Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2 + A1) (1)

We consider a $q$-Painlev\'e III equation and a $q$-Painlev\'e II equation arising from a birational representation of the affine Weyl group of type $(A_2+A_1)^{(1)}$. We study their hypergeometric solutions on the level of $\tau$ functions.


Introduction
We consider a q-analog of the Painlevé III equation (q-P III ) [8,12,13,32] g n+1 = q 2N +1 c 2 f n g n 1 + a 0 q n f n a 0 q n + f n , f n+1 = q 2N +1 c 2 f n g n+1 1 + a 2 a 0 q n−m g n+1 a 2 a 0 q n−m + g n+1 , (1.1) and that of the Painlevé II equation (q-P II ) [12,30,20] X k+1 = q 2N +1 c 2 X k X k−1 1 + a 0 q k/2 X k a 0 q k/2 + X k , (1.2) for the unknown functions f n = f n (m, N ), g n = g n (m, N ), and X k = X k (N ) and the independent variables n, k ∈ Z. Here m, N ∈ Z and a 0 , a 2 , c, q ∈ C × are parameters. These equations arise from a birational representation of the (extended) affine Weyl group of type (A 2 + A 1 ) (1) . Note that substituting m = 0, a 2 = q 1/2 , and putting f k (0, N ) = X 2k (N ), g k (0, N ) = X 2k−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 the τ functions play a crucial role in the theory of integrable systems [19], and it is also possible to introduce them in the theory of Painlevé 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 the τ functions.
The hypergeometric solutions of various Painlevé and discrete Painlevé systems are expressible in the form of ratio of determinants whose entries are given by hypergeometric type functions.
Usually, they are derived by reducing the bilinear equations to the Plücker 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 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. 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-Painlevé systems with E symmetries [17,18] and presented explicit determinant formulae for their hypergeometric solutions.
The purpose of this paper is to apply the above method to the q-Painlevé systems with the affine Weyl group symmetry of type (A 2 + A 1 ) (1) and present the explicit formulae of the hypergeometric τ functions. The hypergeometric τ functions provide not only determinant formulae but also important information originating from the geometry of lattice of the τ 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 Section 2, we first review hypergeometric solutions of q-P III and then those of q-P II . We next introduce a representation of the affine Weyl group of type (A 2 + A 1 ) (1) . In Section 3, we construct the hypergeometric τ functions of q-P III and those of q-P II . We find that the symmetry of the hypergeometric τ functions of q-P III are connected with Heine's transform of the basic hypergeometric series 2 ϕ 1 .
2 q-P III and q-P II 2.1 Hypergeometric solutions of q-P III and q-P II First, we review the hypergeometric solutions of q-P III and q-P II . The hypergeometric solutions of q-P III have been constructed as follows: . The hypergeometric solutions of q-P III , (1.1), with c = 1 are given by and F n,m is an arbitrary solution of the systems The general solution of (2.1) and (2.2) is given by where A n,m and B n,m are periodic functions of period one with respect to n and m, i.e., The explicit form of the hypergeometric solutions of q-P II are given as follows: 20]). The hypergeometric solutions of q-P II , (1.2), with c = 1 are given by

5)
and G k is an arbitrary solution of the system The general solution of (2.6) is given by where A k and B k are periodic functions of period one, i.e., 2.2 Projective reduction from q-P III and q-P II We formulate the family of Bäcklund transformations of q-P III and q-P IV as a birational representation of the affine Weyl group of type (A 2 + A 1 ) (1) . Here, q-P IV is a q-analog of the Painlevé IV equation discussed in [13]. We refer to [21] for basic ideas of this formulation. We define the transformations s i (i = 0, 1, 2) and π on the variables f j (j = 0, 1, 2) and parameters a k (k = 0, 1, 2) by is the Cartan matrix of type A 2 , and the skew-symmetric one represents an orientation of the corresponding Dynkin diagram. We also define the transformations w j (j = 0, 1) and r by

Proposition 2.3 ([13]
). The group of birational transformations s 0 , s 1 , s 2 , π, w 0 , w 1 , r forms the affine Weyl group of type ( ). Namely, the transformations satisfy the fundamental relations and the action of W (A 2 ) = s 0 , s 1 , s 2 , π and that of W (A (1) In general, for a function F = F (a i , f j ), we let an element w ∈ W ((A 2 + A 1 ) (1) ) act as w.F (a i , f j ) = F (a i .w, f j .w), that is, w acts on the arguments from the right. Note that a 0 a 1 a 2 = q and f 0 f 1 f 2 = qc 2 are invariant under the action of W ((A 2 + A 1 ) (1) ) and W (A (1) 2 ), respectively. We define the translations T i (i = 1, 2, 3, 4) by whose action on parameters a i (i = 0, 1, 2) and c is given by Note that T i (i = 1, 2, 3, 4) commute with each other and T 1 T 2 T 3 = 1. The action of T 1 on the f -variables can be expressed as .
which is equivalent to q-P III . Then T 1 and T i (i = 2, 4) are regarded as the time evolution and Bäcklund transformations of q-P III , respectively. We here note that we also obtain q-P IV by identifying T 4 as a time evolution [13]. In order to formulate the symmetrization to q-P II , it is crucial to introduce the transformation R 1 defined by which satisfies Considering the projection of the action of R 1 on the line a 2 = q 1/2 , we have Applying R 1 k T 4 N on (2.11) and putting which is equivalent to q-P II . Then R 1 and T 4 are regarded as the time evolution and a Bäcklund transformation of q-P II , respectively. In general, we can derive various discrete Painlevé 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.
We define the τ function τ n,m We note that .
Let us consider the τ functions for q-P II . We set Note that and In general, it follows that For convenience, we introduce α i , γ, and Q by 3 Hypergeometric τ functions of the q-Painlevé systems of type (A 2 + A 1 ) (1) In this section, we construct the hypergeometric τ functions of q-P III and q-P II . We define the hypergeometric τ functions of q-P III by τ n,m N consistent with the action of T 1 , T 2 , T 3 , T 4 . We also define the hypergeometric τ functions of q-P II by τ k N consistent with the action of R 1 , T 4 . Here, we mean τ (α) consistent with a action of transformation r as r.τ (α) = τ (α.r).
We then regard τ n,m N as function in α 0 and α 2 , i.e., We also regard τ k N as function in α 0 , i.e.,

Hypergeometric τ functions of q-P III
We construct the hypergeometric τ functions of q-P III . By the action of the affine Weyl group, τ n,m  . From (2.8) and Proposition 2.4, we see that the action of T 1 , T 2 , and T 3 are given by 14) where i = 1, 2, 3. Proof . Applying T i−1 on (3.16) and using (3.13) and (3.14), we have We set Here, A n,m and B n,m are periodic functions of period one with respect to n and m.
Proof . We set where c n,m is an arbitrary function. Therefore we have completed the proof.

Hypergeometric τ functions of q-P II
In this section, we construct the hypergeometric τ functions of q-P II by two methods.

Hypergeometric τ functions of q-P II (I)
We construct the hypergeometric τ functions of q-P II by using those of q-P III . We here note that τ n,m N consistent with the action of s 2 , T 1 , T 2 , T 3 , T 4 is also consistent with the action of R 1 because Therefore, we construct τ n,m from (3.28). Moreover, by using (3.29), (3.32) can be rewritten as Therefore we easily obtain the following theorem: we obtain the hypergeometric τ functions of q-P II . Here τ n,m N is given by (3.28).
In general, the entries of determinants of the hypergeometric τ functions of Painlevé systems are expressed by two-parameter family of the functions satisfying the contiguity relations. However the hypergeometric τ functions of q-P II in Theorem 3.2 have only one parameter because of the condition (3.34). In the next section, we construct the hypergeometric τ functions of q-P II which admits two parameters.

Hypergeometric τ functions of q-P II (II)
We construct the hypergeometric τ functions of q-P II whose ratios correspond to the hypergeometric solutions of q-P II in Proposition 2.2. By the action of the affine Weyl group, τ k N is determined as a rational function of τ k 0 and τ k 1 (or τ i and τ i ). Thus, our purpose is determining τ k 0 and τ k 1 consistent with the action of R 1 , T 4 and constructing τ k N under the conditions and the boundary condition First we consider the condition for τ k 0 which follows from the boundary condition (3.36). We use the bilinear equation obtained in [12]: Proposition 3.2. The following bilinear equation holds: By putting N = 0 in (3.37), we get We set We next determine τ k 0 and τ k 1 . From (2.10) and Proposition 2.4, we see that the action of R 1 on τ k 0 and τ k 1 is given by From (2.13), we rewrite (3.42) and (3.45) as respectively. Setting and G k = A k Θ ia 0 q (2k+1)/4 ; q 1/2 1 ϕ 1 0 −q 1/2 ; q 1/2 , −ia 0 q (3+2k)/4 + B k Θ −ia 0 q (2k+1)/4 ; q 1/2 1 ϕ 1 0 −q 1/2 ; q 1/2 , ia 0 q (3+2k)/4 .
Here, A k and B k are periodic functions of period one.
Proof . We set From (3.36), (3.39), and (3.49), we find that which is a variant of the discrete Toda equation. Under the conditions where c k is an arbitrary function. Equation This complete the proof.

Relation between the hypergeometric τ functions of q-P III and Heine's transform
Masuda showed that the consistency of a certain reflection transformation to the hypergeometric τ functions of type E (1) 8 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 type E (1) 7 correspond to limiting case of Bailey's 10 ϕ 9 transformation formula [17]. We here show that the consistency of s 0 to the hypergeometric τ functions of q-P III give rise to a transformation of 1 ϕ 1 which is obtained by Heine's transform for 2 ϕ 1 .
The action of s 0 on τ n,m