[εj6 − δ]
Φ(6)
(
T̃j7(ε)
)
−
∏
l∈C6\{j,k}
[εkl9]
[εk6 − δ]
Φ(6)
(
T̃k7(ε)
)
(5.8)
and
[εjk][εjk9 − δ] Φ(6)(ε) = [εj69 − δ][εj6]Φ(6)
(
T̃−1
j7 (ε)
)
− [εk69 − δ][εk6]Φ(6)
(
T̃−1
k7 (ε)
)
, (5.9)
respectively, for j, k ∈ C6. Similarly, the contiguity relations (5.6) and (5.7) are expressed by
(−1)ω[εk6][εk69][ε79]Φ(6)(ε) =
∏
l∈C6\{k}
[εkl9]
[εk6 − δ]
Φ(6)
(
T̃k7(ε)
)
−
∏
l∈C6
[εl6]
[δ − ε669][−ε669]
Φ(6)
(
T̃67(ε)
)
(5.10)
and
Φ(6)(ε) = Φ(6)(T̃−1
k7 (ε))−
[3δ − ε669][2δ − ε669]
∏
l∈C6\{k}
[εl69 − δ]
[εk6]
∏
l∈C6
[εl6 − δ]
Φ(6)
(
T̃−1
67 (ε)
)
, (5.11)
respectively, for k ∈ C6.
Let us introduce the function Ψ(6)(ε) by∏
i,j∈C6; i<j
1
G+(εij9)
Ψ(6)(ε) = µ(6)(ε)
G+(2δ − ε669)
G+(ε79)
∏
i∈C6
G−(εi6) Φ(6)(ε),
14 T. Masuda
where the right-hand side is the W (D5)-invariant function (5.3). We see that the function Ψ(6)(ε)
is S5-invariant and satisfies the relation∏
i,j∈{1,2,3}; i<j
G+(εij9)
∏
i=1,2,3
G−(εi69) G+(ε459) Ψ(6)(s457(ε)) = G+(ε79)Ψ(6)(ε).
Suppose that the correction factor µ(6)(ε), introduced in the previous subsection, satisfies the
difference equation µ(6)(T̃i7(ε)) = (−1)ωµ(6)(ε) (i ∈ C). Then both of the contiguity rela-
tions (5.8) and (5.10) yield
(−1)ω+1ε2+[εjk][εjk9]Ψ(6)(ε) = Ψ(6)
(
T̃j7(ε)
)
−Ψ(6)
(
T̃k7(ε)
)
for j, k ∈ C. Similarly, we see that (5.9) and (5.11) are reduced to
ε−2
+ [εjk][ε79 − δ]Ψ(6)(ε) =
∏
l∈C\{j,k}
[εkl9 − δ]Ψ(6)
(
T̃−1
k7 (ε)
)
−
∏
l∈C\{j,k}
[εjl9 − δ]Ψ(6)
(
T̃−1
j7 (ε)
)
for j, k ∈ C. It is easy to see that the function Ψ(a)(ε) (a ∈ C6) defined by Ψ(a)(ε) = Ψ(6)(sa6(ε))
satisfies the same contiguity relations as those for Ψ(6)(ε).
Proposition 5.1. Each of the functions Ψ(a)(ε) (a ∈ C) satisfies the contiguity relations
(−1)ω+1ε2+[εjk][εjk9]Ψ(a)(ε) = Ψ(a)
(
T̃j7(ε)
)
−Ψ(a)
(
T̃k7(ε)
)
, (5.12)
ε−2
+ [εjk][ε79 − δ]Ψ(a)(ε)
=
∏
l∈C\{j,k}
[εkl9 − δ]Ψ(a)
(
T̃−1
k7 (ε)
)
−
∏
l∈C\{j,k}
[εjl9 − δ]Ψ(a)
(
T̃−1
j7 (ε)
)
(5.13)
for mutually distinct indices j, k ∈ C.
Here, we give a remark on choice of the correction factor µ(6)(ε). The function µ(6)(ε) in the
form
µ(6)(ε) = ν(6)(ε)
g+(ε79)
∏
i∈C6
g+(δ + ω − εi6)
g+(2δ − ε669)
,
where ν(6)(ε) is a W (D5)-invariant function, is manifestly S5-invariant and satisfies the rela-
tion (5.2). Due to g+(x+δ) = −ε+e(−1
2x)g+(x), what we have to do is to find a W (D5)-invariant
function ν(6)(ε) satisfying the difference equations
ν(6)(T̃i7(ε)) = (−1)ωε−2
+ e
(
1
2(ε79 − εi6 + δ + ω)
)
ν(6)(ε) (i ∈ C6),
ν(6)(T̃67(ε)) = (−1)ωε2+e
(
1
2ε669
)
ν(6)(ε). (5.14)
It is easy to see that the function ν(6)(ε) in the form
ν(6)(ε) = ϕ1(ε79)
∏
i,j∈C6 ; i<j
ϕ1(εij9)
∏
i∈C6
ϕ1(δ + ω − εi69)
∏
i∈C6
ϕ2(εi67)ϕ2(εi6)ϕ3(ε667),
where ϕi(x) (i = 1, 2, 3) are arbitrary functions, is W (D5)-invariant. When ϕi(x) (i = 1, 2, 3)
satisfy ϕi(x + δ) = e(αix + βi)ϕi(x) with α3 = 2α1 − α2, 8α1 + 4α2 = 1 and ε−2
+ e((α1 − α2)δ +
α2ω + (−4β1 + 5β2 + β3)) = 1, the function ν(6)(ε) satisfies the difference equations (5.14).
A typical choice of them is given by ϕi(x) = e(αiδ
(
x/δ
2
)
+ βix/δ). It is possible to determi-
ne µ(6)(ε) according to the choice of the functions ϕi(x) (i = 1, 2, 3) and G+(x). We have
proposed some examples of the functions G+(x) and F+(x) in Appendix of [14].
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 15
5.3 Twelve solutions
Hereafter, we denote Ψ(a)(ε) by Ψ(a;+)(ε). Since the action of the central element wc ∈ W (D6)
on the variables ai (i = 0, 1, . . . , 5) is given by wc(ai) = q/ai, the application of wc to the
contiguity relations (5.12) and (5.13) leads us to
ε2+[εjk][εjk9 − δ]Ψ̌(a;+)(ε) = Ψ̌(a;+)
(
T̃−1
k7 (ε)
)
− Ψ̌(a;+)
(
T̃−1
j7 (ε)
)
,
(−1)ωε−2
+ [εjk][ε79]Ψ̌(a;+)(ε) =
∏
l∈C\{j,k}
[εkl9]Ψ̌(a;+)
(
T̃k7(ε)
)
−
∏
l∈C\{j,k}
[εjl9]Ψ̌(a;+)
(
T̃j7(ε)
)
,
where Ψ̌(a;+)(ε) = Ψ(a;+)(wc(ε)). Let us introduce the function Ψ(a;−)(ε) by
Ψ(a;−)(ε) = E(a;+)(ε)G(a;+)(ε)Ψ̌(a;+)(ε),
E(a;+)(ε) = A′+(ε79)
∏
i,j∈Ca; i<j
A′+(εij9)
∏
i∈Ca
A′+(δ + ω − εia9)
∏
i∈Ca
B′+(εia7)B′+(εia)C′+(εaa7),
G(a;+)(ε) =
∏
i∈Ca
G+(εia9)
∏
i,j∈Ca; i<j
G−(εij9)
G−(ε79)
,
where the functions A′+(x), B′+(x) and C′+(x) are expressed in terms of A+(x), B+(x) and C+(x),
introduced in the previous section, by
A′+(x) =
A+(2δ + ω − x)
A+(δ + ω − x)
, B′+(x) =
B+(−x)
B+(2δ − x)
, C′+(x) =
C+(−δ − x)
C+(3δ − x)
.
When we set dC = dL = (−1)ωε2+, the factors E(a;+)(ε) and G(a;+)(ε) satisfy the difference
equations
E(a;+)
(
T̃i7(ε)
)
= (−1)ω+1E(a:+)(ε), G(a:+)
(
T̃i7(ε)
)
= (−1)ω+1ε4+
∏
l∈Ci
[εil9]
[ε79]
G(a:+)(ε)
for i ∈ C, and we get e(2αδ + 4b + 2c) = (−1)ω+1 and e(αω + 2a) = e(5b + 3c). Thus, we find
that each of the functions Ψ(a;−)(ε) satisfies the contiguity relations
ε−2
+ [εjk][ε79 − δ]Ψ(a;−)(ε) =
∏
l∈C\{j,k}
[εkl9 − δ]Ψ(a;−)
(
T̃−1
k7 (ε)
)
−
∏
l∈C\{j,k}
[εjl9 − δ]Ψ(a;−)
(
T̃−1
j7 (ε)
)
,
(−1)ω+1ε2+[εjk][εjk9]Ψ(a;−)(ε) = Ψ(a;−)
(
T̃j7(ε)
)
−Ψ(a;−)
(
T̃k7(ε)
)
,
which are the same as those for Ψ(a;+)(ε).
Theorem 5.1. Each of the twelve functions Ψ(ε) = Ψ(a;±)(ε) gives rise to the solution of the
contiguity relations
(−1)ω+1ε2+[εjk][εjk9]Ψ(ε) = Ψ
(
T̃j7(ε)
)
−Ψ
(
T̃k7(ε)
)
,
ε−2
+ [εjk][ε79 − δ]Ψ(ε) =
∏
l∈C\{j,k}
[εkl9 − δ]Ψ
(
T̃−1
k7 (ε)
)
−
∏
l∈C\{j,k}
[εjl9 − δ]Ψ
(
T̃−1
j7 (ε)
)
for mutually distinct indices j, k ∈ C.
16 T. Masuda
From these contiguity relations, one can get the q-hypergeometric equation of the second
order. The functions Ψ(a;±)(ε) coincide with the twelve pairwise linearly independent solutions
to the q-hypergeometric equation constructed by Gupta and Masson [2].
Furthermore, we introduce the function E(a;−)(ε) by
E(a;−)(ε) = A′−(ε79)
∏
i,j∈Ca; i<j
A′−(εij9)
∏
i∈Ca
A′−(δ + ω − εia9)
∏
i∈Ca
B′−(εia7)B′−(εia)C′−(εaa7),
where A′−(x), B′−(x) and C′−(x) are defined by A′−(x)A′+(δ +ω−x) = 1, B′−(x)B′+(−x) = 1 and
C′−(x)C′+(−x) = 1, respectively. By construction, we have the following proposition.
Proposition 5.2. The action of W (D6) on the functions Ψ(a;±)(ε) is described as follows:
1. For any permutation σ ∈ S6, we have Ψ(a;±)(σ(ε)) = Ψ(σ(a);±)(ε).
2. Take two mutually distinct indices i, j ∈ C.
(a) If a /∈ {i, j}, then
Ψ(a;±)(sij7(ε)) =
G±(ε79)
G±(εij9)
∏
k,l∈C\{i,j,a};
k<l
G±(εkl9)
∏
k∈C\{i,j,a}
G∓(εka9)
Ψ(a;±)(ε).
(b) If a ∈ {i, j}, then
Ψ(a;±)(sij7(ε)) =
1
E(b;±)(ε)
G∓(ε79)
G±(εij9)
∏
k,l∈C\{i,j}; k<l
G∓(εkl9)
Ψ(b;∓)(ε),
where b is an index such that {a, b} = {i, j}.
3. The action of the central element wc ∈ W (D6) is given by
Ψ(a;∓)(ε) = E(a;±)(ε)
∏
i∈Ca
G±(εia9)
∏
i,j∈Ca; i<j
G∓(εij9)
G∓(ε79)
Ψ(a;±)(wc(ε)).
The set of twelve functions Ψ(a;±)(ε) corresponds to the coset W (D6)/W (D5), as we will see
below. Note that |W (D6)/W (D5)| = 12.
5.4 The τ -functions on M1
Here, we construct the functions τΛ1(ε) (Λ1 ∈ M1) on the basis of the discussion in the previous
subsections. The bilinear equations to be considered are of type (C)0, (C)′0, (C)d0 , (D)1, (D)′1
and (D)d1 , since the functions τΛ0(ε) (Λ0 ∈ M0) are already known.
It is easy to get the following lemma.
Lemma 5.1. If the lattice τ -functions satisfy all the bilinear equations of type (B)d0 and (C)d0
under the boundary condition (4.1), then they also satisfy those of type (C)i0.
From this lemma, we see that it is sufficient for constructing the hypergeometric τ -functions
on M1 to consider the bilinear equations of type (C)d0 .
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 17
Definition 5.1. For each Λ1 ∈ M1, we define the twelve functions τ
(a;±)
Λ1
(ε) by
τ
(a;±)
Λ1
(ε) = N (a;±)
Λ1
(ε) Ψ(a;±)
Λ1
(ε), (5.15)
where N (a;±)
Λ1
(ε) is given by
N (a;±)
Λ1
(ε) = F±(ε79 + (〈e79,Λ1〉+ 1)δ)
∏
i,j∈C; i<j
F±κ
(a)
ij
(εij9 + 〈eij9,Λ1〉δ)
×A±(ε79 + (〈e79,Λ1〉+ 1)δ)
×
∏
i,j∈Ca; i<j
A±(εij9 + (〈eij9,Λ1〉+ 1)δ)
∏
i∈Ca
A∓(εia9 + 〈eia9,Λ1〉δ)
×
∏
i∈Ca
B±(εia7 + 〈eia7,Λ1〉δ)B±(εia + 〈eia,Λ1〉δ)
× C±(εaa7 + (〈eaa7,Λ1〉 − 1)δ),
and Ψ(a;±)
Λ1
(ε) = Ψ(a;±)(ε + 〈e,Λ1〉δ).
Theorem 5.2. The action of W̃
(
D
(1)
6
)
on the functions τ
(a;±)
Λ1
(ε) is described as follows:
1. For any translation operator T ∈ W̃
(
D
(1)
6
)
, we have τ
(a;±)
T.Λ1
(ε) = τ
(a;±)
Λ1
(T (ε)).
2. For any permutation σ ∈ S6, we have τ
(σ(a);±)
σ.Λ1
(ε) = τ
(a;±)
Λ1
(σ(ε)).
3. Take two mutually distinct indices i, j ∈ C.
(a) If a /∈ {i, j}, then τ
(a;±)
sij7.Λ1
(ε) = τ
(a;±)
Λ1
(sij7(ε)).
(b) If a ∈ {i, j}, then τ
(a;±)
sij7.Λ1
(ε) = τ
(b;∓)
Λ1
(sij7(ε)), where b is an index such that {a, b} =
{i, j}.
4. The action of the central element wc ∈ W (D6) is given by τ
(a;∓)
wc.Λ1
(ε) = τ
(a;±)
Λ1
(wc(ε)).
Proof. The first and second statements are obvious from the definition of τ
(a;±)
Λ1
(ε). The third
and fourth statements are guaranteed by Proposition 5.2 and (5.15). �
Corollary 5.1. For the particular element e8 ∈ M1, the set of twelve functions
τ (a;±)
e8 (ε) = F±(ε79 + δ)
∏
i,j∈C; i<j
F±κ
(a)
ij
(εij9)
×A±(ε79 + δ)
∏
i,j∈Ca; i<j
A±(εij9 + δ)
∏
i∈Ca
A∓(εia9)
×
∏
i∈Ca
B±(εia7)B±(εia)× C±(εaa7 − δ)Ψ(a;±)(ε)
is stabilized by W (D6)1. For each label (a;±) ∈ S, the isotropy subgroup of τ
(a;±)
e8 (ε) is isomor-
phic to W (D5);
τ (6;±)
e8 (w(ε)) = τ (6;±)
e8 (ε), w ∈ W (D5) = 〈s12, s23, s34, s45, s127〉
for instance.
1Note that e8 ∈ M1 is W (D6)-invariant.
18 T. Masuda
Let us consider the bilinear equation of type (C)d0
τeiτe0−ei−e9 − τejτe0−ej−e9 + [εij ][εij9]dLτe7τe8 = 0 (5.16)
for mutually distinct indices i, j ∈ C. Substituting (4.11) and (5.15) into (5.16), we get for
Ψ(a;±)(ε) the linear relation
(−1)ω+1ε2+[εij ][εij9]Ψ(a;±)(ε) = Ψ(a;±)(T̃i7(ε))−Ψ(a;±)(T̃j7(ε)).
Similarly, the application of the central element wc ∈ W (D6) to the bilinear equation (5.16)
leads us to
ε−2
+ [εjk][ε79 − δ]Ψ(a;±)(ε) =
∏
l∈C\{j,k}
[εkl9 − δ]Ψ(a;±)
(
T̃−1
k7 (ε)
)
−
∏
l∈C\{j,k}
[εjl9 − δ]Ψ(a;±)(T̃−1
j7 (ε))
for mutually distinct indices i, j ∈ C. These are precisely the contiguity relations in Theorem 5.1.
Also, the set of functions {τ (η)
Λ1
(ε) | η ∈ S, Λ1 ∈ M1} is consistent with respect to the action
of W̃
(
D
(1)
6
)
in the sense of Theorem 5.2. Therefore, we have the following theorem due to
Propositions 3.2, 3.3, 3.4 and Lemma 5.1.
Theorem 5.3. For each label η ∈ S, the family of functions {τ (η)
Λ (x)}Λ∈M0
∐
M1
defined by (4.11)
and (5.15) satisfies all the bilinear equations of type (C)d0, (C)i0, (C)r0, (C)′0, (D)d1, (D)1 and (D)′1
under the conditions (4.1) and (4.3).
Remark 5.1. From this theorem, we see that the bilinear equations of type (C)d0 , (C)i0, (C)r0
and (C)′0 imply the contiguity relations for the q-hypergeometric function 8W7. Also, we get the
quadratic relations for 8W7 from the bilinear equations of type (D)d1 , (D)1 and (D)′1.
Remark 5.2. From the result in this section, one can get an explicit expression for the so-called
Riccati solution to the system of q-difference equations (1.1), in terms of the functions τ
(η)
Λ1
(ε).
When the label η ∈ S is fixed, one can express bi (i = 1, 2, . . . , 8) and t in terms of the parameters
of the q-hypergeometric function 8W7. On the other hand, another expression for the Riccati
solution has been proposed in [10], which is constructed under the condition b1b3 = b5b7; that
is, ε358 ∈ Z. Comparing this with the condition (4.3), we find that these two expressions can be
transformed to each other by a Bäcklund transformation.
6 A determinant formula for the hypergeometric τ -functions
One of the important features of the hypergeometric solutions to the continuous and discrete
Painlevé equations is that they can be expressed in terms of Wronskians or Casorati determi-
nants [4, 12, 7, 3, 16]. In this section, we show that the hypergeometric τ -functions on Mn
(n ∈ Z≥2) are expressed by “two-directional Casorati determinants” of order n.
Let us introduce the auxiliary variables xi (i = 0, 1, . . . , 6) by x0 = δ − ε78 and xi = 1
2εii9
(i ∈ C), where we have x0 + x1 + · · · + x6 = 2δ + 2ω. Under the conditions (4.1) and (4.3),
the functions τΛ(ε) depend on xi (and ω). In what follows, we denote the hypergeometric
τ -functions by τΛ(x) instead of by τΛ(ε) for convenience. Also, we denote a function f (η)(x)
(η ∈ S) by f(η;x).
For each n ∈ Z≥0, we define the twelve functions Kn(η;x) = K
(a;±)
n (x) by the following
“two-directional Casorati determinants”
K2m
(
η;x0 + 2m−1
2 δ, xi + 2m−1
4 δ
) ∣∣∣
xi 7→xi−(m−1)δ (i=1,2,3,4)
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 19
= det
(
Ψ(b−m,m + 1− b, a−m,m + 1− a)
)2m
a,b=1
,
K2m+1
(
η;x0 + mδ, xi + m
2 δ
) ∣∣∣
xi 7→xi−mδ (i=1,2,3,4)
= det
(
Ψ(b−m− 1,m + 1− b, a−m− 1,m + 1− a)
)2m+1
a,b=1
,
where Ψ(m1,m2,m3,m4) = Ψ(η;x)|xi 7→xi+miδ (i=1,2,3,4), and Ψ(η;x) is the hypergeometric func-
tion multiplied by some normalization factors, introduced in Section 5.2 and 5.3. The first some
members of Kn(η;x) are given as follows:
K0(x) = 1, K1(x) = Ψ(x), K2
(
x0 + 1
2δ, xi + 1
4δ
)
=
∣∣∣∣ Ψ24(x) Ψ14(x)
Ψ23(x) Ψ13(x)
∣∣∣∣ ,
K3
(
x0 + δ, xi + 1
2δ
) ∣∣∣
xi 7→xi−δ (i=1,2,3,4)
=
∣∣∣∣∣∣∣
Ψ24
13(x) Ψ4
3(x) Ψ14
23(x)
Ψ2
1(x) Ψ(x) Ψ1
2(x)
Ψ23
14(x) Ψ3
4(x) Ψ13
24(x)
∣∣∣∣∣∣∣ ,
where we omit the label η ∈ S for simplicity, and Ψi1...ir
j1...jr
(x) = Ψ(x)
∣∣∣ xi 7→xi+δ (i=i1,...,ir)
xj 7→xj−δ (j=j1,...,jr)
. By using
Jacobi’s identity, one can easily see that each of the functions Kn(η;x) satisfies the relation
Kn+1(η;x)K(1234)
n−1
(
η;x0 − δ, xi − δ
2
)
= K(24)
n
(
η;x0 − δ
2 , xi − δ
4
)
K(13)
n
(
η;x0 − δ
2 , xi − δ
4
)
−K(14)
n
(
η;x0 − δ
2 , xi − δ
4
)
K(23)
n
(
η;x0 − δ
2 , xi − δ
4
)
,
where K
(i1...ir)
n (η;x) = Kn(η;x) |xi 7→xi+δ (i=i1,...,ir).
Definition 6.1. For each n ∈ Z≥0, we define the twelve functions τn(η;x) by τn(η;x) =
Υn(η;x)Kn(η;x). The normalization factor Υn(η;x) = Υ(a;±)
n (x) is given by
Υ(a;±)
n (x) =
1
cn(x)
F∓
(
x0 + 1−n
2 δ
) ∏
i,j∈C; i<j
F±κ
(a)
ij
(
xi + xj + 1−n
2 δ
)
×A∓
(
x0 + 1−n
2 δ
) ∏
i,j∈Ca; i<j
A±
(
xi + xj + n+1
2 δ
) ∏
i∈Ca
A∓
(
xi + xa + 1−n
2 δ
)
×
∏
i∈Ca
B±(x0 + xi + xa − ω − nδ)B±(xa − xi + (1− n)δ)C±(x0 + 2xa − ω − 2nδ),
where the functions F±(x), A±(x), B±(x) and C±(x) are introduced in Section 4. The factor
cn(x) is defined by
cn(x) = (−1)(ω+1)(n
2)ε
4(n
2)
+
n−1∏
r=1
[x1 − x2 + Irδ][x3 − x4 + Irδ]
×
n−1∏
r=1
[
x1 + x2 +
(
r − n+1
2
)
δ
]r [
x3 + x4 +
(
r − n+1
2
)
δ
]r
,
where Ir (r = 1, 2, . . .) is the subset of Z given by Ir = {−r + 1,−r + 3, . . . , r − 3, r − 1} and
[x + Irδ] =
∏
k∈Ir
[x + kδ].
Proposition 6.1. We have the following bilinear relation
[x1 − x2][x3 − x4]τn+1(η;x)τ (1234)
n−1
(
η;x0 − δ, xi − δ
2
)
20 T. Masuda
=
[
x1 + x4 − n
2 δ
] [
x2 + x3 − n
2 δ
]
τ (24)
n
(
η;x0 − δ
2 , xi − δ
4
)
τ (13)
n
(
η;x0 − δ
2 , xi − δ
4
)
−
[
x2 + x4 − n
2 δ
] [
x1 + x3 − n
2 δ
]
τ (14)
n
(
η;x0 − δ
2 , xi − δ
4
)
τ (23)
n
(
η;x0 − δ
2 , xi − δ
4
)
. (6.1)
This proposition is easily verified by noticing that the normalization factor Υn(η;x) satisfies
the relation
[x1 − x2][x3 − x4]Υn+1(η;x)Υ(1234)
n−1
(
η;x0 − δ, xi − δ
2
)
=
[
x1 + x4 − n
2 δ
] [
x2 + x3 − n
2 δ
]
Υ(24)
n
(
η;x0 − δ
2 , xi − δ
4
)
Υ(13)
n
(
η;x0 − δ
2 , xi − δ
4
)
=
[
x2 + x4 − n
2 δ
] [
x1 + x3 − n
2 δ
]
Υ(14)
n
(
η;x0 − δ
2 , xi − δ
4
)
Υ(23)
n
(
η;x0 − δ
2 , xi − δ
4
)
.
Definition 6.2. For each Λn ∈ Mn (n ∈ Z), we define the twelve functions τΛn(η;x) by
τΛn(η;x) = τn
(
η;x + l(n)δ
)
,
l
(n)
0 = 〈v0,Λn〉+ 1−n
2 , l
(n)
i = 〈vi,Λn〉+ 1−n
4 (i ∈ C), (6.2)
where the vectors vi are defined by v0 = c − e78 and vi = 1
2eii9 (i ∈ C) that correspond to the
variables xi.
We show below that the functions τΛn(η;x) are precisely the hypergeometric τ -functions
on Mn. As a preparation, let us define the action of W̃
(
D
(1)
6
)
on the label set S = {(a, ε) | a ∈
C, ε = ±1}.
Definition 6.3. We define the action of W̃ (D(1)
6 ) on the label η = (a, ε) ∈ S as follows:
1. The label is invariant under the action of any translation.
2. The action of any permutation σ ∈ S6 is defined by σ : (a;±) 7→ (σ(a);±).
3. Take two mutually distinct indices i, j ∈ C. If a /∈ {i, j}, then sij7 : (a;±) 7→ (a;±).
Otherwise, we have sij7 : (a;±) 7→ (b;∓), where b is an index such that {a, b} = {i, j}.
4. The action of the central element wc is defined by wc : (a;±) 7→ (a;∓).
Theorem 6.1.
1. For each η ∈ S, the family of functions {τΛ(η;x)}Λ∈ME7 satisfies all the bilinear equations
for the q-Painlevé system of type E
(1)
7 under the conditions (4.1) and (4.3).
2. For each n ∈ Z, the action of W̃
(
D
(1)
6
)
on the set of functions {τΛn(η;x) | η ∈ S, Λn ∈ Mn}
is described by τw.Λn(w(η);x) = τΛn(η;w(x)) for any w ∈ W̃
(
D
(1)
6
)
.
Let us verify the first statement. We consider the bilinear equations
[ε12][ε34]τLm,1+e8τLm,−1+2e0−e1−e2−e3−e4−e8
= [ε148 −mδ][ε238 −mδ]τLm,0+e0−e2−e4τLm,0+e0−e1−e3
− [ε248 −mδ][ε138 −mδ]τLm,0+e0−e1−e4τLm,0+e0−e2−e3 (6.3)
and
[ε12][ε34]τLm,2+c+e89+e7τLm,0+c+2e0−e1−e2−e3−e4−e8−e9+e7
= [ε148 −mδ][ε238 −mδ]τLm,1+c+e249+e7τLm,1+c+e139+e7
− [ε248 −mδ][ε138 −mδ]τLm,1+c+e149+e7τLm,1+c+e239+e7 , (6.4)
where Lm,n = m(m + n)c + me89 (m ∈ Z), which are of type (B)′2m and (B)′2m+1, respectively.
Substituting (6.2), we see that these bilinear equations are satisfied thanks to (6.1).
In order to verify the second statement of Theorem 6.1, we use the following lemma.
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 21
Lemma 6.1. Suppose that the functions τΛn−1(η;x) and τΛn(η;x) satisfy all the bilinear equa-
tions of type (D)′n and (C)′n−1, and that we have the relations τw.Λn−1(w(η);x) = τΛn−1(η;w(x))
and τw.Λn(w(η);x) = τΛn(η;w(x)) for any w ∈ W (D6). Then the function τΛn+1(η;x) deter-
mined by the bilinear equation (6.3) or (6.4) also satisfies τw.Λn+1(w(η);x) = τΛn+1(η;w(x)) for
any w ∈ W (D6).
Proof. From the assumption, we have the bilinear relations
[x1 − x2][x3 − x4]τ (12)
n (η;x)τ (34)
n (η;x) + (1, 2, 3)-cyclic = 0, (6.5)
[x3 − x4]
[
x1 + x5 − n−1
2 δ
]
τ
(1234)
n−1
(
η;x0 − δ
2 , xi − δ
4
)
τ (25)
n (η;x) + (3, 4, 5)-cyclic = 0,
[x3 − x4]
[
x2 + x5 − n−1
2 δ
]
τ
(1234)
n−1
(
η;x0 − δ
2 , xi − δ
4
)
τ (15)
n (η;x) + (3, 4, 5)-cyclic = 0, (6.6)
and the relations τn−1(w(η);x) = τn−1(η;w(x)) and τn(w(η);x) = τn(η;w(x)) for any w ∈
W (D6) = 〈s12, s23, s34, s45, s56, s127〉. What we have to do is to show that the function τn+1(η;x)
determined by the recurrence relation (6.1) also satisfies
τn+1(w(η);x) = τn+1(η;w(x)) (6.7)
for any w ∈ W (D6). It is obvious that we have (6.7) for w = s12, s34, s56 and s127 under the
assumption. Then, it is sufficient to verify (6.7) for w = s23 and s45. Replacing x by x̃ = s23(x)
in the recurrence relation (6.1), we get
[x1 − x3][x2 − x4]τn+1(η; x̃)τ (1234)
n−1
(
η̃;x0 − δ, xi − δ
2
)
=
[
x1 + x4 − n
2 δ
] [
x2 + x3 − n
2 δ
]
τ (34)
n
(
η̃;x0 − δ
2 , xi − δ
4
)
τ (12)
n
(
η̃;x0 − δ
2 , xi − δ
4
)
−
[
x3 + x4 − n
2 δ
] [
x1 + x2 − n
2 δ
]
τ (14)
n
(
η̃;x0 − δ
2 , xi − δ
4
)
τ (23)
n
(
η̃;x0 − δ
2 , xi − δ
4
)
,
where η̃ = s23(η). Then, the bilinear equation (6.5) yields τn+1(η̃;x) = τn+1(η; x̃). Similarly,
replacing x by x̃ = s45(x) in the recurrence relation (6.1), we get
[x1 − x2][x3 − x5]τn+1(η; x̃)τ (1235)
n−1
(
η̃;x0 − δ, xi − δ
2
)
=
[
x1 + x5 − n
2 δ
] [
x2 + x3 − n
2 δ
]
τ (25)
n
(
η̃;x0 − δ
2 , xi − δ
4
)
τ (13)
n
(
η̃;x0 − δ
2 , xi − δ
4
)
−
[
x2 + x5 − n
2 δ
] [
x1 + x3 − n
2 δ
]
τ (15)
n
(
η̃;x0 − δ
2 , xi − δ
4
)
τ (23)
n
(
η̃;x0 − δ
2 , xi − δ
4
)
,
where η̃ = s45(η). From the bilinear relations (6.6), we get τn+1(η̃;x) = τn+1(η; x̃). �
We already have τw.Λ0(w(η);x) = τΛ0(η;w(x)) and τw.Λ1(w(η);x) = τΛ1(η;w(x)) for any
w ∈ W (D6) from Theorems 4.1 and 5.2. Also, these functions satisfy all the bilinear equations
of type (C)′0 and (D)′1 from Theorem 5.3. Then we have τw.Λ2(w(η);x) = τΛ2(η;w(x)) for any
w ∈ W̃
(
D
(1)
6
)
from Lemma 6.1. Applying Propositions 3.2, 3.3 and 3.4 repeatedly, we can verify
the second statement of Theorem 6.1.
With respect to the system of q-difference equations (1.1), one can get the explicit expression
for the hypergeometric solutions in terms of the functions τn(η;x) introduced in Definition 6.1.
When the label η ∈ S is fixed, one can express bi (i = 1, 2, . . . , 8) and t in terms of the parameters
of the q-hypergeometric function 8W7. For instance, in the case of η = (6;+) ∈ S, we have the
following.
Corollary 6.1. Define the functions fn(x) and gn(x) by
fn(x) =
(
qa0a4a5
a1a2a3
)1/4
t1/2 Nf,n(x)
Df,n(x)
, gn(x) =
(
a1a2a3
qa0a4a5
)1/4
t1/2 Ng,n(x)
Dg,n(x)
22 T. Masuda
with
Nf,n(x) = (a1/a3)1/4τ [1]
n
(
x0 − 3
2δ, xi + δ
4
)
τ (12)
n
(
x0 + 3
2δ, xi − δ
4
)
− (a3/a1)1/4τ [3]
n
(
x0 − 3
2δ, xi + δ
4
)
τ (23)
n
(
x0 + 3
2δ, xi − δ
4
)
,
Df,n(x) = (a1/a3)1/4τ [3]
n
(
x0 − 3
2δ, xi + δ
4
)
τ (23)
n
(
x0 + 3
2δ, xi − δ
4
)
− (a3/a1)1/4τ [1]
n
(
x0 − 3
2δ, xi + δ
4
)
τ (12)
n
(
x0 + 3
2δ, xi − δ
4
)
,
Ng,n(x) = (a2/a3)1/4τ [3]
n
(
x0 − 3
2δ, xi + δ
4
)
τ (13)
n
(
x0 + 3
2δ, xi − δ
4
)
− (a3/a2)1/4τ [2]
n
(
x0 − 3
2δ, xi + δ
4
)
τ (12)
n
(
x0 + 3
2δ, xi − δ
4
)
,
Dg,n(x) = (a2/a3)1/4τ [2]
n
(
x0 − 3
2δ, xi + δ
4
)
τ (12)
n
(
x0 + 3
2δ, xi − δ
4
)
− (a3/a2)1/4τ [3]
n
(
x0 − 3
2δ, xi + δ
4
)
τ (13)
n
(
x0 + 3
2δ, xi − δ
4
)
,
where a0, a1, . . . , a5 are the parameters of the hypergeometric function 8W7(a0; a1, . . . , a5; q, z)
defined by (5.1), τ
(i1i2)
n (x) = τn(x)|xi 7→xi+δ (i=i1,i2) and τ
[i]
n (x) = τn(x)|xi 7→xi−δ. Let bi,n (i =
1, 2, . . . , 8) be the parameters defined by
b1,n = −q1/4a
−3/4
3 (a0a4a5)1/4, b2,n = −q1/4a
−3/4
4 (a0a3a5)1/4,
b3,n = −q1/4a
−3/4
5 (a0a3a4)1/4, b4,n = −q1/4a
−3/4
0 (a3a4a5)1/4,
b5,n = −q1/4(a1a2)−1/2(a0a3a4a5)1/4, b6,n = −q5/4a
5/4
0 (a1a2)−1/2(a3a4a5)−3/4,
b7,n = −qn/2−3/4a
−3/4
0 (a1a2)1/2(a3a4a5)1/4, b8,n = −q−n/2−3/4a
−3/4
0 (a1a2)1/2(a3a4a5)1/4
and t = (a1/a2)1/2. Then, f = fn(x) and g = gn(x) with bi = bi,n give rise to a solution of the
system of q-difference equations (1.1).
A The q-Painlevé system of type E
(1)
7
A.1 Point configurations and Cremona transformations
As mentioned in Section 2, the q-Painlevé system of type E
(1)
7 is a discrete dynamical system
defined on a family of rational surfaces parameterized by nine-point configurations on P2 such
that six points among them are on a conic C and other three are on a line L [17]. In this section,
we set p1, p2, p4, p5, p6, p7 ∈ C and p3, p8, p9 ∈ L so that the standard Cremona transformation
with respect to (p1, p2, p3) is well-defined as a birational action on the configuration space. One
can parameterize the configuration space by [17, 9]
X =
1 1 −u3 1 1 1 1 −u8 −u9 x1
u1 u2 0 u4 u5 u6 u7 0 0 x2
u2
1 u2
2 1 u2
4 u2
5 u2
6 u2
7 1 1 x3
,
where u1, u2, . . . , u9 are parameters satisfying u1u2 · · ·u9 = q−1 (q ∈ C∗) and the tenth column
denotes the coordinates of a general point on P2. The symmetric group SC
6 ×SL
3 with SC
6 =
〈s12, s24, s45, s56, s67〉 and SL
3 = 〈s38, s89〉 naturally acts on the space as σ(uj) = uσ(j) for any
σ ∈ SC
6 ×SL
3 .
Let us normalize X by an action of GL3(C) as
Y =
1 0 0 u14 . . . u19 y1
0 1 0 u24 . . . u29 y2
0 0 1 u34 . . . u39 y3
.
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 23
The coordinates uij can be expressed by
uij =
{
yC
i (uj) (j = 1, 2, 4, 5, 6, 7),
yL
i (uj) (j = 3, 8, 9),
(A.1)
where yC
i (u) and yL
i (u) are defined by
yC
1 (u) =
(u2 − u)(1− u2u3u)
(u2 − u1)(1− u1u2u3)
, yC
2 (u) =
(u1 − u)(1− u1u3u)
(u1 − u2)(1− u1u2u3)
,
yC
3 (u) =
(u1 − u)(u2 − u)
(1− u1u2u3)
and
yL
1 (u) =
u2(u3 − u)
(u2 − u1)(1− u1u2u3)
, yL
2 (u) =
u1(u3 − u)
(u1 − u2)(1− u1u2u3)
,
yL
3 (u) =
1− u1u2u
1− u1u2u3
,
respectively. We further normalize X (or Y ) by an action of PGL3(C) as
Z =
1 0 0 1 v15 . . . v19 z1
0 1 0 1 v25 . . . v29 z2
0 0 1 1 1 . . . 1 1
,
where the coordinates vij and zi are expressed by
vij =
u34 uij
ui4 u3j
, zi =
u34 yi
ui4 y3
(i = 1, 2; j = 5, 6, 7, 8, 9). (A.2)
The action of the standard Cremona transformation with respect to (p1, p2, p3), denoted by s123,
on these variables is given by s123(vij) = 1/vij and s123(zi) = 1/zi. This transformation
together with the symmetric group SC
6 × SL
3 generates the affine Weyl group W (E(1)
7 ) =
〈s12, s24, s45, s56, s67, s38, s89, s123〉 associated with the Dynkin diagram
c c c c c c cc
e89 e38 e123 e24 e45 e56 e67
e12
In this realization of W (E(1)
7 ) ⊂ W (E(1)
8 ) = 〈s12, s23, . . . , s89, s123〉, the automorphism of the
above Dynkin diagram can be expressed by π = s124 s35 s68 s79. The action of the extended
affine Weyl group W̃ (E(1)
7 ) = 〈s12, s24, s45, s56, s67, s38, s89, s123, π〉 on the configuration space is
given by
s12(z1) = z2, s12(z2) = z1, s38(z1) =
z1 − v18
1− v18
, s38(z2) =
z2 − v28
1− v28
,
s123(z1) =
1
z1
, s123(z2) =
1
z2
, s24(z1) =
z2 − z1
z2 − 1
, s24(z2) =
z2
z2 − 1
, (A.3)
s45(z1) =
z1
v15
, s45(z2) =
z2
v25
and
π(vij) =
vik − vi5
vik − 1
(k = 8, 9, 6, 7 for j = 6, 7, 8, 9), π(zi) =
zi − vi5
zi − 1
(A.4)
24 T. Masuda
for i = 1, 2. From this representation, we obtain a family of functional equations
w(vij) = Sw
ij(v), w(zi) = Rw
i (v; z) (A.5)
for each w ∈ W̃ (E(1)
7 ), where Sw
ij(v) and Rw
i (v; z) are some rational functions.
Let us introduce the variables ci = e(εi) (i = 0, 1, . . . , 9), where εi are the coordinate functions
introduced in Section 2, and suppose that the parameters u1, . . . , u9 are expressed by uj = cj/cr
0
(j = 1, 2, 4, 5, 6, 7) and uj = cj/cs
0 (j = 3, 8, 9) with 2r + s = 1. Then, yC
i (u) and yL
i (u) are
expressed by
yC
1 (t/cr
0) =
(1− t/c2)(1− c2c3t/c0)
(1− c1/c2)(1− c1c2c3/c0)
, yC
2 (t/cr
0) =
(1− t/c1)(1− c1c3t/c0)
(1− c2/c1)(1− c1c2c3/c0)
,
yC
3 (t/cr
0) =
c1c2(1− t/c1)(1− t/c2)
c2r
0 (1− c1c2c3/c0)
(A.6)
and
yL
1 (t/cs
0) =
c3(1− t/c3)
cs
0(1− c1/c2)(1− c1c2c3/c0)
, yL
2 (t/cs
0) =
c3(1− t/c3)
cs
0(1− c2/c1)(1− c1c2c3/c0)
,
yL
3 (t/cs
0) =
1− c1c2t/c0
1− c1c2c3/c0
, (A.7)
respectively. These expressions with (A.1) and (A.2) give us
v1j =
(u1 − u4)(1− u2u3uj)
(1− u2u3u4)(u1 − uj)
=
[ε14][ε23j ]
[ε234][ε1j ]
,
v2j =
(u2 − u4)(1− u1u3uj)
(1− u1u3u4)(u2 − uj)
=
[ε24][ε13j ]
[ε134][ε2j ]
(A.8)
for j = 5, 6, 7 and
v1j =
u2(u1 − u4)(u3 − uj)
(1− u2u3u4)(1− u1u2uj)
=
[ε14][ε3j ]
[ε234][ε12j ]
,
v2j =
u1(u2 − u4)(u3 − uj)
(1− u1u3u4)(1− u1u2uj)
=
[ε24][ε3j ]
[ε134][ε12j ]
(A.9)
for j = 8, 9. We find that these expressions satisfy w(vij)(ε) = vij(w(ε)) for any w ∈ W̃
(
E
(1)
7
)
,
namely (A.8) and (A.9) give a solution to the first equation of (A.5). Also, we see that the
functions
z1 =
[ε14]
[ε234]
[ε123 + ε1 − t]
[ε1 − t]
, z2 =
[ε24]
[ε134]
[ε123 + ε2 − t]
[ε2 − t]
(A.10)
and
z1 =
[ε14]
[ε234]
[ε3 − t]
[ε123 + ε3 − t]
, z2 =
[ε24]
[ε134]
[ε3 − t]
[ε123 + ε3 − t]
(A.11)
satisfy zi(w(ε); t) = Rw
i (ε; z(ε; t)) for any w ∈ W̃
(
E
(1)
7
)
. This means that each of (A.10)
and (A.11) provides a one-parameter family of solutions to the second equation of (A.5). These
solutions will be called the canonical solution, which correspond to the vertical solution in the
context of the differential Painlevé equations.
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 25
A.2 The lattice τ -functions and the bilinear equations
Here, we introduce a framework of the lattice τ -functions and show that the action of the
extended affine Weyl group W̃ (E(1)
7 ) is transformed into the bilinear equations for the lattice
τ -functions.
Recall that the lattice τ -functions for the discrete Painlevé system of type E
(1)
8 are indexed
by Λ ∈ M = W
(
E
(1)
8
)
. e1 = {Λ ∈ L | 〈Λ,Λ〉 = 1, 〈c,Λ〉 = −1 } [8]. Let us decompose the central
element c = 3e0 − e1 − e2 − · · · − e9 into two irreducible components by [17]
c = DC +DL,
{
DC = 2e0 − e1 − e2 − e4 − e5 − e6 − e7,
DL = e0 − e3 − e8 − e9
corresponding to the conic C and the line L. Then, we have two W (E(1)
7 )-orbits
MC = {Λ ∈ M | 〈DC ,Λ〉 = −1, 〈DL,Λ〉 = 0 } = W
(
E
(1)
7
)
. e1,
ML = {Λ ∈ M | 〈DC ,Λ〉 = 0, 〈DL,Λ〉 = −1 } = W
(
E
(1)
7
)
. e3,
which are transformed by the action of the Dynkin diagram automorphism π ∈ W̃
(
E
(1)
7
)
to each
other. Hereafter, we consider the lattice τ -functions τΛ for Λ ∈ ME7 = MC
∐
ML = W̃
(
E
(1)
7
)
e1,
on which the action of w ∈ W̃
(
E
(1)
7
)
is defined by w(τΛ) = τw.Λ.
Suppose that the variables yi are expressed by
y1 =
τe2τe3τe0−e2−e3
N1
, y2 =
τe1τe3τe0−e1−e3
N2
, y3 =
τe1τe2τe0−e1−e2
N3
,
where the normalization factors N1, N2 and N3 are certain functions of εi (i = 0, 1, . . . , 9).
Denote the τ -functions for the canonical solution on the conic C by τΛ|C (Λ ∈ ME7);
yC
1 =
τe2 |C τe3 |C τe0−e2−e3 |C
N1
, yC
2 =
τe1 |C τe3 |C τe0−e1−e3 |C
N2
,
yC
3 =
τe1 |C τe2 |C τe0−e1−e2 |C
N3
.
Comparing this expression with (A.6) and (A.7), one can assume that the τ -functions for the
canonical solutions on C and L are expressed by
τej |C =
{
(1− e(t− εj)) e(αεj) (j = 1, 2, 4, 5, 6, 7),
e(βεj) (j = 3, 8, 9),
and
τej |L =
{
e(βεj) (j = 1, 2, 4, 5, 6, 7),
(1− e(t− εj)) e(αεj) (j = 3, 8, 9),
respectively, so that the canonical solutions yC
i and yL
i are transformed by the action of the
Dynkin diagram automorphism π to each other. These requirements lead us to r = 1/4, s = 1/2
and β = α − 1/2, and we get N1 = −c
α−1/2
0 c1[ε12][ε123], N2 = c
α−1/2
0 c2[ε12][ε123] and N3 =
c
α−1/2
0 c
1/2
3 [ε123].
Let us introduce the variables fi (i = 1, 2, 3) by f1 =
τe0−e2−e3
τe1
, f2 =
τe0−e1−e3
τe2
and f3 =
τe0−e1−e2
τe3
. Then, the inhomogeneous coordinates z1 and z2 are expressed by
z1 =
[ε14]
[ε234]
f1
f3
, z2 =
[ε24]
[ε134]
f2
f3
. (A.12)
26 T. Masuda
From (A.3), (A.4) and (A.12), one thus obtain a realization of the extended affine Weyl group
W̃
(
E
(1)
7
)
as a group of birational transformations.
Theorem A.1. The action of W̃
(
E
(1)
7
)
on the variables (f1, f2, f3) and (τe1 , . . . , τe9) is given by
σ(τei) = τeσ(i)
(σ ∈ SC
6 ×SL
3 ),
s123(τe1) = τe1f1, s123(τe2) = τe2f2, s123(τe3) = τe3f3,
s12(f1) = f2, s12(f2) = f1,
s38(f1) =
τe3
τe8
[ε128]f1 − [ε38]f3
[ε123]
, s38(f2) =
τe3
τe8
[ε128]f2 − [ε38]f3
[ε123]
, s38(f3) =
τe3
τe8
f3,
s123(f1) =
1
f1
, s123(f2) =
1
f2
, s123(f3) =
1
f3
,
s24(f1) =
τe2
τe4
[ε14][ε134]f1 − [ε24][ε234]f2
[ε12][ε123]
, s24(f2) =
τe2
τe4
f2,
s24(f3) =
τe2
τe4
[ε134]f3 − [ε24]f2
[ε123]
,
and
π(τe1) =
τe1τe3
τe4
−[ε14]f1 + [ε234]f3
[ε123]
, π(τe2) =
τe2τe3
τe4
−[ε24]f2 + [ε134]f3
[ε123]
,
π(τe3) = τe5 , π(τe4) = τe3f3, π(τe5) = τe3 ,
π(τe6) = τe8 , π(τe7) = τe9 , π(τe8) = τe6 , π(τe9) = τe7 ,
π(f1) =
τe4
τe5
[ε15]f1 − [ε235]f3
[ε14]f1 − [ε234]f3
, π(f2) =
τe4
τe5
[ε25]f2 − [ε135]f3
[ε24]f2 − [ε134]f3
, π(f3) =
τe4
τe5
.
These give rise to a representation of W̃
(
E
(1)
7
)
.
From this theorem, we immediately obtain the bilinear equations (2.2) and (2.3) for mutually
distinct indices i, j, k, l ∈ {1, 2, 4, 5, 6, 7} and r, s ∈ {3, 8, 9}.
B Another representation
In this section we again set C = {1, 2, 3, 4, 5, 6} and L = {7, 8, 9}. The lattice τ -functions τΛ
(Λ ∈ ME7) for the q-Painlevé system of type E
(1)
7 satisfy the following bilinear equations
[εrs]τejτe0−ei−ej = [εijs]τerτe0−ei−er − [εijr]τesτe0−ei−es ,
[εjk]τerτe0−ei−er = [εikr]τejτe0−ei−ej − [εijr]τek
τe0−ei−ek
, (B.1)
and
[εij ][εijr]τek
τe0−ek−er + (i, j, k)-cyclic = 0,
[εij ][εkl]τe0−ei−ejτe0−ek−el
+ (i, j, k)-cyclic = 0,
where i, j, k, l ∈ C and r, s ∈ L.
As discussed in [8, 11], when Λ = de0−ν1e1−· · ·−ν9e9, the τ -function τΛ is characterized by
a homogeneous polynomial of degree d in the homogeneous coordinates of P2 which has a zero
of multiplicity ≥ νj at pj for each j = 1, . . . , 9. From this geometric consideration, we find that
we have the following bilinear equation
τeiτe0−ei−e9 − τejτe0−ej−e9 + [εij ][εij9] dLτe7τe8 = 0 (B.2)
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 27
for i, j ∈ C, which associates with the line passing through the point p9. The factor dL corre-
sponds to the irreducible component of the anti-canonical devisor DL = e0− e7− e8− e9, and is
invariant under the action of W
(
E
(1)
7
)
. The action of the Dynkin diagram automorphism π on
this bilinear equation gives us the second equation in (2.4) with dC = π(dL).
One can get another representation W̃
(
E
(1)
7
)
for the τ -variables, by using the bilinear equa-
tions (B.1) and (B.2).
Theorem B.1. Let us introduce the variables σ and σ̃ by
σ = dr
e(1
4ε23)τe3τe0−e1−e3 − e(−1
4ε23)τe2τe0−e1−e2
[ε23]
,
σ̃ = dl
e(1
4ε23)τe2τe0−e1−e2 − e(−1
4ε23)τe3τe0−e1−e3
[ε23]
,
where the factors dl and dr are given by dl = e
(
1
16αl − 1
16αr
)
and dr = d−1
l with αl = 3ε127 +
2ε78 + ε89 and αr = 3ε34 + 2ε45 + ε56. Then, the action of W̃
(
E
(1)
7
)
on the variables τe3, τe4,
τe5, τe6, τe7, τe8, τe9, τe0−e1−e2, σ and σ̃ is described as follows:
s89 : τe8 ↔ τe9 , s78 : τe7 ↔ τe8 , s127 : τe7 ↔ τe0−e1−e2 ,
s34 : τe3 ↔ τe4 , s45 : τe4 ↔ τe5 , s56 : τe5 ↔ τe6 ,
s23(τe3) =
e(−1
4ε23) d−1
r σ + e(1
4ε23) d−1
l σ̃
τe0−e1−e2
,
s23(τe0−e1−e2) =
e(1
4ε23) d−1
r σ + e(−1
4ε23) d−1
l σ̃
τe3
, (B.3)
s12(σ) =
e(−1
4ε12)τe7τe8τe9τe0−e1−e2 + e(1
4ε12)τe3τe4τe5τe6
σ̃
,
s12(σ̃) =
e(1
4ε12)τe7τe8τe9τe0−e1−e2 + e(−1
4ε12)τe3τe4τe5τe6
σ
, (B.4)
π : τe3 ↔ τe0−e1−e2 , τe4 ↔ τe7 , τe5 ↔ τe8 , τe6 ↔ τe9 , σ ↔ σ̃.
These also give rise to another representation of W̃
(
E
(1)
7
)
.
Proof. We immediately get (B.3) from the definition of σ and σ̃. It is easy to see that σ and σ̃
are invariant under the action of s89, s78, s127, s23, s34, s45 and s56. The bilinear equations (B.1)
and (B.2) yield[
e
(
1
4ε23
)
τe2τe0−e1−e2 − e
(
−1
4ε23
)
τe3τe0−e1−e3
]
×
[
e
(
1
4ε13
)
τe3τe0−e2−e3 − e
(
−1
4ε13
)
τe1τe0−e1−e2
]
= [ε13][ε23]
[
e
(
−1
4ε12
)
τe7τe8τe9τe0−e1−e2 + e
(
1
4ε12
)
τe3τe4τe5τe6
]
,
from which we get the first equation of (B.4). Since we see that π : σ ↔ σ̃ by the definition, we
immediately get the second equation of (B.4). �
Note that this representation coincides with that constructed by Tsuda [18]. The above
theorem gives us the following proposition.
Proposition B.1. Define the variables f and g by
f =
σ̃
σ
e(1
4ε12)τe7τe8τe9τe0−e1−e2 + e(−1
4ε12)τe3τe4τe5τe6
e(−1
4ε12)τe7τe8τe9τe0−e1−e2 + e(1
4ε12)τe3τe4τe5τe6
, g =
σ̃
σ
.
28 T. Masuda
Then, the action of W̃ (E(1)
7 ) on these variables is described by
s12 : f ↔ g, π : f 7→ 1
f
, g 7→ 1
g
, s23 : f 7→ h,
where h is a rational function determined by
h + d2
l e(
1
2ε12)
h + d2
l e(−
1
2ε12)
=
f + d2
l e(
1
2ε13)
f + d2
l e(−
1
2ε13)
g + d2
l e(−
1
2ε23)
g + d2
l e(
1
2ε23)
. (B.5)
Note that the variable f can be expressed by
f = d2
l
e(1
4ε13)τe1τe0−e1−e2 − e(−1
4ε13)τe3τe0−e2−e3
e(1
4ε13)τe3τe0−e2−e3 − e(−1
4ε13)τe1τe0−e1−e2
, (B.6)
and h = s13(g).
C A derivation of the difference equations
By writing down the action of the translation operator T21 ∈ W
(
E
(1)
7
)
on the variables f and g,
we will get the system of q-difference equations (1.1). Hereafter, we denote the time evolution
of a variable x by x = T21(x) and x = T−1
21 (x). Let us introduce the transformation µ by
µ = s12s23s147s158s169. It is easy to see that T21 = µ2 and µ(g) = f . We also introduce the
auxiliary variables k = s247(h) and l = s258(k). Note that we have g = s269(l).
Lemma C.1. We have
f + d2
l e(−
1
2ε13)
f + d2
l e(
1
2ε13)
=
f/g − e(−1
2ε12)
f/g + e(−1
2ε12)
f/h− e(−1
2ε23)
f/h− e(1
2ε23)
,
f + d2
l κ47e(−1
2ε247)
f + d2
l κ47e(1
2ε247)
=
f/h− e(1
2ε23)
f/h− e(−1
2ε23)
f/k − e(−1
2ε347)
f/k − e(1
2ε347)
,
f + d2
l κ58e(−1
2ε258)
f + d2
l κ58e(1
2ε258)
=
f/k − e(1
2ε347)
f/k − e(−1
2ε347)
f/l − e(−1
2ε347 − 1
2ε258)
f/l − e(1
2ε347 + 1
2ε258)
,
f + d2
l κ69e(−1
2ε269)
f + d2
l κ69e(1
2ε269)
=
f/l − e(1
2ε347 + 1
2ε258)
f/l − e(−1
2ε347 − 1
2ε258)
f/g − e(−1
2(ε12 + δ))
f/g − e(1
2(ε12 + δ))
.
Proof. The first equation is reduced to (B.5). The other expressions can be rewritten as
f + d2
l κ47e(−1
2ε247)
f + d2
l κ47e(1
2ε247)
h + d2
l κ47e(1
2ε347)
h + d2
l κ47e(−1
2ε347)
k + d2
l κ47e(−1
2ε23)
k + d2
l κ47e(1
2ε23)
= 1,
f + d2
l κ58e(−1
2ε258)
f + d2
l κ58e(1
2ε258)
k + d2
l κ58e(1
2ε347 + 1
2ε258)
k + d2
l κ58e(−1
2ε347 − 1
2ε258)
l + d2
l κ58e(−1
2ε347)
l + d2
l κ58e(1
2ε347)
= 1,
f + d2
l κ69e(−1
2ε269)
f + d2
l κ69e(1
2ε269)
l + d2
l κ69e(1
2(ε12 + δ))
l + d2
l κ69e(−1
2(ε12 + δ))
g + d2
l κ69e(−1
2ε347 − 1
2ε258)
g + d2
l κ69e(1
2ε347 + 1
2ε258)
= 1, (C.1)
where κ47 = e(1
2ε34 − 1
2ε127), κ58 = e(1
2ε35 − 1
2ε128) and κ69 = e(1
2ε36 − 1
2ε129). From the
expressions (B.6), we get
f + d2
l κ47e(−1
2ε247)
f + d2
l κ47e(1
2ε247)
= e
(
−1
2ε247
) τe7τe0−e2−e7
τe4τe0−e2−e4
.
Hypergeometric τ -Functions of the q-Painlevé System of Type E
(1)
7 29
By applying s13s12 and s247 successively, we also get
h + d2
l κ47e(1
2ε347)
h + d2
l κ47e(−1
2ε347)
= e
(
1
2ε347
) τe4τe0−e3−e4
τe7τe0−e3−e7
,
k + d2
l κ47e(−1
2ε23)
k + d2
l κ47e(1
2ε23)
= e
(
−1
2ε23
) τe0−e2−e4τe0−e3−e7
τe0−e2−e7τe0−e3−e4
,
and then the first equation of (C.1). The second and third equations of (C.1) can be obtained
by a similar way. �
The above lemma immediately gives us
fg − e(1
2ε12)
fg − e(−1
2ε12)
fg − e(1
2(ε12 + δ))
fg − e(−1
2(ε12 + δ))
=
f + d2
l e(
1
2ε13)
f + d2
l e(−
1
2ε13)
f + d2
l κ47e(1
2ε247)
f + d2
l κ47e(−1
2ε247)
f + d2
l κ58e(1
2ε258)
f + d2
l κ58e(−1
2ε258)
f + d2
l κ69e(1
2ε269)
f + d2
l κ69e(−1
2ε269)
,
where we replace g with 1/g. Applying µ−1 to the above equation, we also get
fg − e(−1
2(ε12 − δ))
fg − e(1
2(ε12 − δ))
fg − e(−1
2ε12)
fg − e(1
2ε12)
=
g + d2
re(
1
2ε23)
g + d2
re(−1
2ε23)
g + d2
rκ
−1
47 e(1
2ε147)
g + d2
rκ
−1
47 e(−1
2ε147)
g + d2
rκ
−1
58 e(1
2ε158)
g + d2
rκ
−1
58 e(−1
2ε158)
g + d2
rκ
−1
69 e(1
2ε169)
g + d2
rκ
−1
69 e(−1
2ε169)
.
Let us introduce the parameters bi (i = 1, 2, . . . , 8) and the independent variable t by
b1 = −q1/8d2
l e
(
1
4(ε13 + ε23)
)
, b2 = −q1/8d2
l e
(
1
4(ε14 + ε24) + 1
2ε34
)
,
b3 = −q1/8d2
l e
(
1
4(ε15 + ε25) + 1
2ε35
)
, b4 = −q1/8d2
l e
(
1
4(ε16 + ε26) + 1
2ε36
)
,
b5 = −q1/8d2
l e
(
−1
4(ε13 + ε23)
)
, b6 = −q1/8d2
l e
(
−1
4(ε137 + ε237)− 1
2ε127
)
,
b7 = −q1/8d2
l e
(
−1
4(ε138 + ε238)− 1
2ε128
)
, b8 = −q1/8d2
l e
(
−1
4(ε139 + ε239)− 1
2ε129
)
and t = e(1
2ε12), respectively. Replacing the dependent variables f and g with q−1/8t−1/2f and
q1/8t−1/2g, respectively, we get the system of difference equations (1.1).
Acknowledgements
The author would like to express his sincere thanks to Professors M. Noumi and Y. Yamada
for valuable discussions and comments. Especially, he owes initial steps of this work, including
the formulation by means of the lattice τ -functions and the bilinear equations, to discussions
with them. The author would also thank Professors K. Kajiwara and Y. Ohta for stimulating
discussions.
References
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http://arxiv.org/abs/nlin.SI/0701001
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1)
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http://arxiv.org/abs/nlin.SI/0607065
http://arxiv.org/abs/nlin.SI/0205019
http://arxiv.org/abs/nlin.SI/0303032
http://arxiv.org/abs/nlin.SI/0403036
http://arxiv.org/abs/nlin.SI/0501051
http://arxiv.org/abs/nlin.SI/0411003
http://arxiv.org/abs/nlin.SI/0012063
1 Introduction
2 The q-Painlevé system of type E_7^{(1)}
2.1 The discrete Painlevé system of type E_8^{(1)}
2.2 The q-Painlevé system of type E_7^{(1)}
3 A family of six-dimensional lattices and the bilinear equations
4 The construction of the \tau-functions on M_0
5 The construction of the \tau-functions on M_1
5.1 The q-hypergeometric function {}_8W_7 and its transformation formula
5.2 The contiguity relations for {}_8W_7
5.3 Twelve solutions
5.4 The \tau-functions on M_1
6 A determinant formula for the hypergeometric \tau-functions
A The q-Painlevé system of type E_7^{(1)}
A.1 Point configurations and Cremona transformations
A.2 The lattice \tau-functions and the bilinear equations
B Another representation
C A derivation of the difference equations
References
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