Hypergeometric
τ
Functions
of the
q
-Painlev´
e Systems of Type (
A
2+
A
1)
(1)Nobutaka NAKAZONO
Graduate School of Mathematics, Kyushu University, 744 Motooka, Fukuoka, 819-0395, Japan
E-mail: n-nakazono@math.kyushu-u.ac.jp URL: http://researchmap.jp/nakazono/
Received August 17, 2010, in final form October 08, 2010; Published online October 14, 2010 doi:10.3842/SIGMA.2010.084
Abstract. We consider aq-Painlev´e III equation and aq-Painlev´e II equation arising from a birational representation of the affine Weyl group of type (A2+A1)(1). We study their hypergeometric solutions on the level ofτ functions.
Key words: q-Painlev´e system; hypergeometric function; affine Weyl group;τ function
2010 Mathematics Subject Classification: 33D05; 33D15; 33E17; 39A13
1
Introduction
We consider a q-analog of the Painlev´e III equation (q-PIII) [8,12,13,32]
gn+1=
q2N+1c2 fngn
1 +a0qnfn
a0qn+fn
, fn+1=
q2N+1c2 fngn+1
1 +a2a0qn−mgn+1
a2a0qn−m+gn+1
, (1.1)
and that of the Painlev´e II equation (q-PII) [12,30,20]
Xk+1=
q2N+1c2 XkXk−1
1 +a0qk/2Xk
a0qk/2+Xk
, (1.2)
for the unknown functions fn=fn(m, N), gn =gn(m, N), and Xk =Xk(N) and the
indepen-dent variables n, k ∈Z. Here m, N ∈Z and a0, a2, c, q ∈ C× are parameters. These equations arise from a birational representation of the (extended) affine Weyl group of type (A2+A1)(1).
Note that substituting
m= 0, a2 =q1/2,
and putting
fk(0, N) =X2k(N), gk(0, N) =X2k−1(N),
in (1.1) yield (1.2). This procedure is called a symmetrization of (1.1), which comes from the terminology used for Quispel–Roberts–Thompson (QRT) mappings [28,29].
It is well known that theτfunctions play a crucial role in the theory of integrable systems [19], and it is also possible to introduce them in the theory of Painlev´e systems [5,6,7, 13, 21, 22, 24, 25, 26, 27]. A representation of the affine Weyl groups can be lifted on the level of the τ functions [10, 11, 33], which gives rise to various bilinear equations of Hirota type satisfied theτ functions.
Usually, they are derived by reducing the bilinear equations to the Pl¨ucker relations by using the contiguity relations satisfied by the entries of determinants [2,3,4,8,9,13,14,15,16,20,23,31]. This method is elementary, but it encounters technical difficulties for Painlev´e 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 theτfunctions live (hypergeometric τ functions), so that they are consistent with the action of the affine Weyl groups. 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. Masuda has carried out the calculations for the q-Painlev´e systems with E7(1) and E8(1) symmetries [17, 18] and presented explicit determinant formulae for their hypergeometric solutions.
The purpose of this paper is to apply the above method to theq-Painlev´e systems with the affine Weyl group symmetry of type (A2+A1)(1) and present the explicit formulae of the
hyper-geometric τ functions. The hypergeometric τ functions provide not only determinant formulae but also important information originating from the geometry of lattice of theτ functions. The result has been already announced in [12] and played an essential role in clarifying the mechanism of reduction from hypergeometric solutions of (1.1) to those of (1.2).
This paper is organized as follows: in Section2, we first review hypergeometric solutions of q-PIII and then those ofq-PII. We next introduce a representation of the affine Weyl group of
type (A2+A1)(1). In Section3, we construct the hypergeometricτ functions ofq-PIII and those
of q-PII. We find that the symmetry of the hypergeometric τ functions of q-PIII are connected
with Heine’s transform of the basic hypergeometric series2ϕ1.
We use the following conventions ofq-analysis throughout this paper [1]. q-Shifted factorials:
(a;q)k= k
Y
i=1
1−aqi−1.
Basic hypergeometric series:
sϕr
a1, . . . , as
b1, . . . , br;q, z
=
∞
X
n=0
(a1, . . . , as;q)n
(b1, . . . , br;q)n(q;q)n
(−1)nqn(n−1)/21+r−szn,
where
(a1, . . . , as;q)n= s
Y
i=1
(ai;q)n.
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= kY−1
i,j=0
1−piqja.
It holds that
2
q
-P
IIIand
q
-P
II2.1 Hypergeometric solutions of q-PIII and q-PII
First, we review the hypergeometric solutions ofq-PIII andq-PII. The hypergeometric solutions
of q-PIII have been constructed as follows:
Proposition 2.1 ([8]). The hypergeometric solutions ofq-PIII,(1.1), withc= 1 are given by
and Fn,m is an arbitrary solution of the systems
Fn+1,m−Fn,m=−a02q2nFn,m−1, (2.1)
where An,m andBn,m are periodic functions of period one with respect to nand m, i.e.,
An,m =An+1,m=An,m+1, Bn,m =Bn+1,m=Bn,m+1.
The explicit form of the hypergeometric solutions ofq-PIIare given as follows:
Proposition 2.2 ([20]). The hypergeometric solutions of q-PII, (1.2), with c= 1 are given by
and Gk is an arbitrary solution of the system
The general solution of (2.6) is given by
Gk=AkΘ(ia0q(2k+1)/4;q1/2)1ϕ1
0
−q1/2;q
1/2,−ia
0q(3+2k)/4
+BkΘ(−ia0q(2k+1)/4;q1/2) 1ϕ1
0
−q1/2;q1/2, ia0q(3+2k)/4
, (2.7)
where Ak and Bk are periodic functions of period one, i.e.,
Ak =Ak+1, Bk=Bk+1.
2.2 Projective reduction from q-PIII and q-PII
We formulate the family of B¨acklund transformations of q-PIII and q-PIV as a birational
rep-resentation of the affine Weyl group of type (A2 +A1)(1). Here, q-PIV is a q-analog of the
Painlev´e IV equation discussed in [13]. We refer to [21] for basic ideas of this formulation. We define the transformations si (i = 0,1,2) and π on the variables fj (j = 0,1,2) and
parameters ak (k= 0,1,2) by
si(aj) =ajai−aij, si(fj) =fj
ai+fi
1 +aifi
!uij
,
π(ai) =ai+1, π(fi) =fi+1,
fori, j∈Z/3Z. Here the symmetric 3×3 matrix
A= (aij)2i,j=0 =
2 −1 −1
−1 2 −1
−1 −1 2
,
is the Cartan matrix of type A(1)2 , and the skew-symmetric one
U = (uij)2i,j=0=
0 1 −1
−1 0 1
1 −1 0
,
represents an orientation of the corresponding Dynkin diagram. We also define the transforma-tionswj (j = 0,1) and r by
w0(fi) =
aiai+1(ai−1ai+ai−1fi+fi−1fi)
fi−1(aiai+1+aifi+1+fifi+1)
, w0(ai) =ai,
w1(fi) =
1 +aifi+aiai+1fifi+1
aiai+1fi+1(1 +ai−1fi−1+ai−1aifi−1fi)
, w1(ai) =ai,
r(fi) =
1 fi
, r(ai) =ai,
fori∈Z/3Z.
Proposition 2.3 ([13]). The group of birational transformationshs0, s1, s2, π, w0, w1, ri forms the affine Weyl group of type (A2+A1)(1), denoted by Wf((A2+A1)(1)). Namely, the transfor-mations satisfy the fundamental relations
si2 = (sisi+1)3 =π3 = 1, πsi=si+1π (i∈Z/3Z),
w02 =w12=r2 = 1, rw0=w1r,
In general, for a function F = F(ai, fj), we let an element w ∈ Wf((A2 +A1)(1)) act as
w.F(ai, fj) =F(ai.w, fj.w), that is,wacts on the arguments from the right. Note thata0a1a2 =
q andf0f1f2 =qc2 are invariant under the action ofWf((A2+A1)(1)) andWf(A(1)2 ), respectively.
We define the translations Ti (i= 1,2,3,4) by
T1=πs2s1, T2=s1πs2, T3=s2s1π, T4=rw0, (2.8)
whose action on parameters ai (i= 0,1,2) and cis given by
T1: (a0, a1, a2, c)7→ qa0, q−1a1, a2, c,
T2: (a0, a1, a2, c)7→(a0, qa1, q−1a2, c),
T3: (a0, a1, a2, c)7→ q−1a0, a1, qa2, c,
T4: (a0, a1, a2, c)7→(a0, a1, a2, qc).
Note that Ti (i= 1,2,3,4) commute with each other andT1T2T3 = 1. The action of T1 on the
f-variables can be expressed as
T1(f1) =
qc2
f1f0
1 +a0f0
a0+f0
, T1(f0) =
qc2
f0T1(f1)
1 +a2a0T1(f1)
a2a0+T1(f1)
. (2.9)
Or, applyingT1nT2mT4N (n, m, N ∈Z) on (2.9) and putting
fi,Nn,m =T1nT2mT4N(fi) (i= 0,1,2),
we obtain
f1n,N+1,m= q
2N+1c2
f1n,m,Nf0n,m,N
1 +a0qnf0n,m,N
a0qn+f0n,m,N
, f0n,N+1,m= q
2N+1c2
f0n,m,Nf1n,N+1,m
1 +a2a0qn−mf1n,N+1,m
a2a0qn−m+f1n,N+1,m
,
which is equivalent to q-PIII. ThenT1 and Ti (i= 2,4) are regarded as the time evolution and
B¨acklund transformations of q-PIII, respectively. We here note that we also obtain q-PIV by
identifying T4 as a time evolution [13].
In order to formulate the symmetrization toq-PII, it is crucial to introduce the
transforma-tion R1 defined by
R1=π2s1, (2.10)
which satisfies
R12=T1.
Considering the projection of the action of R1 on the line a2 =q1/2, we have
R1: (a0, a1, c)7→(q1/2a0, q−1/2a1, c),
R1(f0) =
qc2
f0f1
1 +a0f0
a0+f0
, R1(f1) =f0. (2.11)
Applying R1kT4N on (2.11) and putting
fi,Nk =R1kT4N(fi) (i= 0,1,2),
we have
f0k,N+1= q
2N+1c2
f0k,Nf0k,N−1
1 +a0qk/2f0k,N
a0qk/2+f0k,N
which is equivalent toq-PII. ThenR1 andT4 are regarded as the time evolution and a B¨acklund
transformation ofq-PII, respectively.
In general, we can derive various discrete Painlev´e systems from elements of infinite order of affine Weyl groups that are not necessarily translations by taking a projection on a certain subspace of the parameter space. We call such a procedure a projective reduction [12]. The symmetrization is a kind of the projective reduction.
2.3 Birational representation of Wf((A2+A1)(1)) on the τ function
We introduce the new variables τi andτi (i∈Z/3Z) by letting
fi =q1/3c2/3
τi+1τi−1
τi+1τi−1
,
and lift a representation to the affine Weyl group on their level:
Proposition 2.4 ([33]). We define the action of si (i= 0,1,2), π, wj (j = 0,1), and r on τk
and τk (k= 0,1,2)by the following formulae:
si(τi) =
uiτi+1τi−1+τi+1τi−1
ui1/2τi
, si(τj) =τj (i6=j),
si(τi) =
viτi+1τi−1+τi+1τi−1
vi1/2τi
, si(τj) =τj (i6=j),
π(τi) =τi+1, π(τi) =τi+1,
w0(τi) =
ai+11/3(τiτi+1τi+2+ui−1τiτi+1τi+2+ui+1−1τiτi+1τi+2)
ai+21/3τi+1τi+2
, w0(τi) =τi,
w1(τi) =
ai+11/3(τiτi+1τi+2+vi−1τiτi+1τi+2+vi+1−1τiτi+1τi+2)
ai+21/3τi+1τi+2
, w1(τi) =τi,
r(τi) =τi, r(τi) =τi,
with
ui=q−1/3c−2/3ai, vi =q1/3c2/3ai,
where i, j ∈Z/3Z. Then, hs0, s1, s2, π, w0, w1, ri forms the affine Weyl group Wf((A2+A1)(1)).
We define theτ functionτNn,m (n, m, N ∈Z) by
τNn,m =T1nT2mT4N(τ1).
We note that
τ0 =τ0−1,0, τ1=τ00,0, τ2=τ00,1, τ0 =τ1−1,0, τ1 =τ10,0, τ2 =τ10,1, (2.12)
and
f0n,m,N =q(2N+1)/3c2/3 τ
n,m N+1τ
n,m+1
N
τNn,mτNn,m+1+1, f
n,m
1,N =q(2N+1)/3c2/3
τNn,m+1+1τNn−1,m τNn,m+1τNn−+11,m,
f2n,m,N =q(2N+1)/3c2/3 τ
n−1,m N+1 τ
n,m N
Let us consider the τ functions for q-PII. We set
τNk =R1kT4N(τ1).
Note that
τ0 =τ0−2, τ1 =τ00, τ2=τ0−1, τ0 =τ1−2, τ1 =τ10, τ2 =τ1−1, (2.13)
and
f0k,N =q(2N+1)/3c2/3 τ
k N+1τk
−1
N
τNkτNk−+11 .
In general, it follows that
τNn,0 =τN2n, τNn,1 =τ2n−1
N .
For convenience, we introduce αi,γ, andQ by
αi6=ai, γ6=c, Q6=q.
3
Hypergeometric
τ
functions of the
q
-Painlev´
e systems
of type (
A
2+
A
1)
(1)In this section, we construct the hypergeometric τ functions ofq-PIII and q-PII. We define the
hypergeometric τ functions of q-PIII by τNn,m consistent with the action of hT1, T2, T3, T4i. We
also define the hypergeometric τ functions ofq-PIIbyτNk consistent with the action ofhR1, T4i.
Here, we mean τ(α) consistent with a action of transformationr as
r.τ(α) =τ(α.r).
We then regard τNn,m as function in α0 and α2, i.e.,
τNn,m =τN0,0(Qnα0, Q−mα2).
We also regardτNk as function inα0, i.e.,
τNk =τN0(Qk/2α0).
3.1 Hypergeometric τ functions of q-PIII
We construct the hypergeometricτ functions ofq-PIII. By the action of the affine Weyl group,
τNn,m is determined as a rational function in τ0n,m and τ1n,m (orτi and τi). Thus, our purpose is
determining τ0n,m and τ1n,m consistent with the action of hT1, T2, T3, T4i and constructing τNn,m
under the condition
γ = 1, (3.1)
and the boundary condition
τNn,m = 0 (N <0). (3.2)
Proposition 3.1. The following bilinear equations hold:
From (3.6)–(3.8), the following equations hold:
From (2.12), we rewrite (3.15) and (3.16) as follows:
(3.21)–(3.23), respectively. If we assume An,m0 is an arbitrary constant, it does not contradict (3.10)–(3.18). Therefore, we may setAn,m0 = 1.
Finally we constructτNn,m.
Theorem 3.1. Under the assumption (3.1) and (3.2), the hypergeometricτ functions of q-PIII are given as the follows:
τNn,m = (−1)N(N+1)/2Q−2(2n−m)N2+6nN
Here, An,m and Bn,m are periodic functions of period one with respect to nand m.
×Γ(Q2n−m+1α02α2;Q, Q)Γ(Q−n+2m−1α12α0;Q, Q)Γ(Q−n−mα22α1;Q, Q)ψn,mN −1.
From (3.2), (3.9), and (3.27), we find
ψn,mN = 0 (N <0), ψ0n,m = 1, ψn,m1 =Fn,m.
Furthermore, it is easily verified that ψn,mN satisfy
ψn,mN+1ψn,mN−1− ψNn,m2+ψNn+1,mψNn−1,m= 0, (3.30)
from (3.5). In general, (3.30) admits a solution expressed in terms of the Toeplitz type deter-minant
ψn,mN = det (cn−i+j,m)i,j=1,...,N (N >0),
under the boundary conditions
ψn,mN = 0 (N <0), ψ0n,m = 1, ψn,m1 =cn,m,
where cn,m is an arbitrary function. Therefore we have completed the proof.
3.2 Hypergeometric τ functions of q-PII
In this section, we construct the hypergeometric τ functions ofq-PII by two methods.
3.2.1 Hypergeometric τ functions of q-PII (I)
We construct the hypergeometric τ functions of q-PII by using those of q-PIII. We here note
thatτNn,m consistent with the action of hs2, T1, T2, T3, T4i is also consistent with the action ofR1
because
R1=s2T2−1.
Therefore, we construct τNn,m consistent with the action ofhs2, T1, T2, T3, T4i. The action of s2
on τNn,m is
s2(τNn,m) =τ
n−m,−m
N . (3.31)
We consider only τ0n,m and τ1n,m because τNn,m is determined as a rational function in τ0n,m andτ1n,m. It easily verified thatτ0n,m, (3.28) (or (3.9)), is consistent with the action ofs2. When
N = 1, we rewrite (3.31) as
s2(Fn,m−1) =α2−12Q−12m
Θ(α0−12Q12m−12n;Q12)
Θ(α0−12α2−12Q−12n;Q12)Fn−m,−m−1, (3.32)
from (3.28). Moreover, by using (3.29), (3.32) can be rewritten as
s2(An,m)−Bn,m
(α212Q12m;Q12)∞ 1 ϕ1
0
α2−12Q−12m+12;Q 12, α
012Q12n−12m+12
= (An,m−s2(Bn,m))Θ(α0
12Q12n−12m;Q12)
(α2−12Q−12m;Q12)∞Θ(α012α212Q12n;Q12) 1
ϕ1
0
α212Q12m+12;Q 12, α
012α212Q12n+12
,
which implies that τ1n,m is also consistent with the action ofs2 when
Lemma 3.2. Under the assumption (3.33), the hypergeometricτ functions (3.28) are consistent with the action of hs2, T1, T2, T3, T4i.
Therefore we easily obtain the following theorem:
Theorem 3.2. Setting
R1(An,m) =Bn,m, (3.34)
α2=Q1/2,
and putting
τN2n=τNn,0, τ2n−1
N =τ n,1
N ,
we obtain the hypergeometric τ functions of q-PII. Here τNn,m is given by (3.28).
In general, the entries of determinants of the hypergeometricτ functions of Painlev´e systems are expressed by two-parameter family of the functions satisfying the contiguity relations. How-ever the hypergeometricτ functions ofq-PIIin Theorem3.2have only one parameter because of
the condition (3.34). In the next section, we construct the hypergeometric τ functions of q-PII
which admits two parameters.
3.2.2 Hypergeometric τ functions of q-PII (II)
We construct the hypergeometric τ functions of q-PII whose ratios correspond to the
hyper-geometric solutions of q-PII in Proposition 2.2. By the action of the affine Weyl group, τNk is
determined as a rational function ofτk
0 andτ1k(orτiandτi). Thus, our purpose is determiningτ0k
and τ1k consistent with the action ofhR1, T4i and constructing τNk under the conditions
α2=Q1/2, γ = 1, (3.35)
and the boundary condition
τNk = 0 (N <0). (3.36)
First we consider the condition forτ0k which follows from the boundary condition (3.36). We use the bilinear equation obtained in [12]:
Proposition 3.2. The following bilinear equation holds:
τNk+1τNk+1−1−Q(k−4N+1)/2γ−2α0τNk+2τk
−1
N −Q
−k+4N−1γ4α
0−2τNk+1τNk = 0. (3.37)
By puttingN = 0 in (3.37), we get
Q3(k+1)/2α03τ0k+2τ0k−1+τ0k+1τ0k = 0. (3.38)
We set
τ0k= Γ Q(2k+3)/2α02;Q, QΓ Q−k/2α0−1;Q, QΓ Q(−k+3)/2α0−1;Q, QAk1. (3.39)
From (3.38),Ak
1 satisfies
We next determineτ0kand τ1k. From (2.10) and Proposition2.4, we see that the action ofR1
assumeAk1 is an arbitrary constant, it does not contradict (3.40)–(3.46). Therefore, we may put Ak1 = 1.
Finally we present an explicit formula forτk N.
Theorem 3.3. Under the assumption (3.35) and (3.36), the hypergeometricτ functions ofq-PII are given as the follows:
τNk = (−1)N(N−1)/2QN(N−1)(k+N)α02N(N−1)
Proof . We set
τNk = (−1)N(N−1)/2QN(N−1)(k+N)α02N(N−1)
×Γ(Q
(2k+3)/2α
02;Q, Q)Γ(Q−k/2α0−1;Q, Q)Γ(Q(−k+3)/2α0−1;Q, Q)
Θ(Q3k+1α
06;Q3)N
φkN.
From (3.36), (3.39), and (3.49), we find that
φkN = 0 (N <0), φ0k= 1, φk1 =Gk.
From (3.37),φkN satisfies
φNk+1φkN+1−1−φkNφNk+1+φkN+2φkN−1 = 0, (3.50)
which is a variant of the discrete Toda equation. Under the conditions
φkN = 0 (N <0), φ0k= 1, φk1 =ck,
where ck is an arbitrary function. Equation (3.50) admits a solution expressed by
φkN = det (ck+2i−j−1)i,j=1,...,N (N >0).
This complete the proof.
3.3 Relation between the hypergeometric τ functions of q-PIII and Heine’s transform
Masuda showed that the consistency of a certain reflection transformation to the hypergeometric τ functions of type E8(1) correspond to Bailey’s four term transformation formula [18]. It is also shown that the consistency of a certain reflection transformation to the hypergeometric τ functions of typeE7(1) correspond to limiting case of Bailey’s10ϕ9 transformation formula [17].
We here show that the consistency of s0 to the hypergeometric τ functions ofq-PIII give rise to
a transformation of 1ϕ1 which is obtained by Heine’s transform for 2ϕ1.
The action ofs0 on τNn,m is
s0(τNn,m) =τN−n,m−n. (3.51)
We consider only τ0n,m and τ1n,m because τNn,m is determined as a rational function in τ0n,m andτ1n,m. It easily verified thatτ0n,m, (3.28) (or (3.9)), is consistent with the action ofs0. When
N = 1, (3.51) implies
s0(Fn,m−1) =
Θ(α2−12Q−12n+12m;Q12)
Θ(α0−12α2−12Q12m;Q12)
F−n,m−n−1, (3.52)
from (3.28). Moreover, by using (3.29), (3.52) can be rewritten as
s0(An,m)1ϕ1
0
α012α212Q−12m+12;Q 12, α
212Q12n−12m+12
−An,m
(α212Q12n−12m+12;Q12)∞ (α012α212Q−12m+12;Q12)∞ 1
ϕ1
0
α212Q12n−12m+12;Q 12, α
212α012Q−12m+12
−Bn,mα012Q−12n
× 1ϕ1
Equation (3.54) corresponds to a specialization of Heine’s transform. Actually, by putting
a=b−1c, d=b−1z,
which is equivalent to (3.54).
Acknowledgements
The author would like to express sincere thanks to Professor T. Masuda for fruitful discussions and valuable suggestions. I acknowledge continuous encouragement by Professors K. Kajiwara and T. Tsuda. This work has been partially supported by the JSPS Research Fellowship.
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