Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation
We consider a q-Painlevé IV equation which is the A₄⁽¹⁾-surface type in the Sakai's classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric functions. Two of them are expressed by ₂φ₁ basic hypergeometric series a...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2014 |
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Інститут математики НАН України
2014
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| Цитувати: | Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation / N. Nakazono // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 41 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859817437277454336 |
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| author | Nakazono, N. |
| author_facet | Nakazono, N. |
| citation_txt | Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation / N. Nakazono // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 41 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider a q-Painlevé IV equation which is the A₄⁽¹⁾-surface type in the Sakai's classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric functions. Two of them are expressed by ₂φ₁ basic hypergeometric series and the other is given by ₂ψ₂ bilateral basic hypergeometric series.
|
| first_indexed | 2025-12-07T15:23:54Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 090, 23 pages
Hypergeometric Solutions
of the A
(1)
4 -Surface q-Painlevé IV Equation
Nobutaka NAKAZONO
School of Mathematics and Statistics, The University of Sydney,
New South Wales 2006, Australia
E-mail: nobua.n1222@gmail.com
Received June 06, 2013, in final form August 14, 2014; Published online August 22, 2014
http://dx.doi.org/10.3842/SIGMA.2014.090
Abstract. We consider a q-Painlevé IV equation which is the A
(1)
4 -surface type in the
Sakai’s classification. We find three distinct types of classical solutions with determinantal
structures whose elements are basic hypergeometric functions. Two of them are expressed
by 2ϕ1 basic hypergeometric series and the other is given by 2ψ2 bilateral basic hypergeo-
metric series.
Key words: q-Painlevé equation; basic hypergeometric function; affine Weyl group; τ -func-
tion; projective reduction; orthogonal polynomial
2010 Mathematics Subject Classification: 33D05; 33D15; 33D45; 33E17; 39A13
1 Introduction
The focus of this paper is on the following single second-order ordinary difference equation:
(Xn+1Xn − 1)(Xn−1Xn − 1)
= q(−N+2n−m−1)/2a0a
3/2
1 a22
(
Xn + q(N−m)/2a
1/2
1
)(
Xn + q(−N+m)/2a
−1/2
1
)
Xn + q(−N+n−m)/2a
1/2
1 a2
, (1.1)
where n ∈ Z is the independent variable, Xn = Xn(m,N) is the dependent variable and m,N ∈
Z and a0, a1, a2, q (|q| < 1) ∈ C∗ are parameters. Equation (1.1) is known as a q-discrete analog
of the Painlevé IV equation (q-PIV) [35].
In 2001, Sakai introduced a geometric approach to the theory of the Painlevé and discrete
Painlevé equations (Painlevé systems) and showed the classifications of Painlevé systems by the
rational surface [37]. The rational surface can be identified with the space of initial condition,
and the group of Cremona isometries associated with the surface generate the affine Weyl group.
He also showed that the translation part of the affine Weyl group gives rise to various discrete
Painlevé equations. Then, such discrete Painlevé equations are said to have the affine Weyl
group symmetries.
In 2004, q-PIV (1.1) was generalized to the following simultaneous first-order ordinary diffe-
rence equations by the singularity confinement criterion [38]:
(yk+1xk − 1)(ykxk − 1)
= q(−N+4k−m+2l−2)/2a0a
3/2
1 a22a3
(
xk + q(N−m)/2a
1/2
1
)(
xk + q(−N+m)/2a
−1/2
1
)
xk + q(−N+2k−m+2l)/2a
1/2
1 a2
, (1.2a)
(ykxk − 1)(ykxk−1 − 1)
= q(−N+4k−m+2l−4)/2a0a
3/2
1 a22a3
(
yk + q(N−m)/2a
1/2
1
)(
yk + q(−N+m)/2a
−1/2
1
)
yk + q(−N+2k−m−2)/2a
1/2
1 a2a3
, (1.2b)
mailto:nobua.n1222@gmail.com
http://dx.doi.org/10.3842/SIGMA.2014.090
2 N. Nakazono
where k ∈ Z is the independent variable, xk = xk(l,m,N) and yk = yk(l,m,N) are dependent
variables and l,m,N ∈ Z and a0, a1, a2, a3, q (|q| < 1) ∈ C∗ are parameters. System (1.2) is
known as a q-discrete analog of the Painlevé V equation (q-PV). It is also known that q-PV (1.2)
is the A
(1)
4 -surface type in the Sakai’s classification and has the affine Weyl group symmetry of
type A
(1)
4 .
Conversely, q-PIV (1.1) can be recovered from q-PV (1.2) by putting
a3 = q1/2, l = 0,
and replacing the independent variable and the dependent variables by
2k = n, xk = Xn, yk = Xn−1.
This procedure is referred to as “symmetrization” of q-PV (1.2), which comes from the ter-
minology of the Quispel–Roberts–Thompson (QRT) mapping [33, 34]. After this terminology,
q-PV (1.2) is sometimes called the “asymmetric discrete Painlevé equation”, and q-PIV (1.1) is
called the “symmetric discrete Painlevé equation”. It appears as though the symmetrization is
a simple specialization on the level of the equation, but the following problems were known:
(i) According to Sakai’s theory [37], the discrete Painlevé equations arise as the birational
mappings corresponding to the translations of the affine Weyl groups. The asymmetric
discrete Painlevé equations are characterized in this manner, however, it was not known
how to characterize the symmetric discrete Painlevé equations as the action of affine Weyl
groups;
(ii) Painlevé systems admit the particular solutions expressible in terms of the hypergeometric
type functions (hypergeometric solutions) when some of the parameters take special values
(see, for example, [13, 14] and references therein). However, the hypergeometric solutions
to the symmetric discrete Painlevé equation cannot be obtained by the näıve specialization
of those to the corresponding asymmetric equation.
In [17], the mechanism of the symmetrization was investigated and the nontrivial incon-
sistency among the hypergeometric solutions were explained in detail by taking an example
of q-Painlevé equation with the affine Weyl group symmetry of type (A2 + A1)
(1). The key
to characterize the symmetric discrete Painlevé equation as the action of affine Weyl group is
taking the half-step translation instead of a translation as a time evolution. In general, various
discrete dynamical systems of Painlevé type can be obtained from elements of infinite order that
are not necessarily translations in the affine Weyl group by taking the projection on appropriate
subspaces of the parameter spaces. Such a procedure is called a “projective reduction”.
It is well known that the τ -functions play a crucial role in the theory of integrable systems [25],
and it is also possible to introduce them in the theory of Painlevé systems [6, 7, 8, 27, 29, 30, 31,
32]. A representation of the affine Weyl groups can be lifted on the level of the τ -functions [12,
15, 39], which gives rise to various bilinear equations of Hirota type satisfied the τ -functions.
Usually, the hypergeometric solutions are derived by reducing the bilinear equations to the
Plücker relations by using the contiguity relations satisfied by the entries of determinants [2, 3,
9, 10, 11, 18, 19, 20, 21, 28, 36]. This method is elementary, but it encounters technical difficulties
for Painlevé systems with large symmetries. In order to overcome this difficulty, Masuda has
proposed a method of constructing hypergeometric solutions under a certain boundary condition
on the lattice where the τ -functions live (hypergeometric τ -functions), so that they are consistent
with the action of the affine Weyl groups [23, 24, 26]. Although this requires somewhat complex
calculations, the merit is that it is systematic and that it can be applied to the systems with
large symmetries.
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 3
In [16], the list of the simplest hypergeometric solutions to the symmetric q-Painlevé equa-
tions are shown. In general, hypergeometric solutions of Painlevé systems can be expressed by
determinants whose entries are given by hypergeometric type functions. Therefore, it is natural
to be curious about the determinant formulae of them. The purpose of this paper is to obtain
the determinant formulae of the hypergeometric solutions to the q-PIV via the construction of
the hypergeometric τ -functions and the theory of orthogonal polynomials.
This paper is organized as follows: in Section 2, we first introduce a representation of the
affine Weyl group of type A
(1)
4 . Next, we show how q-PV (1.2) and q-PIV (1.1) can be de-
rived from the representation. In Section 3, we construct the hypergeometric τ -functions for
the q-PIV and obtain the hypergeometric solutions of the q-PIV which are expressed by basic
hypergeometric series (see Theorems 1 and 2). In Section 4, we obtain the hypergeometric so-
lutions of the q-PIV which are expressed by bilateral basic hypergeometric series via the theory
of orthogonal polynomials (see Theorem 3). Some concluding remarks are given in Section 5.
We use the following conventions of q-analysis with |q|, |p| < 1 throughout this paper [1, 22]:
• q-shifted factorials:
(a; q)∞ =
∞∏
i=1
(
1− aqi−1
)
, (a; q)λ =
(a; q)∞
(aqλ; q)∞
,
where λ ∈ C;
• Jacobi theta function:
Θ(a; q) = (a; q)∞
(
qa−1; q
)
∞;
• Elliptic gamma function:
Γ(a; p, q) =
(
pqa−1; p, q
)
∞
(a; p, q)∞
,
where
(a; p, q)k =
k−1∏
i,j=0
(
1− piqja
)
;
• Basic hypergeometric series:
sϕr
(
a1, . . . , as
b1, . . . , br
; q, z
)
=
∞∑
n=0
(a1, . . . , as; q)n
(b1, . . . , br; q)n(q; q)n
[
(−1)nqn(n−1)/2
]1+r−s
zn,
where
(a1, . . . , as; q)n =
s∏
j=1
(aj ; q)n;
• Bilateral basic hypergeometric series:
sψr
(
a1, . . . , as
b1, . . . , br
; q, z
)
=
∞∑
n=−∞
(a1, . . . , as; q)n
(b1, . . . , br; q)n
[
(−1)nqn(n−1)/2
]r−s
zn;
4 N. Nakazono
• Bilateral q-integral:∫ ∞
−∞
f(t)dqt = (1− q)
∞∑
n=−∞
(
f(qn) + f(−qn)
)
qn.
We note that the following formulae hold:
(a; q)λ+1
(a; q)λ
= 1− aqλ, Θ(qa; q)
Θ(a; q)
= −a−1, Γ(qa; q, q)
Γ(a; q, q)
= Θ(a; q).
2 Affine Weyl group of type A
(1)
4
2.1 Birational representation of the affine Weyl group of type A
(1)
4
In this section, we formulate the family of Bäcklund transformations of q-PV (1.2) as a birational
representation of the affine Weyl group of type A
(1)
4 .
Let si (i = 0, 1, 2, 3, 4), σ and ι be transformations of the parameters ak (k = 0, 1, 2, 3, 4)
and the variables fj (j = 0, 1, 2, 3, 4). The action of the transformations on the parameters is
given by
si(aj) = aja
−aij
i , σ(ai) = ai+1,
ι : (a0, a1, a2, a3, a4) 7→
(
a−10 , a−14 , a−13 , a−12 , a−11
)
,
where i, j ∈ Z/5Z and the symmetric 5× 5 matrix
A = (aij)
4
i,j=0 =
2 −1 0 0 −1
−1 2 −1 0 0
0 −1 2 −1 0
0 0 −1 2 −1
−1 0 0 −1 2
is the Cartan matrix of type A
(1)
4 . Moreover, the action on the variables is given by
si(fi+2) =
ai+3ai+4(aiai+1 + ai+3fi)
a2i+1fi+3
, si(fi+4) =
ai+4(ai+2 + ai+4aifi+1)
aiai+1a2i+2fi+3
,
si(fj) = fj , j 6= i+ 2, i+ 4, σ(fi) = fi+1,
ι : (f0, f1, f2, f3, f4) 7→ (f1, f0, f4, f3, f2),
where i ∈ Z/5Z. Note that the variables satisfy the following conditions:
a2i+3ai+4fi = ai+1(aiai+1fi+2fi+3 − ai+3ai+4),
where i ∈ Z/5Z. The conditions above look like five, but they are essentially three. Therefore,
variables fi are essentially two.
Proposition 1 ([2, 37, 39]). The group of birational transformations W̃
(
A
(1)
4
)
= 〈s0, s1, s2, s3,
s4, σ, ι〉 gives a representation of the (extended) affine Weyl group of type A
(1)
4 . Namely, the
transformations satisfy the fundamental relations
s2i = 1, (sisi±1)
3 = 1, (sisj)
2 = 1, j 6= i± 1, σ5 = 1, σsi = si+1σ,
ι2 = 1, ιs0 = s0ι, ιs1 = s4ι, ιs2 = s3ι,
where i, j ∈ Z/5Z.
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 5
In general, for a function F = F (ai, fj), we let an element w ∈ W̃
(
A
(1)
4
)
act as w.F (ai, fj) =
F (w.ai, w.fj), that is, w acts on the arguments from the left. Note that q = a0a1a2a3a4 is in-
variant under the action of 〈s0, s1, s2, s3, s4, σ〉. We define the translations Ti (i = 0, 1, 2, 3, 4) by
T0 = σs4s3s2s1, T1 = σs0s4s3s2, T2 = σs1s0s4s3,
T3 = σs2s1s0s4, T4 = σs3s2s1s0, (2.1)
whose action on the parameters is given by
T0 : (a0, a1, a2, a3, a4) 7→
(
qa0, q
−1a1, a2, a3, a4
)
,
T1 : (a0, a1, a2, a3, a4) 7→
(
a0, qa1, q
−1a2, a3, a4
)
,
T2 : (a0, a1, a2, a3, a4) 7→
(
a0, a1, qa2, q
−1a3, a4
)
,
T3 : (a0, a1, a2, a3, a4) 7→
(
a0, a1, a2, qa3, q
−1a4
)
,
T4 : (a0, a1, a2, a3, a4) 7→
(
q−1a0, a1, a2, a3, qa4
)
.
Note that Ti (i = 0, 1, 2, 3, 4) commute with each other and T0T1T2T3T4 = 1.
2.2 Derivations of the q-Painlevé equations
In this section, we derive the q-Painlevé equations from W̃
(
A
(1)
4
)
. The action of T23 = T2T3 on
f -variables can be expressed as
(T23(y)x− 1)(yx− 1) = q−1a0a
3/2
1 a22a3
(
x+ a
1/2
1
)(
x+ a
−1/2
1
)
x+ a
1/2
1 a2
, (2.2)
(yx− 1)
(
yT−123 (x)− 1
)
= q−2a0a
3/2
1 a22a3
(
y + a
1/2
1
)(
y + a
−1/2
1
)
y + q−1a
1/2
1 a2a3
, (2.3)
where
x = a0a
1/2
1 a−13 f2, y = a
−1/2
1 a−12 a−14 s4(f1).
Applying T k23T
l
2T
m
0 T
N
1 on equations (2.2) and (2.3) and putting
xk(l,m,N) = T k23T
l
2T
m
0 T
N
1 (x), yk(l,m,N) = T k23T
l
2T
m
0 T
N
1 (y),
we obtain q-PV (1.2). Then, we can regard T23 and Ti (i = 0, 1, 2, 3, 4) as the time evolution and
the Bäcklund transformations of q-PV (1.2), respectively. We note that considering the action
of T0:
T0(g)g =
(
f + a
−3/4
0 a
1/4
1 a
−1/4
3
)(
f + a
−3/4
0 a
1/4
1 a
−1/4
3 a−14
)
1 + a
−1/4
0 a
−1/4
1 a
1/4
3 f
,
T−10 (f)f =
(
g + a
−1/4
0 a
3/4
1 a
1/4
3
)(
g + a
−1/4
0 a
3/4
1 a2a
1/4
3
)
1 + a
1/4
0 a
1/4
1 a
−1/4
3 g
,
where
f = a
−3/4
0 a
−3/4
1 a
3/4
3 f0, g = a
3/4
0 a
3/4
1 a
−3/4
3 f2,
we obtain another q-discrete analog of Painlevé V equation [37]:
gn+1gn =
(
fn+q−n+
k+l−m
4 a
− 3
4
0 a
1
4
1 a
− 1
4
3
)(
fn+q−n+
k+l+3m
4 a
− 3
4
0 a
1
4
1 a
− 1
4
3 a−14
)
1+q
−k−l+m
4 a
− 1
4
0 a
− 1
4
1 a
1
4
3 fn
, (2.4a)
6 N. Nakazono
fn+1fn =
(
gn+1+q−n−1+
3k−l+m
4 a
− 1
4
0 a
3
4
1 a
1
4
3
)(
gn+1 + q−n−1+
−k+3l+m
4 a
− 1
4
0 a
3
4
1 a2a
1
4
3
)
1+q
k+l−m
4 a
1
4
0 a
1
4
1 a
− 1
4
3 gn+1
, (2.4b)
where
fn = fn(k, l,m) = Tn0 T
k
1 T
l
2T
m
3 (f), gn = gn(k, l,m) = Tn0 T
k
1 T
l
2T
m
3 (g).
Thus, q-PV (1.2) and equation (2.4) are the Bäcklund transformations each other.
In order to derive q-PIV (1.1), we factorize T23 as follows
T23 = R2
23,
where R23 is given by
R23 = σs1s0s4. (2.5)
The action of R23 on the parameters is given by
R23 : (a0, a1, a2, a3, a4) 7→
(
a0, a1, a2a3, qa
−1
3 , q−1a3a4
)
.
Let us consider the projection of the action of R23 on the line
a3 = q1/2. (2.6)
Then, the action on the parameters becomes translational motion:
R23 : (a0, a1, a2, a4) 7→
(
a0, a1, q
1/2a2, q
−1/2a4
)
.
Since the action of R23 on the variable f2 is given by
R23(f2) =
q
a20a1f2
(
1 +
a0
(
a0f2 + q1/2
)(
a0a1f2 + q1/2
)
q1/2
(
q1/2a2 + a0f2
)
f4
)
,
R−123 (f2) =
q
a20a1f2
(
1 +
a1a
2
2
q1/2
f4
)
,
we obtain
(
R23(X)X − 1
)(
R−123 (X)X − 1
)
= q−1/2a0a
3/2
1 a22
(
X + a
1/2
1
)(
X + a
−1/2
1
)
X + a
1/2
1 a2
, (2.7)
where
X = q−1/2a0a
1/2
1 f2. (2.8)
Applying Rn23T
m
0 T
N
1 on equation (2.7) and putting
Xn(m,N) = Rn23T
m
0 T
N
1 (X),
we obtain q-PIV (1.1). Note that R23 commute with Ti (i = 0, 1, 4) and T0T1R
2
23T4 = 1. Then,
R23 and Ti (i = 0, 1, 4) are regarded as the time evolution and the Bäcklund transformations
of q-PIV (1.1), respectively.
3 Hypergeometric solutions of the q-PIV (I)
In this section, we obtain the hypergeometric solutions of q-PIV (1.1) by constructing the hy-
pergeometric τ -functions for the q-PIV.
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 7
3.1 τ -functions
In this section, we define the τ -functions.
We introduce the new variables τi (i = 1, 2, . . . , 7) with
f2 =
τ4τ5
τ6τ7
, f4 =
τ1τ2
τ3τ7
, (3.1)
and lift the representation of W̃
(
A
(1)
4
)
on their level:
s0(τ1) =
a4(a0τ3τ4τ5 + a2a3τ1τ2τ6 + a0a3τ3τ6τ7)
a20a1a2τ4τ7
, s0(τi) = τi, i = 2, 3, 5, 6,
s0(τ4) =
a0a4(a0τ3τ4τ5 + a2a3τ1τ2τ6 + a3τ3τ6τ7)
a1a2τ1τ7
,
s0(τ7) =
a4
(
a20τ3τ4τ5 + a3a0τ3τ6τ7 + a2a3τ1τ2τ6
)
a0a1a2τ1τ4
, s1(τ1) = τ2, s1(τ2) = τ1,
s1(τi) = τi, i = 3, . . . , 7, s2(τ1) =
a0a1(a0τ4τ5 + a2a3τ6τ7)
a23τ3
,
s2(τ3) =
a0a1(a0τ4τ5 + a3τ6τ7)
a2a23τ1
, s2(τi) = τi, i = 2, 4, 5, 6, 7,
s3(τ4) =
a2(a2a3τ1τ2 + a0τ3τ7)
a20a3a4τ6
, s3(τ6) =
a2a3(a2τ1τ2 + a0τ3τ7)
a20a4τ4
,
s3(τi) = τi, i = 1, 2, 3, 5, 7, s4(τ4) = s4(τ5), s4(τ5) = s4(τ4),
s4(τi) = τi, i = 1, 2, 3, 6, 7, ι : (τ1, τ2, τ3, τ4, τ5, τ6, τ7) = (τ4, τ5, τ6, τ1, τ2, τ3, τ7),
σ(τ1) =
a0a1 (a0τ4τ5 + a3τ6τ7)
a2a23τ1
, σ(τ2) = τ3, σ(τ3) = τ6,
σ(τ4) =
a4
(
a20τ3τ4τ5 + a3a0τ3τ6τ7 + a2a3τ1τ2τ6
)
a0a1a2τ1τ4
, σ(τ5) = τ7,
σ(τ6) = τ5, σ(τ7) = τ2.
Then, we get the following proposition:
Proposition 2 ([39]). The transformations: s0, s1, s2, s3, s4, σ, ι, on the variables τi (i =
1, 2, . . . , 7) also realize the (extended) affine Weyl group of type A
(1)
4 .
Let us define the τ -functions τn,mN (n,m,N ∈ Z) by
τn,mN = Rn23T
m
0 T
N
1 (τ4).
By definition, every τ -function can be determined by a rational function in 7 initial variables τi
(i = 1, 2, . . . , 7). We note that the 7 initial variables are expressed by the τ -functions as the
following (see Fig. 1):
τ1 = τ1,01 , τ2 = τ1,10 , τ3 = τ1,11 , τ4 = τ0,00 ,
τ5 = τ3,11 , τ6 = τ2,11 , τ7 = τ1,00 . (3.2)
3.2 Hypergeometric τ -functions for the q-PIV
The aim of this section is to construct the hypergeometric τ -functions for the q-PIV. We define
the hypergeometric τ -functions for the q-PIV by τn,mN consistent with the action of 〈R23, T0〉.
Here, we mean τ = τ(α) consistent with an action of transformation r as
r.τ = τ(r.α).
8 N. Nakazono
T1 T0 R23�4 = �0;00 �7 = �1;00
�2 = �1;10
�1 = �1;01 �3 = �1;11 �6 = �2;11 �5 = �3;11
Figure 1. Configuration of the τ -functions on the 3D-lattice.
Hereinafter, we impose the condition (2.6), and then regard τ -functions τn,mN as the functions
in a0 and a2 consistent with the action of 〈R23, T0〉, i.e.,
τn,mN = τ0,0N
(
qma0, q
n/2a2
)
.
By definition, every τ -function τn,mN is determined by a rational function in τn,m0 and τn,m1
(or τ1, . . . , τ7). Thus, our purpose is determining τn,m0 and τn,m1 consistent with the action of
〈R23, T0〉 and constructing the closed-form expressions of τn,mN (N ≥ 2) under the condition
a0a1 = q, (3.3)
and the boundary condition
τn,mN = 0, N < 0. (3.4)
Henceforth, we construct the hypergeometric τ -functions for the q-PIV in the following four
steps.
Step 1. Conditions of τn,m
0
In the first step, we obtain the condition of τn,m0 , which follows from the boundary condi-
tion (3.4).
Lemma 1. The following bilinear equations hold:
τn,mN+1τ
n−1,m−1
N−1 − q(−n−4m+4N+7)/2a−20 a−12 τn,m−1N τn−1,mN
− q(−n−2m+4N+4)/2a−10 a−12 τn,mN τn−1,m−1N = 0, (3.5)
τn,mN+1τ
n−1,m
N−1 + q2N−n+1a−22
(
q(−n+2N+1)/2a−12 − 1
)
τn−1,mN τn,mN
− q(−3n+6N+3)/2a−32 τn−2,mN τn+1,m
N = 0, (3.6)
τn,mN+1τ
n,m
N−1 + q(−2n+6N+1)/2a−22
(
1− qN−m+1a−10
)(
τn,mN
)2
− q4N−4m+4a−40 τn,m−1N τn,m+1
N = 0. (3.7)
Proof. Application of T1 on τ3 yields the following bilinear equations:
T1(τ3)τ4 − q2a−10 a1a
−1
2 τ1R
−1
23 (τ3)− q3/2a1a−12 τ3R
−1
23 (τ1) = 0, (3.8)
T1(τ3)τ2 + q3/2a0a1a
−2
2 (1− a1)(τ3)2 − a41τ1T0(τ3) = 0. (3.9)
Applying Rn−123 Tm−10 TN−11 on equations (3.8) and (3.9) and substituting condition (3.3) in them,
we obtain equations (3.5) and (3.7), respectively. Similarly, application of T1 on τ6 yields
T1(τ6)τ2 + qa−22
(
q1/2a−12 − 1
)
τ3τ6 − q3/2a−32 R−123 (τ3)τ5 = 0. (3.10)
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 9
Then, applying Rn−223 Tm−10 TN−11 on equation (3.10) and substituting condition (3.3) in it, we
obtain equation (3.6). Although we do not write the action of R23 and T0 on the variables τi
here, it will be described in the next step. �
Putting N = 0 in equations (3.5)–(3.7) and using the boundary condition (3.4), we get the
following conditions:
τn+1,m
0 τn,m+1
0
τn,m0 τn+1,m+1
0
= −q(2m−1)/2a0, (3.11)
τn,m0 τn+3,m
0
τn+1,m
0 τn+2,m
0
= 1− q(n+1)/2a2, (3.12)
τn,m0 τn,m+2
0(
τn,m+1
0
)2 = q(−2n+8m+1)/2a40a
−2
2
(
1− q−ma−10
)
. (3.13)
Step 2. Conditions of τn,m
1
In the second step, we shall get the conditions of τn,m1 from the consistency with the action of
〈R23, T0〉. By definitions (2.1) and (2.5) and Proposition 2, the action of T0 and R23 are given
by the follows:
T0(τ1) = τ3, T0(τ7) = τ2, R23(τ3) = τ6, R23(τ4) = τ7, R23(τ6) = τ5,
T0(τ2) =
q3/2a0a
−1
1 a−12 τ2T0(τ6) + qa20a
−1
2 T0(τ4)T0(τ5)
τ6
, (3.14)
T0(τ3) =
a20a1T0(τ4)T0(τ5) + q−1/2a0a1a2T0(τ6)τ2
R23(τ2)
, (3.15)
T0(τ4) =
a2τ2τ6 + qa−10 a−21 τ3R23(τ2)
τ5
, (3.16)
T0(τ5) =
qa−20 a−21 a−12 τ2τ6 + q3/2a−20 a−31 a−12 τ3R23(τ2)
τ4
, (3.17)
T0(τ6) =
qa−21 a−12 τ2τ6 + q3/2a−10 a−31 a−12 τ3R23(τ2)
τ7
, (3.18)
T−10 (τ1) =
q−2a20a1a
−1
2 T−10 (τ4)T
−1
0 (τ5) + q−1/2a0a1a
−1
2 T−10 (τ6)T
−1
0 (τ7)
R23(τ7)
, (3.19)
T−10 (τ4) =
qa−20 a−21 a−12 τ7T
−1
0 (τ6) + q1/2a−20 a−31 a−12 τ1R23(τ7)
τ5
, (3.20)
T−10 (τ5) =
a2τ7T
−1
0 (τ6) + a−10 a−21 τ1R23(τ7)
τ4
, (3.21)
T−10 (τ6) =
q−1a20a
−1
2 τ4τ5 + q−1/2a0a
−1
1 a−12 τ6τ7
τ2
, (3.22)
T−10 (τ7) =
q−1a−21 a−12 τ7T
−1
0 (τ6) + q−1/2a−10 a−31 a−12 τ1R23(τ7)
τ6
, (3.23)
R23(τ1) =
q−1a20a
−1
2 τ4τ5 + q−1/2a0a
−1
1 a−12 τ6τ7
τ2
, (3.24)
R23(τ2) =
q−1a20a1a
−1
2 τ4τ5 + q−1/2a0a1a
−1
2 τ6τ7
τ1
, (3.25)
R23(τ5) =
q3/2a−20 a22R23(τ1)R23(τ2) + qa−20 a−11 τ6R23(τ7)
τ4
, (3.26)
10 N. Nakazono
R23(τ7) =
q−1a20a1τ4τ5 + q−1/2a0a1a2τ6τ7
τ3
, (3.27)
R−123 (τ1) =
q−1/2a20a1a
−1
2 R−123 (τ4)τ6 + a0a1a
−1
2 τ3τ4
τ2
, (3.28)
R−123 (τ2) =
q−1/2a20a
−1
2 R−123 (τ4)τ6 + a0a
−1
1 a−12 τ3τ4
τ1
, (3.29)
R−123 (τ3) =
q−1a20a1R
−1
23 (τ4)τ6 + q−1a0a1a2τ3τ4
τ7
, (3.30)
R−123 (τ4) =
q1/2a−20 a22τ1τ2 + qa−20 a−11 τ3τ7
τ5
. (3.31)
Using notation (3.2) and condition (3.3), we can rewrite equations (3.14)–(3.31) as
a2τ
2,1
1 τ1,20 = q1/2a20τ
1,1
0 τ2,21 + qa20τ
0,1
0 τ3,21 , (3.32)
τ2,10 τ1,21 = qa0τ
0,1
0 τ3,21 + q1/2a2τ
2,2
1 τ1,10 , (3.33)
qτ3,11 τ0,10 = qa2τ
1,1
0 τ2,11 + a0τ
1,1
1 τ2,10 , (3.34)
q3/2a2τ
0,0
0 τ3,21 = q1/2τ1,10 τ2,11 + a0τ
1,1
1 τ2,10 , (3.35)
q3/2a2τ
1,0
0 τ2,21 = q1/2a20τ
1,1
0 τ2,11 + a20τ
1,1
1 τ2,10 , (3.36)
a2τ
2,0
0 τ1,−11 = q−1a0τ
0,−1
0 τ3,01 + q1/2τ2,01 τ1,−10 , (3.37)
q5/2a2τ
3,1
1 τ0,−10 = q3/2τ1,00 τ2,01 + a0τ
1,0
1 τ2,00 , (3.38)
q2τ0,00 τ3,01 = q2a2τ
1,0
0 τ2,01 + a0τ
1,0
1 τ2,00 , (3.39)
q3/2a2τ
1,1
0 τ2,01 = q1/2a20τ
0,0
0 τ3,11 + a20τ
2,1
1 τ1,00 , (3.40)
q7/2a2τ
2,1
1 τ1,−10 = q1/2a20τ
1,0
0 τ2,01 + a20τ
1,0
1 τ2,00 , (3.41)
q3/2a2τ
1,1
0 τ2,01 = q1/2a20τ
0,0
0 τ3,11 + a20τ
2,1
1 τ1,00 , (3.42)
a2τ
1,0
1 τ2,10 = a0τ
0,0
0 τ3,11 + q1/2τ2,11 τ1,00 , (3.43)
a20τ
0,0
0 τ4,11 = q3/2a22τ
2,0
1 τ2,10 + a0τ
2,1
1 τ2,00 , (3.44)
τ1,11 τ2,00 = a0τ
0,0
0 τ3,11 + q1/2a2τ
2,1
1 τ1,00 , (3.45)
a2τ
1,1
0 τ0,01 = q1/2a0τ
−1,0
0 τ2,11 + qτ1,11 τ0,00 , (3.46)
qa2τ
1,0
1 τ0,10 = q1/2a20τ
−1,0
0 τ2,11 + a20τ
1,1
1 τ0,00 , (3.47)
τ1,00 τ0,11 = a0τ
−1,0
0 τ2,11 + a2τ
1,1
1 τ0,00 , (3.48)
a20τ
3,1
1 τ−1,00 = q1/2a22τ
1,0
1 τ1,10 + a0τ
1,1
1 τ1,00 , (3.49)
respectively. By setting
τn,m1 =
(
q(n−1)/2a2; q
1/2
)
∞τ
n,m
0 Fn,m, (3.50)
and using conditions (3.11)–(3.13), equations (3.32)–(3.49) can be reduced to the following
contiguity relations:
Fn+2,m − q(n−1)/2a2Fn+1,m − qm−2a0
(
1− q(n−1)/2a2
)
Fn,m = 0, (3.51)
Fn+1,m+1 − qm−1a0Fn,m+1 − q(n−2)/2a2Fn,m = 0, (3.52)
q1/2Fn+2,m+1 − q1/2Fn+1,m+1 + q(n−1)/2a2
(
1− q(n−1)/2a2
)
Fn,m = 0, (3.53)(
1− qm−1a0
)
Fn+1,m+1 − q(n−1)/2a2Fn+1,m − qn/2−1a2
(
1− q(n−1)/2a2
)
Fn,m = 0, (3.54)
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 11
q3/2
(
1− qm−1a0
)
Fn+2,m+1 − q(n+2)/2a2Fn+1,m
− q(2m+n−1)/2a0a2
(
1− q(n−1)/2a2
)
Fn,m = 0, (3.55)
qFn+2,m+1 − qma0Fn,m+1 − q(2n−1)/2a22Fn,m = 0. (3.56)
The correspondence between equations (3.32)–(3.49) and equations (3.51)–(3.56) is established
as follows:
(3.33), (3.34), (3.39), (3.45), (3.48)⇒ (3.51),
(3.32), (3.40), (3.42), (3.47)⇒ (3.52),
(3.37), (3.43), (3.46)⇒ (3.53),
(3.36), (3.41)⇒ (3.54),
(3.35), (3.38)⇒ (3.55),
(3.44), (3.49)⇒ (3.56).
Step 3. Determination of τn,m
0 and τn,m
1
In this step, we determine τn,m0 and τn,m1 , i.e., we solve equations (3.11)–(3.13) and equa-
tions (3.51)–(3.56). It is easily verified that the function
τn,m0 =
(
qma0; q, q
)
∞
(
q(n+1)/2a2; q
1/2, q
)
∞Γ
(
q(2n+2m−3)/4a
1/2
0 a2; q
1/2, q1/2
)
×
Γ
(
q(n−m+1)/4a
−1/4
0 a
1/2
2 ; q1/4, q1/4
)
Γ
(
q(n−m)/4a
−1/4
0 a
1/2
2 ; q1/4, q1/4
)
Γ
(
−q3m−1a30; q3, q3
)
Γ
(
−q2na42; q2, q2
) , (3.57)
is the solution of equations (3.11)–(3.13). Therefore, the aim of this step is to solve the equa-
tions (3.51)–(3.56). Since equations (3.51)–(3.56) are overdetermined system, let us first consider
the essential conditions of Fn,m.
Lemma 2. Equations (3.51) and (3.52) are essential conditions for Fn,m.
Proof. Eliminating Fn,m+1 from equations (3.51)m→m+1 and (3.52), we obtain equation (3.53).
In a similar manner, equations (3.54)–(3.56) can be proven by the following procedures: elimi-
nating Fn+2,m+1 from equations (3.52)n→n+1 and (3.53), we obtain equation (3.54); elimina-
ting Fn+1,m+1 from equations (3.52)n→n+1 and (3.53), we obtain equation (3.55); eliminating
Fn+1,m+1 from equations (3.52) and (3.53), we obtain equation (3.56). These calculations mean
that if Fn,m satisfies conditions (3.51) and (3.52), then it also satisfies conditions (3.53)–(3.56).
Therefore we have completed the proof. �
Next, we solve equations (3.51) and (3.52).
Lemma 3. The general solution of contiguity relations (3.51) and (3.52) is given by
Fn,m = An,m
Θ
(
qn/2a2; q
1/2
)
Θ
(
q(2m−1)/2a0; q
)(
q(m−1)/2a
1/2
0 ; q1/2
)
∞
Θ
(
q(n+m−2)/2a
1/2
0 a2; q1/2
)
× 2ϕ1
(
0, q(−m+2)/2a
−1/2
0
−q1/2
; q1/2, q(n−1)/2a2
)
+Bn,m
Θ
(
qn/2a2; q
1/2
)
Θ
(
q(2m−1)/2a0; q
)(
−q(m−1)/2a1/20 ; q1/2
)
∞
Θ
(
−q(n+m−2)/2a1/20 a2; q1/2
)
× 2ϕ1
(
0,−q(−m+2)/2a
−1/2
0
−q1/2
; q1/2, q(n−1)/2a2
)
,
12 N. Nakazono
where An,m and Bn,m are periodic functions of period one for n and m, i.e.,
An,m = An+1,m = An,m+1, Bn,m = Bn+1,m = Bn,m+1.
Proof. For convenience, we put
t = qn/2a2, α = qma0, Fn,m = F (t, α).
Then, equations (3.51) and (3.52) can be rewritten as
F (qt, α)− q−1/2tF
(
q1/2t, α
)
− q−2α
(
1− q−1/2t
)
F (t, α) = 0, (3.58)
F
(
q1/2t, qα
)
− q−1αF (t, qα)− q−1tF (t, α) = 0, (3.59)
respectively. Substituting
F (t, α) = D(t, α)
∞∑
k=0
Ck(α)tk,
in equation (3.58), we obtain
D(qt, α) = q−2αD(t, α), (3.60)
Ck(α) =
(
q2D
(
q1/2t, α
)
D(t, α)−1α−1; q1/2
)
k
qk/2
(
−q1/2, q1/2; q1/2
)
k
C0(α).
Therefore, we obtain the solution of equation (3.58):
F (t, α) = D1(t, α) 2ϕ1
(
0, qα−1/2
−q1/2 ; q1/2, q−1/2t
)
+D2(t, α) 2ϕ1
(
0,−qα−1/2
−q1/2 ; q1/2, q−1/2t
)
, (3.61)
where D1(t, α) and D2(t, α) are the solutions of (3.60) which satisfy
D1
(
q1/2t, α
)
= q−1α1/2D1(t, α), (3.62)
D2
(
q1/2t, α
)
= −q−1α1/2D2(t, α), (3.63)
respectively. Substituting (3.61) in equation (3.59), we can obtain the following equations:
q−1/2α1/2
2ϕ1
(
0, q1/2α−1/2
−q1/2 ; q1/2, t
)
− q−1α 2ϕ1
(
0, q1/2α−1/2
−q1/2 ; q1/2, q−1/2t
)
− q−1t D1(t, α)
D1(t, qα)
2ϕ1
(
0, qα−1/2
−q1/2 ; q1/2, q−1/2t
)
= 0, (3.64)
q−1/2α1/2
2ϕ1
(
0,−q1/2α−1/2
−q1/2 ; q1/2, t
)
+ q−1α 2ϕ1
(
0,−q1/2α−1/2
−q1/2 ; q1/2, q−1/2t
)
+ q−1t
D2(t, α)
D2(t, qα)
2ϕ1
(
0,−qα−1/2
−q1/2 ; q1/2, q−1/2t
)
= 0. (3.65)
By the definition of basic hypergeometric series 2ϕ1, it is easily verified that
2ϕ1
(
0, a
c
; q1/2, z
)
− a−1 2ϕ1
(
0, a
c
; q1/2, q−1/2z
)
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 13
−
(
1− a−1
)
2ϕ1
(
0, q1/2a
c
; q1/2, q−1/2z
)
= 0. (3.66)
Therefore, we obtain the following conditions from equations (3.64) and (3.65) by using equa-
tion (3.66):
D1(t, qα) = − t
α(1− q1/2α−1/2)
D1(t, α), (3.67)
D2(t, qα) = − t
α(1 + q1/2α−1/2)
D2(t, α). (3.68)
By setting
D1(t, α) =
Θ
(
t; q1/2
)
Θ
(
q−1/2α; q
)(
q−1/2α1/2; q1/2
)
∞
Θ
(
q−1α1/2t; q1/2
) A(t, α),
D2(t, α) =
Θ
(
t; q1/2
)
Θ
(
q−1/2α; q
)(
−q−1/2α1/2; q1/2
)
∞
Θ
(
−q−1α1/2t; q1/2
) B(t, α),
equations (3.62), (3.63), (3.67) and (3.68) can be rewritten as
A
(
q1/2t, α
)
= A(t, α), B
(
q1/2t, α
)
= B(t, α),
A(t, qα) = A(t, α), B(t, qα) = B(t, α),
respectively. This completes the proof. �
It was shown that hypergeometric solutions of a symmetric discrete Painlevé equation, which
can be obtained by projective reduction, have two expressions and there are the following
differences between the two expressions (see [17, Section 2.3]):
(i) the bases of hypergeometric series appearing in the solutions are different;
(ii) the periodicities of periodic functions appearing in the solutions are different.
The differences between these two expressions can be explained by the factorization of the linear
difference operators associated with the three-term relation of the hypergeometric functions
(see [17, Section 3.2]). Namely, we can see these differences by comparing Lemmas 3 and 6. To
get another expression, we first reselect essential conditions of Fn,m.
Lemma 4. Equations (3.52) and
qm−1a0
(
1− qma0
)
Fn,m+2 − q(n−5)/2a2
((
1 + q1/2
)
qma0 − qn/2a2
)
Fn,m+1
− q(2n−5)/2a22Fn,m = 0, (3.69)
are essential conditions of Fn,m.
Proof. Eliminating Fn,m from equations (3.52) and (3.69), we obtain
qm−2a0
(
1− qm−1a0
)
Fn,m+1 − q(n−3)/2a2Fn+1,m
− q(n−5)/2a2
(
q(2m−1)/2a0 − qn/2a2
)
Fn,m = 0. (3.70)
Similarly, eliminating Fn,m+1 from equations (3.52) and (3.70), we obtain(
1− qm−1a0
)
Fn+1,m+1 − q(n−1)/2a2Fn+1,m − q(n−2)/2a2
(
1− q(n−1)/2a2
)
Fn,m = 0. (3.71)
Finally, eliminating Fn+1,m+1 from equations (3.70)n→n+1 and (3.71), we obtain equation (3.51).
This result together with Lemma 2 complete the proof. �
14 N. Nakazono
By setting
Fn,m =
Θ
(
qma0; q
)
Θ
(
qn/2a2; q
1/2
)
Θ
(
−qn/2a2; q1/2
) Gn−3,m−1, (3.72)
equations (3.52) and (3.69) can be rewritten as
q−m+1a−10 Gn−2,m +Gn−3,m − qn/2a2Gn−3,m−1 = 0, (3.73)
Gn,m−2 +
(
q−m+1a−10 −
(
1 + q1/2
)
q(−n−3)/2a−12
)
Gn,m−1
− q(−2n−5)/2a−22
(
q−m+1a−10 − 1
)
Gn,m = 0, (3.74)
respectively. Before solving equations (3.73) and (3.74), we prepare the following lemma:
Lemma 5. The following recurrence relations hold:
2ϕ1
(
a, b
c
; q, z
)
− 2ϕ1
(
a, b
c
; q, qz
)
=
(1− a)(1− b)z
1− c 2ϕ1
(
qa, qb
qc
; q, z
)
, (3.75)
(
q−1c− 1
)
2ϕ1
(
a, b
q−1c
; q, z
)
+ 2ϕ1
(
a, b
c
; q, z
)
− q−1c 2ϕ1
(
a, b
c
; q, qz
)
= 0. (3.76)
Proof. Substituting
2ϕ1
(
a, b
c
; q, z
)
= 1 +
∞∑
n=0
(qa, qb; q)n
(qc, q; q)n
(1− a)(1− b)
(1− c)
(
1− qn+1
)zn+1,
in the left-hand side of equation (3.75), we obtain the right-hand side. Equation (3.76) can be
easily verified as the following:
2ϕ1
(
a, b
q−1c
; q, z
)
=
∞∑
n=0
(a, b; q)n
(c, q; q)n
1− qn−1c
1− q−1c
zn
=
1
1− q−1c 2ϕ1
(
a, b
c
; q, z
)
− q−1c
1− q−1c 2ϕ1
(
a, b
c
; q, qz
)
.
Therefore we have completed the proof. �
Using Lemma 5, we obtain the following lemma:
Lemma 6. The general solution of equations (3.73) and (3.74) is given by
Gn,m = Λn,m
Θ
(
q(n−2m+2)/2a−10 a2; q
)
Θ
(
q−ma−10 ; q
)
Θ
(
qn/2a2; q
)(
q(n+3)/2a2; q
)
∞
(
q−1/2; q
)
∞
× 2ϕ1
(
0, q(n+3)/2a2
q3/2
; q, q−m+1a−10
)
+ Λn+1,m
q1/2Θ
(
q(n−2m+3)/2a−10 a2; q
)
Θ
(
q−ma−10 ; q
)
Θ
(
q(n+1)/2a2; q
)(
q(n+2)/2a2; q
)
∞
(
q1/2; q
)
∞
× 2ϕ1
(
0, q(n+2)/2a2
q1/2
; q, q−m+1a−10
)
,
where Λn,m is a periodic function of period two for n and period one for m, i.e.,
Λn+2,m = Λn,m+1 = Λn,m.
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 15
Proof. For convenience, we put
t = q−m+1a−10 , α = qn/2a2, Gn,m = G(t, α).
Then, equations (3.73) and (3.74) can be rewritten as
tG
(
t, q−1α
)
+G
(
t, q−3/2α
)
− αG
(
qt, q−3/2α
)
= 0, (3.77)
G
(
q2t, α
)
+
(
t−
(
1 + q1/2
)
q−3/2α−1
)
G(qt, α)− q−5/2α−2(t− 1)G(t, α) = 0, (3.78)
respectively. Substituting
G(t, α) = D(t, α)
∞∑
k=0
Ck(α)tk,
in equation (3.78), we obtain
G(t, α) = D1(t, α) 2ϕ1
(
0, q3/2α
q3/2
; q, t
)
+D2(t, α) 2ϕ1
(
0, qα
q1/2
; q, t
)
, (3.79)
where D1(t, α) and D2(t, α) satisfy
D1(qt, α) = q−1α−1D1(t, α), (3.80)
D2(qt, α) = q−3/2α−1D2(t, α), (3.81)
respectively. Substituting (3.79) in equation (3.77), we can obtain the following equations:
2ϕ1
(
0, q−1/2α
q1/2
; q, t
)
− 2ϕ1
(
0, q−1/2α
q1/2
; q, qt
)
= −t
D1
(
t, q−1α
)
D2
(
t, q−3/2α
) 2ϕ1
(
0, q1/2α
q3/2
; q, t
)
,(3.82)
t
D2
(
t, q−1α
)
D1
(
t, q−3/2α
) 2ϕ1
(
0, α
q1/2
; q, t
)
+ 2ϕ1
(
0, α
q3/2
; q, t
)
− q1/22ϕ1
(
0, α
q3/2
; q, qt
)
= 0. (3.83)
Therefore, we obtain
D1
(
t, q1/2α
)
= − 1− qα
1− q1/2
D2(t, α), (3.84)
D2
(
t, q1/2α
)
= −1− q1/2
t
D1(t, α), (3.85)
from equations (3.82) and (3.83) by using equations (3.75) and (3.76), respectively. By setting
D1(t, α) =
Θ(αt; q)(
q−1/2; q
)
∞
(
q3/2α; q
)
∞Θ
(
q−1t; q
)
Θ(α; q)
Λ(t, α),
D2(t, α) =
q1/2Θ(q1/2αt; q)(
q1/2; q
)
∞(qα; q)∞Θ
(
q−1t; q
)
Θ
(
q1/2α; q
)Λ
(
t, q1/2α
)
,
equations (3.80), (3.81), (3.84) and (3.85) can be reduced to
Λ(t, qα) = Λ(qt, α) = Λ(t, α).
This completes the proof. �
16 N. Nakazono
Step 4. Constructing the closed-form expressions of τn,m
N (N ≥ 2)
In this final step, constructing the closed-form expressions of τn,mN (N ≥ 2), we obtain the
hypergeometric τ -functions for the q-PIV.
Let
τn,mN = (−1)N(N−1)/2q3N(N−1)(N−n+1)/4a
−3N(N−1)/2
2
(
N∏
k=1
(
q(n−2k+1)/2a2; q
1/2
)
∞
)
×
(
qma0; q, q
)
∞
(
q(n+1)/2a2; q
1/2, q
)
∞Γ
(
q(2n+2m−3)/4a
1/2
0 a2; q
1/2, q1/2
)
×
Γ
(
q(n−m+1)/4a
−1/4
0 a
1/2
2 ; q1/4, q1/4
)
Γ
(
q(n−m)/4a
−1/4
0 a
1/2
2 ; q1/4, q1/4
)
Γ
(
−q3m−1a30; q3, q3
)
Γ
(
−q2na42; q2, q2
) φn,mN .
From (3.4), (3.50) and (3.57), we get
φn,mN = 0, N < 0, φn,m0 = 1, φn,m1 = Fn,m.
Moreover, it is easily verified that equation (3.6) can be rewritten as
φn,mN+1φ
n−1,m
N−1 − φ
n−1,m
N φn,mN + φn−2,mN φn+1,m
N = 0. (3.86)
In general, equation (3.86) admits a solution expressed in terms of Jacobi–Trudi type determi-
nant
φn,mN = det(cn−2i+j+1,m)i,j=1,...,N , N > 0,
under the boundary conditions
φn,mN = 0, N < 0, φn,m0 = 1, φn,m1 = cn,m,
where cn,m is an arbitrary function. Therefore, we obtain the following lemma:
Lemma 7. Under the assumptions
a0a1 = q, τn,mN = 0, N < 0,
the hypergeometric τ -functions for the q-PIV are given as
τn,mN = (−1)N(N−1)/2q3N(N−1)(N−n+1)/4a
−3N(N−1)/2
2
(
N∏
k=1
(
q(n−2k+1)/2a2; q
1/2
)
∞
)
×
(
qma0; q, q
)
∞
(
q(n+1)/2a2; q
1/2, q
)
∞Γ
(
q(2n+2m−3)/4a
1/2
0 a2; q
1/2, q1/2
)
×
Γ
(
q(n−m+1)/4a
−1/4
0 a
1/2
2 ; q1/4, q1/4
)
Γ
(
q(n−m)/4a
−1/4
0 a
1/2
2 ; q1/4, q1/4
)
Γ
(
−q3m−1a30; q3, q3
)
Γ
(
−q2na42; q2, q2
) φn,mN ,
where
φn,mN =
∣∣∣∣∣∣∣∣∣
Fn,m Fn+1,m . . . Fn+N−1,m
Fn−2,m Fn−1,m . . . Fn+N−3,m
...
...
. . .
...
Fn−2N+2,m Fn−2N+3,m . . . Fn−N+1,m
∣∣∣∣∣∣∣∣∣ , φn,m0 = 1, φn,m−N = 0, N > 0.
Here, Fn,m is given in Lemma 3.
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 17
We also show another expression of the hypergeometric τ -functions for the q-PIV. From
relation (3.72), φn,mN can be rewritten as
φn,mN = Θ
(
qma0; q
)N (N−1∏
k=0
Θ
(
q(n+k)/2a2; q
1/2
)
Θ
(
−q(n+k)/2a2; q1/2
))ψn−3,m−1N ,
where
ψn,mN =
∣∣∣∣∣∣∣∣∣
Gn,m Gn+1,m . . . Gn+N−1,m
Gn−2,m Gn−1,m . . . Gn+N−3,m
...
...
. . .
...
Gn−2N+2,m Gn−2N+3,m . . . Gn−N+1,m
∣∣∣∣∣∣∣∣∣ , ψn,m0 = 1, ψn,m−N = 0, N > 0.
This gives the following lemma:
Lemma 8. Under the assumptions
a0a1 = q, τn,mN = 0, N < 0,
the hypergeometric τ -functions for the q-PIV are given as
τn,mN = (−1)N(N−1)/2q3N(N−1)(N−n+1)/4a
−3N(N−1)/2
2
(
N∏
k=1
(
q(n−2k+1)/2a2; q
1/2
)
∞
)
×
(
qma0; q, q
)
∞
(
q(n+1)/2a2; q
1/2, q
)
∞Γ
(
q(2n+2m−3)/4a
1/2
0 a2; q
1/2, q1/2
)
×
Γ
(
q(n−m+1)/4a
−1/4
0 a
1/2
2 ; q1/4, q1/4
)
Γ
(
q(n−m)/4a
−1/4
0 a
1/2
2 ; q1/4, q1/4
)
Γ
(
−q3m−1a30; q3, q3
)
Γ
(
−q2na42; q2, q2
)
×Θ
(
qma0; q
)N (N−1∏
k=0
Θ
(
q(n+k)/2a2; q
1/2
)
Θ
(
−q(n+k)/2a2; q1/2
))ψn−3,m−1N ,
where
ψn,mN =
∣∣∣∣∣∣∣∣∣
Gn,m Gn+1,m . . . Gn+N−1,m
Gn−2,m Gn−1,m . . . Gn+N−3,m
...
...
. . .
...
Gn−2N+2,m Gn−2N+3,m . . . Gn−N+1,m
∣∣∣∣∣∣∣∣∣ , ψn,m0 = 1, ψn,m−N = 0, N > 0.
Here, Gn,m is given in Lemma 6.
3.3 Hypergeometric solutions of the q-PIV
In this section, we show the hypergeometric solutions of q-PIV (1.1).
From relations (2.8) and (3.1), the variable for q-PIV (1.1) is expressed by the τ -functions as
the following:
Xn(m,N) = q(m+N−1)/2a0a
1/2
1
τn,mN τn+3,m+1
N+1
τn+2,m+1
N+1 τn+1,m
N
.
Therefore, from Lemmas 7 and 8, we obtain the following theorems:
18 N. Nakazono
Theorem 1. The hypergeometric solutions of q-PIV (1.1) with
a0a1 = q, N ≥ 0, (3.87)
is given by
Xn(m,N) = −q(−2N−m+1)/2a
−1/2
0
φn,mN φn+3,m+1
N+1
φn+2,m+1
N+1 φn+1,m
N
,
where
φn,mN =
∣∣∣∣∣∣∣∣∣
Fn,m Fn+1,m . . . Fn+N−1,m
Fn−2,m Fn−1,m . . . Fn+N−3,m
...
...
. . .
...
Fn−2N+2,m Fn−2N+3,m . . . Fn−N+1,m
∣∣∣∣∣∣∣∣∣ , φn,m0 = 1.
Here, Fn,m is given in Lemma 3.
Theorem 2. The hypergeometric solutions of q-PIV (1.1) with the condition (3.87) is given by
Xn(m,N) = q(−2N−m+1)/2a
−1/2
0
ψn−3,m−1N ψn,mN+1
ψn−1,mN+1 ψn−2,m−1N
,
where
ψn,mN =
∣∣∣∣∣∣∣∣∣
Gn,m Gn+1,m . . . Gn+N−1,m
Gn−2,m Gn−1,m . . . Gn+N−3,m
...
...
. . .
...
Gn−2N+2,m Gn−2N+3,m . . . Gn−N+1,m
∣∣∣∣∣∣∣∣∣ , ψn,m0 = 1.
Here, Gn,m is given in Lemma 6.
4 Hypergeometric solutions of the q-PIV (II)
In this section, we show that q-PIV (1.1) also has the hypergeometric solutions expressed by
bilateral basic hypergeometric series.
First, we recall the definitions of orthogonal polynomials.
Definition 1. A polynomial sequence (Pn(t))∞n=0 which satisfies the following conditions is
called an orthogonal polynomial sequence over the field K, and each term Pn(t) is called an
orthogonal polynomial over the field K.
(i) deg(Pn(t)) = n;
(ii) there exists a linear functional L : K(t)→ K which holds the orthogonal condition:
L[tkPn(t)] = hnδn,k, n ≥ k,
where δn,k is Kronecker’s symbol. Here, hn and µn = L[tn] are called a normalization
factor and a moment, respectively.
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 19
Definition 2. An orthogonal polynomial sequence whose leading coefficient is 1 is called a monic
orthogonal polynomial sequence (MOPS). Let (Pn(t))∞n=0 be a MOPS. Then, polynomial Pn(t)
and its normalization factor hn are expressed by the moment µn as the following:
Pn =
1
Φn
∣∣∣∣∣∣∣∣∣∣∣
µ0 µ1 . . . µn−1 µn
µ1 µ2 . . . µn µn+1
...
...
. . .
...
...
µn−1 µn . . . µ2n−2 µ2n−1
1 t . . . tn−1 tn
∣∣∣∣∣∣∣∣∣∣∣
, P0 = 1, hn =
Φn+1
Φn
, h0 = µ0, (4.1)
where Φn is the Hankel determinant given by
Φn =
∣∣∣∣∣∣∣∣∣
µ0 µ1 . . . µn−1
µ1 µ2 . . . µn
...
...
. . .
...
µn−1 µn . . . µ2n−2
∣∣∣∣∣∣∣∣∣ . (4.2)
We assume that (Pn)∞n=0 = (Pn(t))∞n=0 and (P̂n)∞n=0 = (P̂n(t))∞n=0 are MOPSs which satisfy
the following orthogonal conditions:
L
[
tkPn(t)
]
= hnδn,k, n ≥ k, L̂
[
tkP̂n(t)
]
= ĥnδn,k, n ≥ k,
respectively. In addition, we put the case that Pn and P̂n are related by the Christoffel trans-
formation (or Geronimus transformation), that is, the linear functionals satisfy the following
relation for an arbitrary function f(t):
L[f(t)] = L̂
[
f(t)
t− c0
+ δ(t− c0)
]
,
where δ(x) is the Dirac delta function and c0 ∈ C is an additional parameter. For these MOPSs,
the following relations hold [4, 40, 41]:
(t− c0)P̂n = Pn+1 +
ĥn
hn
Pn, (4.3)
Pn = P̂n +
hn
ĥn−1
P̂n−1. (4.4)
Eliminating Pn from equation (4.3) by using equation (4.4), we obtain the following three-term
recurrence relation:
tP̂n = P̂n+1 +
(
hn+1
ĥn
+
ĥn
hn
+ c0
)
P̂n +
ĥn
ĥn−1
P̂n−1. (4.5)
Let
P̂n(t) =
h̃n(c1t; p)
cn1
, c1 > 0,
where h̃n(t; q) is the discrete q-Hermite II polynomial:
h̃n(t; q) = tn2ϕ1
(
q−n, q−n+1
0
; q2,−q
2
t2
)
.
20 N. Nakazono
Then, the linear functional, the normalization factor and the three-term recurrence relation
for P̂n are given by
L̂[f(t)] =
∫ ∞
−∞
f(t)(
−c21t2; p2
)
∞
dpt,
ĥn =
2
pn2c2n1
(p; p)n
(
p2; p2
)
∞Θ
(
−pc21; p2
)(
p3; p2
)
∞Θ
(
−c21; p2
) , (4.6)
tP̂n = P̂n+1 + p−2n+1
(
1− pn
)
c−21 P̂n−1, (4.7)
respectively. We note that these properties of q-Hermite II polynomials are given in [22]. We
here impose the condition c0 6= pa for all a ∈ Z since the linear functional for Pn is given by
L[f(t)] =
∫ ∞
−∞
f(t)
(t− c0)
(
−c21t2; p2
)
∞
dpt.
In addition, the moment µn can be obtained by
µn = −1− p
c20
∞∑
k=−∞
(
1− (−1)n
)
pk +
(
1 + (−1)n
)
c0(
1− p2kc−20
)(
−c21p2k; p2
)
∞
pk(n+1)
=
2(1− p)(
1− c20
)(
−c21; p2
)
∞
∞∑
k=−∞
(
−c21, c
−2
0 ; p2
)
k(
p2c−20 ; p2
)
k
(
1− (−1)n
2
pk +
1 + (−1)n
2
c0
)
pk(n+1)
=
2(1− p)(
1− c20
)(
−c21; p2
)
∞
2ψ2
(
−c21, c
−2
0
0, p2c−20
; p2, p2k+1
)
, n = 2k − 1,
2c0(1− p)(
1− c20
)(
−c21; p2
)
∞
2ψ2
(
−c21, c
−2
0
0, p2c−20
; p2, p2k+1
)
, n = 2k.
(4.8)
Comparing the coefficients of equations (4.5) and (4.7), we obtain the following equations:
hn+1
ĥn
+
ĥn
hn
+ c0 = 0, (4.9)
ĥn
ĥn−1
= p−2n+1
(
1− pn
)
c−21 . (4.10)
From equations (4.9) and (4.10), we obtain
ĥn
hn+1
= − hn
p−2n+1(1− pn)c−21 ĥn−1 + c0hn
. (4.11)
By setting
Xn = i
1− pn+1
pnc1
ĥn
hn+1
, (4.12)
equation (4.11) can be rewritten as the following discrete Riccati equation:
Xn =
1− pn+1
Xn−1 + ipnc1c0
. (4.13)
Since in the case of
a
1/2
0 a
1/2
1 = q1/2, a2 = 1, N = −1,
Hypergeometric Solutions of the A
(1)
4 -Surface q-Painlevé IV Equation 21
q-PIV (1.1) admits the reduction to
Xn =
1− q(n+1)/2
Xn−1 + q(n−m+1)/2a
−1/2
0
,
which is equivalent to equation (4.13) with the following correspondence:
qm/2a
1/2
0 = −iq1/2c−10 c−11 , q1/2 = p,
(4.1), (4.2), (4.6), (4.8) and (4.12) give the hypergeometric solutions of q-PIV (1.1). Therefore,
we finally obtain the following theorem:
Theorem 3. In the case of
a
1/2
0 a
1/2
1 = q1/2, a2 = 1, N = −1, n ≥ 0,
q-PIV (1.1) with
qm/2a
1/2
0 = −iq1/2c−10 c−11 ,
admits the following hypergeometric solution:
Xn =
2i
(
1− q(n+1)/2
)(
q1/2; q1/2
)
n
(q; q)∞Θ
(
−q1/2c21; q
)
qn(n+1)/2c2n+1
1
(
q3/2; q
)
∞Θ
(
−c21; q
) Φn+1
Φn+2
,
where
Φn =
∣∣∣∣∣∣∣∣∣
µ0 µ1 . . . µn−1
µ1 µ2 . . . µn
...
...
. . .
...
µn−1 µn . . . µ2n−2
∣∣∣∣∣∣∣∣∣ ,
µn =
2
(
1− q1/2
)(
1− c20
)(
−c21; q
)
∞
2ψ2
(
−c21, c
−2
0
0, qc−20
; q, q(2k+1)/2
)
, n = 2k − 1,
2c0
(
1− q1/2
)(
1− c20
)(
−c21; q
)
∞
2ψ2
(
−c21, c
−2
0
0, qc−20
; q, q(2k+1)/2
)
, n = 2k.
Here, c0 6= qa/2 for all a ∈ Z.
5 Concluding remarks
In this paper, we have constructed the hypergeometric solutions of q-PIV (1.1) via the construc-
tion of the hypergeometric τ -functions and the theory of orthogonal polynomials. We showed
that the hypergeometric solutions of the q-PIV can be expressed by the three expressions. We
note that the hypergeometric solutions of Painlevé systems expressed by the determinants whose
sizes do not depend on the independent variable are called the lattice type solutions, while
those expressed by the determinants whose sizes depend on the independent variable are called
molecule type solutions. Thus, the hypergeometric solutions given in Theorems 1 and 2 are
lattice type solutions whereas those given in Theorem 3 are molecule type solutions.
Before closing, we mention the bilateral type hypergeometric solutions here. It is well known
that the coalescence cascade of hypergeometric functions, from the Gauss’s hypergeometric func-
tion to the Airy function, corresponds to the diagram of degeneration of the Painlevé equations,
22 N. Nakazono
from the Painlevé VI equation to the Painlevé II equation, in the sense of the hypergeometric
solutions [5]:
PVI → PV → PIII
Gauss Kummer Bessel
↓ ↓
PIV → PII
Weber Airy
Similarly, the relations between basic hypergeometric series and q-Painlevé equations are also
investigated [14, 16]. However, the hypergeometric solutions described by bilateral basic hyper-
geometric series have not been considered. It might be an interesting future problem to make
a list of the bilateral basic hypergeometric series that appear as the solutions of the q-Painlevé
equations.
Acknowledgments
The author would like to thank Professors K. Kajiwara, S. Kakei, H. Miki, M. Noumi, and
S. Tsujimoto for the useful comments. He also appreciates the valuable comments from the
referees which have improved the quality of this paper. This work has been supported by JSPS
Grant-in-Aid for Scientific Research No. 22·4366 and the Australian Research Council grant
DP130100967.
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1 Introduction
2 Affine Weyl group of type A4(1)
2.1 Birational representation of the affine Weyl group of type A4(1)
2.2 Derivations of the q-Painlevé equations
3 Hypergeometric solutions of the q-PIV (I)
3.1 -functions
3.2 Hypergeometric -functions for the q-PIV
3.3 Hypergeometric solutions of the q-PIV
4 Hypergeometric solutions of the q-PIV (II)
5 Concluding remarks
References
|
| id | nasplib_isofts_kiev_ua-123456789-146608 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:23:54Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Nakazono, N. 2019-02-10T09:57:23Z 2019-02-10T09:57:23Z 2014 Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation / N. Nakazono // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 41 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D05; 33D15; 33D45; 33E17; 39A13 DOI:10.3842/SIGMA.2014.090 https://nasplib.isofts.kiev.ua/handle/123456789/146608 We consider a q-Painlevé IV equation which is the A₄⁽¹⁾-surface type in the Sakai's classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric functions. Two of them are expressed by ₂φ₁ basic hypergeometric series and the other is given by ₂ψ₂ bilateral basic hypergeometric series. The author would like to thank Professors K. Kajiwara, S. Kakei, H. Miki, M. Noumi, and S. Tsujimoto for the useful comments. He also appreciates the valuable comments from the referees which have improved the quality of this paper. This work has been supported by JSPS Grant-in-Aid for Scientific Research No. 22·4366 and the Australian Research Council grant DP130100967. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation Article published earlier |
| spellingShingle | Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation Nakazono, N. |
| title | Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation |
| title_full | Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation |
| title_fullStr | Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation |
| title_full_unstemmed | Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation |
| title_short | Hypergeometric Solutions of the A₄⁽¹⁾-Surface q-Painlevé IV Equation |
| title_sort | hypergeometric solutions of the a₄⁽¹⁾-surface q-painlevé iv equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146608 |
| work_keys_str_mv | AT nakazonon hypergeometricsolutionsofthea41surfaceqpainleveivequation |