Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations
We derive series representations for the tau functions of the q-Painlevé V, III₁, III₂, and III₃ equations, as degenerations of the tau functions of the q-Painlevé VI equation in [Jimbo M., Nagoya H., Sakai H., J. Integrable Syst. 2 (2017), xyx009, 27 pages]. Our tau functions are expressed in terms...
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| Cite this: | Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations / Y. Matsuhira, H. Nagoya // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 28 назв. — англ. |
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| citation_txt | Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations / Y. Matsuhira, H. Nagoya // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 28 назв. — англ. |
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| description | We derive series representations for the tau functions of the q-Painlevé V, III₁, III₂, and III₃ equations, as degenerations of the tau functions of the q-Painlevé VI equation in [Jimbo M., Nagoya H., Sakai H., J. Integrable Syst. 2 (2017), xyx009, 27 pages]. Our tau functions are expressed in terms of q-Nekrasov functions. Thus, our series representations for the tau functions have explicit combinatorial structures. We show that general solutions to the q-Painlevé V, III₁, III₂, and III₃ equations are written by our tau functions. We also prove that our tau functions for the q-Painlevé III₁, III₂, and III₃ equations satisfy the three-term bilinear equations for them.
|
| first_indexed | 2025-12-07T21:24:47Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 074, 17 pages
Combinatorial Expressions for the Tau Functions
of q-Painlevé V and III Equations
Yuya MATSUHIRA and Hajime NAGOYA
School of Mathematics and Physics, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan
E-mail: y.matsu0727@gmail.com, nagoya@se.kanazawa-u.ac.jp
Received November 24, 2018, in final form September 13, 2019; Published online September 23, 2019
https://doi.org/10.3842/SIGMA.2019.074
Abstract. We derive series representations for the tau functions of the q-Painlevé V, III1,
III2, and III3 equations, as degenerations of the tau functions of the q-Painlevé VI equation
in [Jimbo M., Nagoya H., Sakai H., J. Integrable Syst. 2 (2017), xyx009, 27 pages]. Our tau
functions are expressed in terms of q-Nekrasov functions. Thus, our series representations
for the tau functions have explicit combinatorial structures. We show that general solutions
to the q-Painlevé V, III1, III2, and III3 equations are written by our tau functions. We
also prove that our tau functions for the q-Painlevé III1, III2, and III3 equations satisfy the
three-term bilinear equations for them.
Key words: q-Painlevé equations; tau functions; q-Nekrasov functions; bilinear equations
2010 Mathematics Subject Classification: 39A13; 33E17; 05A30
1 Introduction
The q-Painlevé equations [17, 24] are q-difference analogs of the Painlevé equations, which were
introduced as new special functions beyond elliptic functions and the hypergeometric functions
more than one hundred years ago [12, 22, 23], and are now considered as important special
functions with many applications both in mathematics and physics.
Similarly, as for other integrable systems, tau functions play a crucial role in the studies of
the Painelvé equations. The recent discovery by [10] states that the tau function of the sixth
Painlevé equation is a Fourier transform of Virasoro conformal blocks with c = 1, which admit
explicit combinatorial formulas by AGT correspondence [1]. Series representations of the tau
functions of other types are also studied in [8, 11, 20, 21] for differential cases, [5, 6, 15] for
q-difference cases.
In [15], a general solution (y, z) to the q-Painlevé VI equation [16] was expressed by the tau
functions having q-Nekrasov type expressions, and it was conjectured that the tau functions
satisfy the bilinear equations for the q-Painlevé VI equation. In this paper, we give explicit
expressions for general solutions to the q-Painlevé V, III1, III2, and III3 equations using degen-
erations of the tau functions of the q-Painlevé VI equation. We also give conjectures on the
bilinear equations satisfied by the tau functions of the q-Painlevé V equation and prove that the
tau functions of the q-Painlevé III1, III2, and III3 equations satisfy the bilinear equations.
Our q-difference equations are as follows.
(i) the q-Painlevé VI equation:
yy
a3a4
=
(z − b1t)(z − b2t)
(z − b3)(z − b4)
,
zz
b3b4
=
(y − a1t)(y − a2t)
(y − a3)(y − a4)
.
(ii) the q-Painlevé V equation:
yy
a3a4
= −(z − b1t)(z − b2t)
z − b3
,
zz
b3
= −(y − a1t)(y − a2t)
a4(y − a3)
.
mailto:y.matsu0727@gmail.com
mailto:nagoya@se.kanazawa-u.ac.jp
https://doi.org/10.3842/SIGMA.2019.074
2 Y. Matsuhira and H. Nagoya
(iii) the q-Painlevé III1 equation:
yy
a3a4
= −z(z − b2t)
z − b3
,
zz
b3
= − y(y − a1t)
a4(y − a3)
.
(iv) the q-Painlevé III2 equation:
yy
a3a4
= − z2
z − b3
,
zz
b3
= − y(y − a2t)
a4(y − a3)
.
(v-1) the q-Painlevé III3 equation of surface type A
(1)′
7 :
yy
a3
= z2, zz = −y(y − a2t)
y − a3
.
(v-2) the q-Painlevé III3 equation of surface type A
(1)
7 :
yy
a3
= − z2
z − b3
, zz =
y(y − a2t)
a2
.
Here, y, z are functions of t, y = y(qt), z = z(qt), and ai, bi (i = 1, 2, 3, 4) are parameters.
From the point of view of Sakai’s classification for the discrete Painlevé equations [25], the
q-Painlevé VI, V, III1, III2 and III3 equations are derived from the symmetries/surfaces of type
D
(1)
5 /A
(1)
3 , A
(1)
4 /A
(1)
4 , E
(1)
2 /A
(1)
5 , E
(1)
2 /A
(1)
6 and A
(1)
1 /A
(1)
7 , respectively.
The degeneration scheme of Painlevé equations is as follows
PVI PV PIII1 PIII2 PIII3
PIV PII PI.
- -
Q
Q
QQs
-
Q
Q
QQs
-
Q
Q
QQs
- -
The degeneration pattern of the q-Painlevé equations we use is similar to the one in [19] but
not exactly the same. Rather, our limiting procedure is a q-version for the one used in [11] in
order to derive combinatorial expressions of tau functions of PV, PIII1 , PIII2 , and PIII3 from the
Nekrasov type expression of the tau function of PVI [10].
For the case of the q-Painlevé III3 equation of surface type A
(1)′
7 , a series representation for
the tau function was proposed in [5], which are expressed by q-Virasoro Whittaker conformal
blocks which equal Nekrasov partition functions for pure SU(2) 5d theory [3, 28]. A Fredholm
determinant representation of the tau function by topological strings/spectral theory duality is
proposed in [7]. For the q-Painlevé III3 equation of surface type A
(1)
7 , a series representation
for the tau function was proposed in [4]. Our tau functions for the q-Painlevé III3 equations
obtained by the degeneration are equivalent to them.
Our plan is as follows. In Section 2, we recall the result on q-Painlevé VI equation in [15]. In
Sections 3–6, we compute limits of tau functions and derive combinatorial expressions of general
solutions and bilinear equations for q-Painlevé V, III1, III2 and III3 equations.
Notations. Throughout the paper we fix q ∈ C× such that |q| < 1. We set
[u] =
(
1− qu
)
/(1− q), (a; q)N =
N−1∏
j=0
(
1− aqj
)
,
(a1, . . . , ak; q)∞ =
k∏
j=1
(aj ; q)∞, (a; q, q)∞ =
∞∏
j,k=0
(
1− aqj+k
)
.
Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations 3
We use the q-Gamma function and q-Barnes function defined by
Γq(u) =
(q; q)∞
(qu; q)∞
(1− q)1−u, Gq(u) =
(qu; q, q)∞
(q; q, q)∞
(q; q)u−1
∞ (1− q)−(u−1)(u−2)/2,
which satisfy Γq(1) = Gq(1) = 1 and
Γq(u+ 1) = [u]Γq(u), Gq(u+ 1) = Γq(u)Gq(u). (1.1)
A partition is a finite sequence of positive integers λ = (λ1, . . . , λl) such that λ1 ≥ · · · ≥ λl > 0.
Denote the length of the partition by `(λ) = l. The conjugate partition λ′ = (λ′1, . . . , λ
′
l′) is
defined by λ′j = ]{i |λi ≥ j}, l′ = λ1. We regard a partition as a Young diagram. Namely,
we regard a partition λ also as the subset {(i, j) ∈ Z2 | 1 ≤ j ≤ λi, i ≥ 1} of Z2, and denote
its cardinality by |λ|. We denote the set of all partitions by Y. For � = (i, j) ∈ Z2
>0 we set
aλ(�) = λi − j (the arm length of �) and `λ(�) = λ′j − i (the leg length of �). In the last
formulas we set λi = 0 if i > `(λ) (resp. λ′j = 0 if j > `(λ′)). For a pair of partitions (λ, µ) and
u ∈ C we set
Nλ,µ(u) =
∏
�∈λ
(
1− q−`λ(�)−aµ(�)−1u
) ∏
�∈µ
(
1− q`µ(�)+aλ(�)+1u
)
,
which we call a Nekrasov factor.
2 Results on q-PVI from [15]
In this section, we recall the results of [15] on the q-Painlevé VI equation. Define the tau function
by
τVI
[
θ1 θt
θ∞ θ0
∣∣∣s, σ, t] =
∑
n∈Z
snt(σ+n)2−θ2t−θ20C
[
θ1 θt
θ∞ θ0
∣∣∣σ + n
]
Z
[
θ1 θt
θ∞ θ0
∣∣∣σ + n, t
]
,
with the definition
C
[
θ1 θt
θ∞ θ0
∣∣∣σ] =
∏
ε,ε′=±
Gq(1 + εθ∞ − θ1 + ε′σ)Gq(1 + εσ − θt + ε′θ0)
Gq(1 + 2σ)Gq(1− 2σ)
,
Z
[
θ1 θt
θ∞ θ0
∣∣∣σ, t] =
∑
λ=(λ+,λ−)∈Y2
t|λ| ·
∏
ε,ε′=±
N∅,λε′
(
qεθ∞−θ1−ε
′σ
)
Nλε,∅
(
qεσ−θt−ε
′θ0
)
∏
ε,ε′=±
Nλε,λε′
(
q(ε−ε′)σ
) .
Put
τVI
1 = τVI
[
θ1 θt
θ∞ + 1
2 θ0
∣∣∣s, σ, t] , τVI
2 = τVI
[
θ1 θt
θ∞ − 1
2 θ0
∣∣∣s, σ, t] ,
τVI
3 = τVI
[
θ1 θt
θ∞ θ0 + 1
2
∣∣∣s, σ + 1
2 , t
]
, τVI
4 = τVI
[
θ1 θt
θ∞ θ0 − 1
2
∣∣∣s, σ − 1
2 , t
]
,
τVI
5 = τVI
[
θ1 − 1
2 θt
θ∞ θ0
∣∣∣s, σ, t] , τVI
6 = τVI
[
θ1 + 1
2 θt
θ∞ θ0
∣∣∣s, σ, t] ,
τVI
7 = τVI
[
θ1 θt − 1
2
θ∞ θ0
∣∣∣s, σ + 1
2 , t
]
, τVI
8 = τVI
[
θ1 θt + 1
2
θ∞ θ0
∣∣∣s, σ − 1
2 , t
]
.
Here and after we write f(t) = f(qt), f(t) = f(t/q).
4 Y. Matsuhira and H. Nagoya
Theorem 2.1 ([15]). The functions y and z defined by
y = q−2θ1−1t · τ
VI
3 τVI
4
τVI
1 τVI
2
, z =
τVI
1 τVI
2 − τVI
1 τVI
2
q1/2+θ∞τVI
1 τVI
2 − q1/2−θ∞τVI
1 τVI
2
(2.1)
are solutions to the q-Painlevé VI equation
yy
a3a4
=
(z − b1t)(z − b2t)
(z − b3)(z − b4)
,
zz
b3b4
=
(y − a1t)(y − a2t)
(y − a3)(y − a4)
, (2.2)
with the parameters
a1 = q−2θ1−1, a2 = q−2θt−2θ1−1, a3 = q−1, a4 = q−2θ1−1,
b1 = q−θ0−θt−θ1 , b2 = qθ0−θt−θ1 , b3 = qθ∞−1/2, b4 = q−θ∞−1/2.
The formula for y above can be regarded as an extension of Mano’s asymptotic expansion
to all orders for the solution of q-PVI [18]. Theorem 2.1 was obtained by constructing the
fundamental solution of the Lax-pair for q-PVI in [16], in terms of q-conformal blocks in [2].
The method of construction of the fundamental solution is a q-analogue of the CFT approach
used in [14]. In the derivation of Theorem 2.1 convergence of the fundamental solution was
assumed and it has not been proved. Recently, analyticity of K-theoretic Nekrasov functions in
cases different from our case was discussed in [9]. We remark that the convergence of the pure
q-Nekrasov partition function with q1q2 = 1 on C is proved in [5].
Conjecture 2.2 ([15]). The tau functions τVI
i (i = 1, . . . , 8) satisfy the following bilinear equa-
tions
τVI
1 τVI
2 − q−2θ1tτVI
3 τVI
4 −
(
1− q−2θ1t
)
τVI
5 τVI
6 = 0, (2.3)
τVI
1 τVI
2 − tτVI
3 τVI
4 −
(
1− q−2θtt
)
τVI
5 τVI
6 = 0, (2.4)
τVI
1 τVI
2 − τVI
3 τVI
4 +
(
1− q−2θ1t
)
q2θtτVI
7 τVI
8 = 0, (2.5)
τVI
1 τVI
2 − q2θtτVI
3 τVI
4 +
(
1− q−2θtt
)
q2θtτVI
7 τVI
8 = 0, (2.6)
τVI
5 τVI
6 + q−θ1−θ∞+θt−1/2tτVI
7 τVI
8 − τVI
1 τVI
2 = 0, (2.7)
τVI
5 τVI
6 + q−θ1+θ∞+θt−1/2tτVI
7 τVI
8 − τVI
1 τVI
2 = 0, (2.8)
τVI
5 τVI
6 + qθ0+2θtτVI
7 τVI
8 − qθtτVI
3 τVI
4 = 0, (2.9)
τVI
5 τVI
6 + q−θ0+2θtτVI
7 τVI
8 − qθtτVI
3 τVI
4 = 0. (2.10)
Then, the function y, z
y = q−2θ1−1t
τVI
3 τVI
4
τVI
1 τVI
2
, z = −qθt−θ1−1t
τVI
7 τVI
8
τVI
5 τVI
6
(2.11)
solves q-PVI (2.2).
The function y in Conjecture 2.2 is expressed in the same form in Theorem 2.1, while the
function z in Conjecture 2.2 is not. By the bilinear equations (2.7) and (2.8), we obtain the
expression of z in (2.11) from the expression of z in (2.1).
We note that in [15] we have a Lax pair with respect to the shift t→ qt, namely, a fundamental
solution of the linear q-difference equations
Y (qx, t) = A(x, t)Y (x, t), Y (x, qt) = B(x, t)Y (x, t) (2.12)
for certain 2 by 2 matrices A(x, t) and B(x, t) was constructed in terms of q-Nekrasov functions.
From (2.12) we obtain the four-term bilinear equation in [15, Remark 3.5]:
τVI
1 τVI
2 − τVI
1 τVI
2 =
q1/2+θ∞ − q1/2−θ∞
q−θ0 − qθ0
q−θ1−1t
(
τVI
3 τVI
4 − τVI
3 τVI
4
)
. (2.13)
Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations 5
3 From q-PVI to q-PV
In this section, we take a limit of the tau functions of q-PVI to q-PV. Define the tau function by
τV(θ∗, θt, θ0 | s, σ, t) =
∑
n∈Z
snt(σ+n)2−θ2t−θ20CV[θ∗, θt, θ0 |σ + n]ZV[θ∗, θt, θ0 |σ + n, t],
with
CV [θ∗, θt, θ0 |σ] = (q − 1)−σ
2
∏
ε=±
Gq(1− θ∗ + εσ)
Gq(1 + 2εσ)
∏
ε,ε′=±
Gq(1 + εσ − θt + ε′θ0),
ZV [θ∗, θt, θ0 |σ, t] =
∑
(λ+,λ−)∈Y2
t|λ+|+|λ−|
∏
ε=±
N∅,λε(q
−θ∗−εσ)fλε(q
εσ)
∏
ε,ε′=±
Nλε,∅(qεσ−θt−ε
′θ0)∏
ε,ε′=±
Nλε,λε′ (q
(ε−ε′)σ)
,
where
fλ(u) =
∏
�∈λ
(
− q`λ(�)+a∅(�)+1u−1
)
.
We remark that the factor fλ(u) corresponds to the five-dimensional Chern–Simons term. The
Chern–Simons term in [27] reads as
exp
(
−β
∑
k
∑
(i,j)∈Yk
(ak + ε(i− j))
)
,
where β, ak are parameters and Y1, . . . , YN are Young tableaux labelling the fixed points. See [27]
for the details. Since∑
�∈λ
`λ(�) + a∅(�) + 1 =
∑
(i,j)∈λ
λ′j − i− j + 1 =
∑
(i,j)∈λ
i− j,
they coincide when N = 2. It is possible to remove fλε(q
εσ) from ZV[θ∗, θt, θ0 |σ, t] by change
of variables. Because if we set
ZCS=0
V [θ∗, θt, θ0 |σ, t] =
∑
(λ+,λ−)∈Y2
t|λ+|+|λ−|
∏
ε=±
N∅,λε
(
q−θ∗−εσ
) ∏
ε,ε′=±
Nλε,∅
(
qεσ−θt−ε
′θ0
)
∏
ε,ε′=±
Nλε,λε′ (q
(ε−ε′)σ)
,
then we have
ZV[θ∗, θt, θ0 |σ, t] = ZCS=0
V
[
−θ∗,−θt, θ0 |σ, q−θ∗−2θtt
]
from the relations N∅,λ(u) = fλ
(
u−1
)
Nλ,∅
(
u−1
)
, Nλ,∅(u) = fλ(u)−1N∅,λ
(
u−1
)
, and Nλ,µ(u) =
Nµ′,λ′(u) [15, Lemma A.2].
We define tau functions for q-PV by
τV
1 = τV
(
θ∗ − 1
2 , θt, θ0 | s, σ, t/
√
q
)
, τV
2 = τV
(
θ∗ + 1
2 , θt, θ0 | s, σ,
√
qt
)
,
τV
3 = τV
(
θ∗, θt, θ0 + 1
2 | s, σ + 1
2 , t
)
, τV
4 = τV
(
θ∗, θt, θ0 − 1
2 | s, σ −
1
2 , t
)
,
τV
5 = τV
(
θ∗, θt − 1
2 , θ0 | s, σ + 1
2 , t
)
, τV
6 = τV
(
θ∗, θt + 1
2 , θ0 | s, σ − 1
2 , t
)
.
Let
C1 = C6 = (q − 1)−σ
2
q−(Λ+1/2)(σ2−θ2t−θ20)
∏
ε=±
Gq
(
1
2 − Λ + εσ
)−1
,
6 Y. Matsuhira and H. Nagoya
C2 = C5 = (q − 1)−σ
2
q−(Λ−1/2)(σ2−θ2t−θ20)
∏
ε=±
Gq
(
3
2 − Λ + εσ
)−1
,
C3 = (q − 1)−(σ+1/2)2q−Λ((σ+1/2)2−θ2t−(θ0+1/2)2)
∏
ε=±
Gq
(
1− Λ + ε
(
σ + 1
2
))−1
,
C4 = (q − 1)−(σ−1/2)2q−Λ((σ−1/2)2−θ2t−(θ0−1/2)2)
∏
ε=±
Gq
(
1− Λ + ε
(
σ − 1
2
))−1
,
C7 = (q − 1)−(σ+1/2)2q−Λ((σ+1/2)2−(θt−1/2)2−θ20)
∏
ε=±
Gq
(
1− Λ + ε
(
σ + 1
2
))−1
,
C8 = (q − 1)−(σ−1/2)2q−Λ((σ−1/2)2−(θt+1/2)2−θ20)
∏
ε=±
Gq
(
1− Λ + ε
(
σ − 1
2
))−1
.
Proposition 3.1. Set
θ1 + θ∞ = Λ, θ1 − θ∞ = θ∗, t = qΛt1,
s = s̃(q − 1)−2σq−2σΛ
∏
ε=±
Γq
(
1
2 − Λ + εσ
)−ε
. (3.1)
Then we have
Ciτ
VI
i (θ∞, θ1, θt, θ0 | s, σ, t)→ τV
i (θ∗, θt, θ0 | s̃, σ, t1), i = 1, 2, 3, 4,
C5τ
VI
5 (θ∞, θ1, θt, θ0 | s, σ, t)→ τV
1 (θ∗, θt, θ0 | s̃, σ, qt1),
C6τ
VI
6 (θ∞, θ1, θt, θ0 | s, σ, t)→ τV
2 (θ∗, θt, θ0 | s̃, σ, t1/q),
Ciτ
VI
i (θ∞, θ1, θt, θ0 | s, σ, t)→ τV
i−2(θ∗, θt, θ0 | s̃, σ, t1), i = 7, 8,
as Λ→∞. Here, we denote by τVI
i (θ∞, θ1, θt, θ0 | s, σ, t) the tau functions of q-PVI presented in
the previous section.
Proof. First, we verify the limit of the series part. For any partition λ we have
N∅,λ
(
q−Λu
)
qΛ|λ| =
∏
�∈λ
(
qΛ − q`λ(�)+a∅(�)+1u
)
→ fλ
(
u−1
)
, Λ→∞.
Hence, the series Z
[
θ1 θt
θ∞ θ0
∣∣∣σ, t] goes to ZV[θ∗, θt, θ0 |σ, t] as Λ→∞.
Second, we examine the limits of the coefficients of Z. By the identities (1.1) on q-Gamma
function and q-Barnes function, for n ∈ Z we have
∏
ε=±
Gq(1− x+ ε(σ + n)) =
∏
ε=±
Gq(1− x+ εσ)Γq(−x+ εσ)εn
|n|−1∏
i=0
[
−x+
|n|
n
σ
]
×
|n|−1∏
i=0
i∏
j=1
[−x+ σ + j]
|n|−1∏
i=0
i∏
j=1
[−x− σ − j]. (3.2)
Using the identity above, we compute the coefficient of Z in τVI
1 multiplied by C1 as follows
C1s
nC
[
θ1 θt
θ∞ + 1
2 θ0
∣∣∣σ + n
]
t(σ+n)2−θ2t−θ20
= s̃n(q − 1)σ
2−2σnq−(σ2−θ2t−θ20)/2t
(σ+n)2−θ2t−θ20
1 qΛn2
∏
ε=±
(
Γq
(
−Λ− 1
2 + εσ
)
Γq
(
−Λ + 1
2 + εσ
))εn
Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations 7
×
|n|−1∏
i=0
[
−Λ− 1
2 +
|n|
n
σ
] |n|−1∏
i=0
i∏
j=1
[−Λ− 1
2 + σ + j]
|n|−1∏
i=0
i∏
j=1
[
−Λ− 1
2 − σ − j
]
×
∏
ε=±
Gq(1− θ∗ − 1
2 + ε(σ + n))
∏
ε,ε′=±
Gq(1 + ε(σ + n)− θt + ε′θ0)
Gq(1 + 2(σ + n))Gq(1− 2(σ + n))
.
Then we have as Λ→∞ by the definition of q-number
qΛn2
|n|−1∏
i=0
[
−Λ− 1
2 +
|n|
n
σ
] |n|−1∏
i=0
i∏
j=1
[
−Λ− 1
2 + σ + j
] |n|−1∏
i=0
i∏
j=1
[
−Λ− 1
2 − σ − j
]
→ (q − 1)−n
2
|n|−1∏
i=0
q−1/2+|n|σ/n
|n|−1∏
i=0
i∏
j=1
q−1/2+σ+j
|n|−1∏
i=0
i∏
j=1
q−1/2−σ−j
= (q − 1)−n
2
q−n
2/2+σn,
and by the identity (1.1) of q-Gamma function
∏
ε=±
(
Γq(−Λ− 1
2 + εσ)
Γq(−Λ + 1
2 + εσ)
)εn
=
(
[−Λ− 1
2 − σ]
[−Λ− 1
2 + σ]
)n
→ q−2σn.
Therefore we obtain
C1s
nC
[
θ1 θt
θ∞ + 1
2 θ0
∣∣∣σ + n
]
t(σ+n)2−θ2t−θ20
→ s̃n
(
t1/
√
q
)(σ+n)2−θ2t−θ20CV
[
θ∗ − 1
2 , θt, θ0 |σ + n
]
as Λ→∞. Similarly, we can compute the coefficients of Z in the other tau functions and obtain
the desired results. �
In what follows, we abbreviate τV
i (θ∗, θt, θ0 | s, σ, t) to τi.
Theorem 3.2. The functions
y = q−θ∗−1(q − 1)1/2t
τ3τ4
τ1τ2
, z = −
τ1τ2 − τ1τ2
qθ∗/2+1/2τ1τ2
solves the q-Painlevé V equation
yy
a3a4
= −(z − b1t)(z − b2t)
z − b3
,
zz
b3
= −(y − a1t)(y − a2t)
a4(y − a3)
(3.3)
with the parameters
a1 = q−θ∗−1, a2 = q−2θt−θ∗−1, a3 = q−1, a4 = q−3θ∗/2−1/2,
b1 = q−θ0−θt−θ∗/2, b2 = qθ0−θt−θ∗/2, b3 = q−θ∗/2−1/2.
Proof. By definition we have
C1C2 = (q − 1)1/2C3C4.
Hence, by (3.1) the solution (y, z) of the q-Painlevé VI equation has the following limit
y → y1 = q−θ∗−1(q − 1)1/2t1
τ3τ4
τ1τ2
, q−Λ/2z → z1 = −
τ1τ2 − τ1τ2
qθ∗/2+1/2τ1τ2
, Λ→∞.
8 Y. Matsuhira and H. Nagoya
Substituting (3.1) into the q-Painlevé VI equation (2.2), we get
yy
q−Λ−θ∗−2
=
(
z − q−θ0−θt+(Λ−θ∗)/2t1
)(
z − qθ0−θt+(Λ−θ∗)/2t1
)(
z − q(Λ−θ∗−1)/2
)(
z − q−(Λ+θ∗+1)/2)
) , (3.4)
zz
q−1
= −
(
y − q−θ∗−1t1
)(
y − q−2θt−θ∗−1t1
)(
y − q−1
)(
y − q−Λ−θ∗−1
) . (3.5)
Hence, since y → y1, q−Λ/2z → z1 as Λ→∞, the system (3.4), (3.5) degenerate to the q-Painle-
vé V equation (3.3) for y = y1 and z = z1 as Λ→∞. �
Since we also have
C5C6 = (q − 1)1/2C7C8, C1C2 = C5C6,
we obtain the following conjecture.
Conjecture 3.3. The tau functions τi (i = 1, . . . , 6) satisfy the following bilinear equations
τ1τ2 − q−θ∗(q − 1)1/2tτ3τ4 −
(
1− q−θ∗t
)
τ1τ2 = 0, (3.6)
(q − 1)−1/2τ1τ2 − τ3τ4 +
(
1− q−θ∗t
)
q2θtτ5τ6 = 0, (3.7)
(q − 1)−1/2τ1τ2 − q2θtτ3τ4 + q2θtτ5τ6 = 0, (3.8)
τ1τ2 + qθt−1/2(q − 1)1/2tτ5τ6 − τ1τ2 = 0, (3.9)
(q − 1)−1/2τ1τ2 + qθ0+2θtτ5τ6 − qθtτ3τ4 = 0, (3.10)
(q − 1)−1/2τ1τ2 + q−θ0+2θtτ5τ6 − qθtτ3τ4 = 0. (3.11)
Then the functions
y = q−θ∗−1(q − 1)1/2t
τ3τ4
τ1τ2
, z = −qθt−θ∗/2−1(q − 1)1/2t
τ5τ6
τ1τ2
solves q-PV (3.3).
The four-term bilinear equation (2.13) admits the following limit.
Proposition 3.4. We have
τ1τ2 − τ1τ2 =
q−1/2(q − 1)1/2
qθ0 − q−θ0
t(τ3τ4 − τ3τ4). (3.12)
Proof. The identity (3.12) is a direct consequence of (2.13) by the limit (3.1) as Λ→∞. �
We remark that tau functions without the Chern–Simons term is also obtained by the limit
θ1 + θ∞ = −Λ, θ1 − θ∞ = θ∗, s = s̃(q − 1)−2σ
∏
ε=±
Γq
(
1
2 + Λ + εσ
)−ε
, Λ→∞
from the tau functions of q-PVI.
Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations 9
4 From q-PV to q-PIII1
In this section, we take a limit of the tau functions of q-PV to q-PIII1 . Define the tau function
by
τ III1(θ∗, θ? | s, σ, t) =
∑
n∈Z
snt(σ+n)2CIII1 [θ∗, θ? |σ + n]ZIII1 [θ∗, θ? |σ + n, t],
with
CIII1 [θ∗, θ? |σ] = (q − 1)−2σ2
∏
ε=±
Gq(1− θ∗ + εσ)Gq(1 + εσ − θ?)
Gq(1 + 2εσ)
,
ZIII1 [θ∗, θ? |σ, t] =
∑
(λ+,λ−)∈Y2
t|λ+|+|λ−|
∏
ε=±
N∅,λε
(
q−θ∗−εσ
)
Nλε,∅
(
qεσ−θ?
)
∏
ε,ε′=±
Nλε,λε′
(
q(ε−ε′)σ
) .
Let us define the tau functions for q-PIII1 by
τ III1
1 = τ III1
(
θ∗ − 1
2 , θ? | s, σ, t/
√
q
)
, τ III1
2 = τ III1
(
θ∗ + 1
2 , θ? | s, σ,
√
qt
)
,
τ III1
3 = τ III1
(
θ∗, θ? − 1
2 | s, σ + 1
2 , t/
√
q
)
, τ III1
4 = τ III1
(
θ∗, θ? + 1
2 | s, σ −
1
2 ,
√
qt
)
.
Put
C1 = (q − 1)−σ
2
q−Λσ2−(θ2t+θ20)/2tθ
2
t+θ20
∏
ε=±
Gq(1− Λ + εσ)−1,
C2 = (q − 1)−σ
2
q−Λσ2+(θ2t+θ20)/2tθ
2
t+θ20
∏
ε=±
Gq(1− Λ + εσ)−1,
C3 = (q − 1)−(σ+1/2)2q−(Λ+1/2)(σ+1/2)2tθ
2
t+(θ0+1/2)2
∏
ε=±
Gq
(
1
2 − Λ + ε
(
σ + 1
2
))−1
,
C4 = (q − 1)−(σ−1/2)2q−(Λ−1/2)(σ−1/2)2tθ
2
t+(θ0−1/2)2
∏
ε=±
Gq
(
3
2 − Λ + ε
(
σ − 1
2
))−1
,
C5 = (q − 1)−(σ+1/2)2q−(Λ−1/2)(σ+1/2)2t(θt−1/2)2+θ20
∏
ε=±
Gq
(
3
2 − Λ + ε
(
σ + 1
2
))−1
,
C6 = (q − 1)−(σ−1/2)2q−(Λ+1/2)(σ−1/2)2t(θt+1/2)2+θ20
∏
ε=±
Gq
(
1
2 − Λ + ε
(
σ − 1
2
))−1
.
Proposition 4.1. Set
θt + θ0 = Λ, θt − θ0 = θ?, t = qΛt1,
s = s̃(q − 1)−2σq−σ(2Λ+1)
∏
ε=±
Γq(−Λ + εσ)−ε. (4.1)
Then we have
Ciτ
V
i (θ∗, θt, θ0 | s, σ, t)→ τ III1
i (θ∗, θ? | s̃, σ, t1), i = 1, 2, 3, 4,
Ciτ
V
i (θ∗, θt, θ0 | s, σ, t)→ τ III1
i−2 (θ∗, θ? | s̃, σ, qt1), i = 5, 6,
as Λ→∞.
Proof. For any partition λ we have
Nλ,∅
(
q−Λu
)
qΛ|λ| =
∏
�∈λ
(
qΛ − q−`λ(�)−a∅(�)−1u
)
→ fλ(u)−1, Λ→∞.
10 Y. Matsuhira and H. Nagoya
Hence, the series ZV[θ∗, θt, θ0 |σ, t] goes to ZIII1 [θ∗, θ? |σ, t1] as Λ → ∞. The coefficients of ZV
are computed in the same way as in the proof of Proposition 3.1 using (3.2) and we obtain the
desired results. �
In what follows, we abbreviate τ III1
i (θ∗, θ? | s, σ, t) to τi. Fortunately, the four-term bilinear
equation (3.12) degenerates to a three-term bilinear equation.
Proposition 4.2. We have
τ1τ2 − τ1τ2 = q−1/4t1/2τ3τ4. (4.2)
Proof. By definition and (4.1) we have
C1C2 = C1C2 =
(
q−Λ − qσ
)
(q − 1)−1/2t
−1/2
1 qθ0+1/4C3C4
=
(
q−Λ − qσ
)
(q − 1)−1/2t
−1/2
1 q−θ0+1/4C3C4.
Hence from the four-term bilinear equation (3.12) degenerates to the three-term bilinear equa-
tion (4.2) by (4.1) as Λ→∞. �
Theorem 4.3. The functions
y = q−θ∗−1t1/2
τ3τ4
τ1τ2
, z = q−θ∗/2−3/4t1/2
τ3τ4
τ1τ2
(4.3)
solves the q-Painlevé III1 equation
yy
a3a4
= −z(z − b2t)
z − b3
,
zz
b3
= − y(y − a2t)
a4(y − a3)
(4.4)
with the parameters
a2 = q−θ?−θ∗−1, a3 = q−1, a4 = q−3θ∗/2−1/2, b2 = q−θ∗/2, b3 = q−θ∗/2−1/2.
Furthermore, the tau functions τi (i = 1, . . . , 4) satisfy the following bilinear equations.
τ1τ2 − q−θ∗t1/2τ3τ4 − τ1τ2 = 0, (4.5)
τ1τ2 − qθ?t−1/2τ3τ4 + qθ?t−1/2τ3τ4 = 0, (4.6)
τ1τ2 + q−1/4t1/2τ3τ4 − τ1τ2 = 0, (4.7)
τ1τ2 + q1/4t−1/2τ3τ4 − q1/4t−1/2τ3τ4 = 0. (4.8)
Proof. By definition and (4.1) we have
C1C2 =
(
q−Λ − qσ
)
(q − 1)−1/2t
−1/2
1 C3C4.
Hence, by (4.1) and (4.2) the solution (y, z) of the q-Painlevé V equation degenerates to
y → y1 = q−θ∗−1t
1/2
1
τ3τ4
τ1τ2
, z → z1 = q−θ∗/2−3/4t
1/2
1
τ3τ4
τ1τ2
, Λ→∞.
Also, the q-Painlevé V equation (3.3) degenerates to the q-Painlevé III1 equation (4.4) for y = y1
and z = z1 as Λ→∞.
Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations 11
Next we prove the bilinear equations (4.5)–(4.8). The bilinear equation (4.7) is (4.2). The
identity (4.8) is obtained by substituting the expression (4.3) of (y, z) into the q-Painlevé III1
equation
yy
a3a4
= −z(z − b2t)
z − b3
,
and using the bilinear equation (4.7).
In order to prove (4.5) and (4.6), we use the following transformation(
θ̃∗, θ̃?, σ̃, s̃, t̃
)
=
(
− θ?,−θ∗, σ − 1
2 , Cs, q
−θ∗−θ?+1/2t
)
, (4.9)
where
C = q(σ−1)(2θ∗+2θ?+1)
∏
ε,ε′=±
Γq
(
1
2 + εθ∗ + ε′(σ − 1)
)−εε′
Γq
(
1
2 + ε
(
θ? + 1
2
)
+ ε′(σ − 1)
)−εε′
.
From the definition of the Nekrasov factor, for a partition λ we have
N∅,λ(u)Nλ,∅(w) = (uw)|λ|N∅,λ
(
w−1
)
Nλ,∅
(
u−1
)
.
By the identity above, the series part Z of the tau functions τ1, . . . , τ4 transform to
ZIII1
[
θ̃∗ − 1
2 , θ̃? | σ̃, t̃/
√
q
]
= ZIII1
[
θ∗, θ? + 1
2 |σ −
1
2 ,
√
qt
]
,
ZIII1
[
θ̃∗ + 1
2 , θ̃? | σ̃,
√
qt̃
]
= ZIII1
[
θ∗, θ? − 1
2 |σ −
1
2 ,
√
qt
]
,
ZIII1
[
θ̃∗, θ̃? − 1
2 | σ̃ + 1
2 , t̃/
√
q
]
= ZIII1
[
θ∗ + 1
2 , θ? |σ,
√
qt
]
,
ZIII1
[
θ̃∗, θ̃? + 1
2 | σ̃ −
1
2 , t̃
]
= ZIII1
[
θ∗ − 1
2 , θ? |σ − 1,
√
qt
]
,
respectively. Using the identity
Gq(1 + x+ n)Gq(1− x)
Gq(1− x− n)Gq(1 + x)
= (−1)n(n+1)/2qn(n+1)x/2+(n−1)n(n+1)/6Γq(x)nΓq(1− x)n
for n ∈ Z, we can compute the coefficients CIII1 and obtain
τ̃1 = K
[
θ∗, θ? + 1
2 , σ −
1
2
]
τ4, τ̃2 = sK
[
θ∗, θ? − 1
2 , σ −
1
2
]
τ3,
τ̃3 = K
[
θ∗ + 1
2 , θ?, σ
]
τ2, τ̃4 = sK
[
θ∗ − 1
2 , θ?, σ − 1
]
τ1,
where we denote by τ̃i the tau functions with parameters (θ̃∗, θ̃?, σ̃, s̃, t̃) and by τi the tau functions
with parameters (θ∗, θ?, σ, s, t), and
K[θ∗, θ?, σ] = q−(θ∗+θ?)σ2
∏
ε,ε′=±
Gq(1 + εθ∗ + ε′σ)εGq(1 + εθ? + ε′σ)ε.
By definition we have
K
[
θ∗, θ? + 1
2 , σ −
1
2
]
K
[
θ∗, θ? − 1
2 , σ −
1
2
]
K
[
θ∗ + 1
2 , θ?, σ
]
K
[
θ∗ − 1
2 , θ?, σ − 1
] = −q(θ?−θ∗)/2. (4.10)
Applying the transformation (4.9) to the bilinear equations (4.7) and (4.8) and using the rela-
tion (4.10), we obtain the identities (4.5) and (4.6). �
We note that the bilinear equations (3.6), (3.8), (3.9), and (3.10) for the tau functions of q-PV
degenerate to (4.5), (4.6), (4.7), and (4.8), respectively.
12 Y. Matsuhira and H. Nagoya
5 From q-PIII1 to q-PIII2
In this section, we take a limit of the tau functions of q-PIII1 to q-PIII2 . Define the tau function
by
τ III2(θ∗ | s, σ, t) =
∑
n∈Z
snt(σ+n)2CIII2 [θ∗ |σ + n]ZIII2 [θ∗ |σ + n, t],
with
CIII2 [θ∗ |σ] = (q − 1)−3σ2
∏
ε=±
Gq(1− θ∗ + εσ)
Gq(1 + 2εσ)
,
ZIII2 [θ∗ |σ, t] =
∑
(λ+,λ−)∈Y2
t|λ+|+|λ−|
∏
ε=±
N∅,λε
(
q−θ∗−εσ
)
fλε(q
εσ)−1∏
ε,ε′=±
Nλε,λε′
(
q(ε−ε′)σ
) .
In the same way as in Section 3, it is possible to remove fλε(q
εσ)−1 from ZIII2 [θ∗ |σ, t] by
change of variables. Because if we set
ZCS=0
III2 [θ∗ |σ, t] =
∑
(λ+,λ−)∈Y2
t|λ+|+|λ−|
∏
ε=±
N∅,λε
(
q−θ∗−εσ
)
∏
ε,ε′=±
Nλε,λε′
(
q(ε−ε′)σ
) ,
then we have
ZIII2 [θ∗ |σ, t] = ZCS=0
III2
[
−θ∗ |σ, q−θ∗t
]
.
Let us define the tau functions for q-PIII2 by
τ III2
1 = τ III2
(
θ∗ − 1
2 | s, σ, t/
√
q
)
, τ III2
2 = τ III2
(
θ∗ + 1
2 | s, σ + 1,
√
qt
)
,
τ III2
3 = τ III2
(
θ∗ | s, σ + 1
2 , t
)
.
Put
C1 = (q − 1)−σ
2
q−Λσ2
∏
ε=±
Gq(1− Λ + εσ)−1,
C2 = C1,
C3 = (q − 1)−(σ+1/2)2q−(Λ−1/2)(σ+1/2)2
∏
ε=±
Gq
(
3
2 − Λ + ε
(
σ + 1
2
))−1
,
C4 = (q − 1)−(σ−1/2)2q−(Λ+1/2)(σ−1/2)2
∏
ε=±
Gq
(
1
2 − Λ + ε
(
σ − 1
2
))−1
.
Proposition 5.1. Set
θ? = Λ, t = qΛt1, s = s̃(q − 1)−2σq−σ(2Λ+1)
∏
ε=±
Γq(−Λ + εσ)−ε.
Then we have
Ciτ
III1
i (θ∗, θ? | s, σ, t)→ τ III2
i (θ∗ | s̃, σ, t1), i = 1, 3,
C2τ
III1
2 (θ∗, θ? | s, σ, t)→ s̃τ III2
2 (θ∗ | s̃, σ, t1),
C4τ
III1
4 (θ∗, θ? | s, σ, t)→ s̃τ III2
3 (θ∗ | s̃, σ, t1)
as Λ→∞.
Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations 13
In what follows, we abbreviate τ III2
i (θ∗ | s, σ, t) to τi. Since we have the relation
C1C2 = (q − 1)−1/2
(
q−Λ/2 − qΛ/2−σ)C3C4,
we obtain the following theorem by the degeneration.
Theorem 5.2. The functions
y = q−θ∗−1(q − 1)−1/2t1/2
τ2
3
τ1τ2
, z = q−θ∗/2−3/4(q − 1)−1/2t1/2
τ3τ3
τ1τ2
solves the q-Painlevé III2 equation
yy
a3a4
= − z2
z − b3
,
zz
b3
= − y(y − a2t)
a4(y − a3)
with the parameters
a2 = q−θ∗−1, a3 = q−1, a4 = q−3θ∗/2−1/2, b2 = q−θ∗/2, b3 = q−θ∗/2−1/2.
Furthermore, the tau functions τi (i = 1, 2, 3) satisfy the following bilinear equations.
τ1τ2 − q−θ∗(q − 1)−1/2t1/2τ2
3 − τ1τ2 = 0, (5.1)
τ1τ2 − (q − 1)−1/2t−1/2τ2
3 + (q − 1)−1/2t−1/2τ3τ3 = 0, (5.2)
τ1τ2 + q−1/4(q − 1)−1/2t1/2τ3τ3 − τ1τ2 = 0. (5.3)
We note that the bilinear equations (4.5), (4.6), and (4.7) for the tau functions of q-PIII1
degenerate to (5.1), (5.2), and (5.3), respectively.
6 From q-PIII2 to q-PIII3
In this section, we take a limit of the tau functions of q-PIII2 to q-PIII3 . Define the tau function
by
τ III3(s, σ, t) =
∑
n∈Z
snt(σ+n)2CIII3 [σ + n]ZIII3 [σ + n, t],
with
CIII3 [σ] = (q − 1)−4σ2
∏
ε=±
1
Gq(1 + 2ε(σ + n))
,
ZIII3 [σ, t] =
∑
(λ+,λ−)∈Y2
t|λ+|+|λ−|
1∏
ε,ε′=±
Nλε,λε′
(
q(ε−ε′)σ
) .
Let us define the tau functions for q-PIII3 by
τ III3
1 = τ III3(s, σ, t), τ III3
2 = τ III3
(
s, σ + 1
2 , t
)
.
Put
C1 = (q − 1)−σ
2
q−(Λ−1/2)σ2
∏
ε=±
Gq
(
3
2 − Λ + εσ
)−1
,
C2 = (q − 1)−(σ+1)2q−(Λ+1/2)(σ+1)2
∏
ε=±
Gq
(
1
2 − Λ + ε(σ + 1)
)−1
,
C3 = (q − 1)−(σ+1/2)2q−Λ(σ+1/2)2
∏
ε=±
Gq
(
1− Λ + ε
(
σ + 1
2
))−1
.
14 Y. Matsuhira and H. Nagoya
Proposition 6.1. Set
θ∗ = Λ, t = qΛt1, s = s̃(q − 1)−2σq−2σΛ
∏
ε=±
Γq
(
1
2 − Λ + εσ
)−ε
.
Then we have
C1τ
III2
1 (θ∗ | s, σ, t)→ τ III3
1 (s̃, σ, t1),
C2τ
III2
2 (θ∗ | s, σ, t)→ τ III3
1 (s̃, σ, t1)/s̃,
C3τ
III2
3 (θ∗ | s, σ, t)→ τ III3
2 (s̃, σ, t1),
as Λ→∞.
In what follows, we abbreviate τ III3
i (s, σ, t) to τi. Since we have the relation
C1C2 = (q − 1)1/2 q
−σ−1/2+Λ/2
q−σ−1/2 − qΛ
C2
3 ,
we obtain the following theorem by the degeneration.
Theorem 6.2. The functions
y = t1/2
sτ2
2
τ2
1
, z = q−3/4t1/2
sτ2τ2
τ1τ1
solves the q-Painlevé III3 equation
yy
a3
= z2, zz = −y(y − a2t)
y − a3
(6.1)
with the parameters
a2 = q−1, a3 = q−1.
Furthermore, the tau functions τ1, τ2 satisfy the following bilinear equations.
st1/2τ2
2 − τ2
1 + τ1τ1 = 0, (6.2)
s−1t1/2τ2
1 − τ2
2 + τ2τ2 = 0. (6.3)
We note that the bilinear equations (5.1), (5.2) for the tau functions of q-PIII2 degenerate
to (6.2), (6.3), respectively. As suggested in [5, equations (2.9)–(2.11)], the bilinear equation (6.3)
is derived from (6.2) by the transformation σ → σ + 1/2.
Remark 6.3. The tau function Tc
(
q2σ, s; q | t
)
proposed in [5] for the q-Painlevé III3 equation
are related to our tau functions by
Tc
(
q2σ, s; q | t
)
= (−1)−2σ2
τ III3
(
(−1)−4σs, σ, t
)
.
Remark 6.4. q-P (A′7) in [19] (or q-P
(
A
(1)
1 /A
(1)
7
)
in [17, equation (8.14)]) is
yy
a4
= −z(z − b2t)
z − b3
,
zz
b3
=
y2
a4
,
where y = y(t), z = z(t), and a4, b1, b2, b3 are complex parameters. Replacing y, z in (6.1)
by z, y, we obtain q-P (A′7) with a4 = 1, b2 = 1, and b3 = q−1.
Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations 15
The bilinear equations (6.2), (6.3) are also proved by using the Nakajima–Yoshioka blow-up
equations [6]. There exists another q-difference equation admitting PIII3 and PI as limits [13],
which corresponds to the q-difference Painlevé equation of the surface type A
(1)
7 [25]. Its standard
form (see equation (2.44) in [26]) is
gg2g = t2(1− g), (6.4)
where g = g(t). A series expansion of the tau function for q-P
(
A
(1)
7
)
(6.4) was proposed and
conjectured to satisfy its bilinear form in [4]. Later, it was proved in [6]. Below, we show that
their tau function for q-P
(
A
(1)
7
)
(6.4) is also obtained as another limit of the tau function for
q-PIII2 .
Redefine the tau function by
τ III3(s, σ, t) =
∑
n∈Z
snt(σ+n)2CIII3 [σ + n]ZIII3 [σ + n, t],
with
CIII3 [σ] = (−1)n
2
(q − 1)−4σ2
∏
ε=±
1
Gq(1 + 2ε(σ + n))
,
ZIII3 [σ, t] =
∑
(λ+,λ−)∈Y2
t|λ+|+|λ−|
∏
ε=±
fλε(q
εσ)−1∏
ε,ε′=±
Nλε,λε′
(
q(ε−ε′)σ
) .
Let us define the tau functions for q-P
(
A
(1)
7
)
by
τ III3
1 = τ III3
(
s, σ, t/
√
q
)
, τ III3
2 = τ III3
(
s, σ + 1
2 , t
)
.
Put
C1 = (q − 1)−σ
2
∏
ε=±
Gq
(
3
2 + Λ + εσ
)−1
,
C2 = (q − 1)−(σ+1)2
∏
ε=±
Gq
(
1
2 + Λ + ε(σ + 1)
)−1
,
C3 = (q − 1)−(σ+1/2)2
∏
ε=±
Gq
(
1 + Λ + ε
(
σ + 1
2
))−1
.
Proposition 6.5. Set
θ∗ = −Λ, s = s̃(q − 1)−2σ
∏
ε=±
Γq
(
1
2 + Λ + εσ
)−ε
.
Then we have
C1τ
III2
1 (θ∗ | s, σ, t)→ τ III3
1 (s̃, σ, t),
C2τ
III2
2 (θ∗ | s, σ, t)→ τ III3
1 (s̃, σ, qt)/s̃,
C3τ
III2
3 (θ∗ | s, σ, t)→ τ III3
2 (s̃, σ, t),
as Λ→∞.
In what follows, we abbreviate τ III3
i (s, σ, t) to τi. Since we have the relation
C1C2 =
(q − 1)1/2
1− qΛ−σ+1/2
C2
3 ,
we obtain the following theorem by the degeneration.
16 Y. Matsuhira and H. Nagoya
Theorem 6.6. The functions
y = −q−1t1/2
sτ2
2
τ1τ1
, z = −q−3/4t1/2
sτ2τ2
τ1τ1
solves
yy = −q−3/2 z2
z − q−1/2
, zz = y(qy − t). (6.5)
Furthermore, the tau functions τ1, τ2 satisfy the following bilinear equations.
s−1t1/2τ1τ1 − τ2
2 + τ2τ2 = 0, (6.6)
τ2
1 − sq−1/4t1/2τ2τ2 − τ1τ1 = 0. (6.7)
We note that the bilinear equations (5.2), (5.3) for the tau functions of q-PIII2 degenerate
to (6.6), (6.7), respectively. By the change of variables t → √qt, σ → σ + 1/2, the bilinear
equation (6.7) transforms (6.6). The bilinear equation (6.6) is equivalent to the bilinear equa-
tion (4.20) for N = 2, m = 1 in [6], which is for q-P
(
A
(1)
7
)
. Following [4, Example 3.5], we
take a time evolution T as T (f(σ, t)) = f(σ + 1/2,
√
qt). Then the bilinear equation (6.7) is
equivalent to
τ2 − t1/2ττ − ττ = 0,
where τ = τ III3(s, σ, t), τ = T (τ), τ = T−1(τ). Let g = t1/2τττ−2, then g satisfies q-
P
(
A
(1)
7
)
(6.4).
Acknowledgements
This work is partially supported by JSPS KAKENHI Grant Number JP15K17560. The authors
thank the referees for valuable suggestions and comments.
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1 Introduction
2 Results on q-PVI from JNS
3 From q-PVI to q-PV
4 From q-PV to q-PIII1
5 From q-PIII1 to q-PIII2
6 From q-PIII2 to q-PIII3
References
|
| id | nasplib_isofts_kiev_ua-123456789-210221 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:47Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Matsuhira, Y. Nagoya, H. 2025-12-04T13:00:56Z 2019 Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations / Y. Matsuhira, H. Nagoya // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 39A13; 33E17; 05A30 arXiv: 1811.03285 https://nasplib.isofts.kiev.ua/handle/123456789/210221 https://doi.org/10.3842/SIGMA.2019.074 We derive series representations for the tau functions of the q-Painlevé V, III₁, III₂, and III₃ equations, as degenerations of the tau functions of the q-Painlevé VI equation in [Jimbo M., Nagoya H., Sakai H., J. Integrable Syst. 2 (2017), xyx009, 27 pages]. Our tau functions are expressed in terms of q-Nekrasov functions. Thus, our series representations for the tau functions have explicit combinatorial structures. We show that general solutions to the q-Painlevé V, III₁, III₂, and III₃ equations are written by our tau functions. We also prove that our tau functions for the q-Painlevé III₁, III₂, and III₃ equations satisfy the three-term bilinear equations for them. This work is partially supported by JSPS KAKENHI Grant Number JP15K17560. The authors thank the referees for their valuable suggestions and comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations Article published earlier |
| spellingShingle | Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations Matsuhira, Y. Nagoya, H. |
| title | Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations |
| title_full | Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations |
| title_fullStr | Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations |
| title_full_unstemmed | Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations |
| title_short | Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations |
| title_sort | combinatorial expressions for the tau functions of q-painlevé v and iii equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210221 |
| work_keys_str_mv | AT matsuhiray combinatorialexpressionsforthetaufunctionsofqpainlevevandiiiequations AT nagoyah combinatorialexpressionsforthetaufunctionsofqpainlevevandiiiequations |