Hypergeometric $\tau$ Functions of the $q$-Painlev\'e Systems of Types $A_4^{(1)}$ and $(A_1+A_1')^{(1)}$

We consider $q$-Painlev\'e equations arising from birational representations of the extended affine Weyl groups of $A_4^{(1)}$- and $(A_1+A_1)^{(1)}$-types. We study their hypergeometric solutions on the level of $\tau$ functions.


Introduction 1.Purpose
The purpose of this paper is to construct the hypergeometric τ functions associated with q-Painlevé equations of A (1) 4 -and A (1) 6 -surface types in Sakai's classification [56].As a corollary, we obtain the hypergeometric solutions of the corresponding q-Painlevé equations.

Background
Discrete Painlevé equations are nonlinear ordinary difference equations of second order, which include discrete analogues of the six Painlevé equations, and are classified by types of rational surfaces connected to affine Weyl groups [56].They admit particular solutions, so called hypergeometric solutions, which are expressible in terms of the hypergeometric type functions, when some of the parameters take special values (see, for example, [30,31,33] and references therein).Together with the Painlevé equations, discrete Painlevé equations are now regarded as one of the most important classes of equations in the theory of integrable systems (see, e.g., [14,35]).
It is well known that the τ functions play a crucial role in the theory of integrable systems [42], and it is also possible to introduce them in the theory of Painlevé systems [20,21,22,45,47,48,49,50]. A representation of the affine Weyl groups can be lifted on the level of the τ functions [25,26,29,32,40,41,58,59], which gives rise to various bilinear equations of Hirota type satisfied by the τ functions.
Usually, the hypergeometric solutions of discrete Painlevé equations are derived by reducing the bilinear equations to the Plücker relations by using the contiguity relations satisfied by the entries of determinants [16,17,23,27,28,34,36,37,38,46,55].This method is elementary, but it encounters technical difficulties for discrete Painlevé equations 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, so that they are consistent with the action of the affine Weyl groups.We call such hypergeometric solutions hypergeometric τ functions [40,41,43].Although this requires somewhat complex calculations, the merit is that it is systematic and can be applied to the systems with large symmetries.
Some discrete Painlevé equations have been found in the studies of random matrices [11,19,51].As one such example, let us consider the partition function of the Gaussian Unitary Ensemble of an n × n random matrix: where g 2 > 0 and ∆(t 1 , . . ., t n ) is Vandermonde's determinant.Letting we obtain the following difference equation [11,13,15,53] n corresponds to hypergeometric τ functions.Such relations between discrete Painlevé equations and random matrices are well investigated.Moreover, in recent years, the relations between τ functions of Painlevé systems and a certain class of integrable partial difference equations introduced by Adler-Bobenko-Suris and Boll [1, 2, 8, 9, 10], which include a discrete analogue of the Korteweg-de Vries equation, are well investigated [7,18,24,25,26].Throughout these relations and by using the hypergeometric τ functions, a discrete analogue of the power function was derived and its properties, such as discrete analogue of the Riemann surface and circle packing, were shown in [3,4,5,6,7,44].These results consolidate the importance of the studies of the hypergeometric τ function for applications of Painlevé systems.
In [16,17], the hypergeometric solutions of the q-Painlevé equations (2.32) and (3.1) (or (3.4)) are constructed by solving the minimum required bilinear equations to obtain those equations.In this paper, we solve all bilinear equations arising from the actions of the translation subgroups of W A (1) 4 and W (A 1 + A 1 ) (1) , that is, the hypergeometric τ functions given in Theorems 2.7 and 3.1 are for not only the hypergeometric solutions of the q-Painlevé equations (2.32) and (3.1) but also those of other q-Painlevé equations, e.g., (2.33), (3.2) and (3.3) (see Corollaries 2.9 and 3.2).Moreover, as mentioned above we can derive the various integrable partial difference equations from the τ functions of discrete Painlevé equations (see, for example, [18,25,26]).Therefore, the hypergeometric τ functions constructed in this paper also give the hypergeometric solutions of the partial difference equations appeared in [25,26].

Plan of the paper
This paper is organized as follows: in Section 2, we first introduce τ functions with the representation of the affine Weyl group W A (1) of W A (1) 4 -type (see Theorem 2.7).Finally, we obtain the hypergeometric solutions of the q-Painlevé equations of A (1) 4 -surface type (see Corollary 2.9).In Section 3, we summarize the result for the W (A 1 + A 1 ) (1) -type (or, A (1) 6 -surface type).

q-Special functions
We use the following conventions of q-analysis with |p|, |q| < 1 throughout this paper [12].
We note that the following formulae hold where n ∈ Z >0 .

-type
In this section, we construct the hypergeometric τ functions of W A (1) 4 -type.
To iterate each variable τ (j) i , we need the translations T i , i = 0, . . ., 4, defined by ) The action of translations on the parameters is given by where i ∈ Z/5Z.Note that T i , i = 0, . . ., 4, commute with each other and We define τ functions by where l i ∈ Z.We note that τ (1)

Hypergeometric τ functions
The aim of this section is to construct the hypergeometric τ functions of W A (1) 4 -type.Hereinafter, we consider the τ functions τ l 0 ,l 2 ,l 3 l 1 satisfying the following conditions: (i) τ l 0 ,l 2 ,l 3 l 1 satisfy the action of the translation subgroup of W A (1) 4 , T 0 , T 1 , T 2 , T 3 , T 4 ; (ii) τ l 0 ,l 2 ,l 3 l 1 are functions in a 0 , a 2 and a 4 consistent with the action of T 0 , T 2 , T 3 , i.e., τ l 0 ,l 2 ,l 3 l 1 = τ l 1 q l 0 a 0 , q l 2 a 2 , q −l 3 a 4 ; (iii) τ l 0 ,l 2 ,l 3 l 1 satisfy the following boundary conditions: for l 1 < 0; under the conditions of parameters We here call such functions τ l 0 ,l 2 ,l 3 l 1 hypergeometric τ functions of W A (1) 4 -type.
From the condition (i), every τ l 0 ,l 2 ,l 3 l 1 can be given by a rational function of ten variables τ (j) i (or, τ l 0 ,l 2 ,l 3 0 l i ∈Z and τ l 0 ,l 2 ,l 3 1 l i ∈Z ).Therefore, our purpose in this section is to obtain the explicit formulae for τ l 0 ,l 2 ,l 3 0 l i ∈Z and τ l 0 ,l 2 ,l 3 1 l i ∈Z , satisfying the condition (ii) under the condition (iii) and construct the closed-form expressions of τ l 0 ,l 2 ,l 3 l 1 l i ∈Z, l 1 ≥2 .
(2.10n)Moreover, by using the action of T 1 , we obtain the following lemma.
Step 2. In this step, we get the explicit formulae for τ l 0 ,l 2 ,l 3 0 and τ l 0 ,l 2 ,l 3

1
. Letting where H l 0 ,l 2 ,l 3 = H q l 0 a 0 , q l 2 a 2 , q −l 3 a 4 , we obtain the following lemma.
Step 3. In this final step, we give the hypergeometric τ functions of W A (1) 4 -type.Substituting , in equation (2.11a), we obtain the following bilinear equation In general, equation (2.28) admits a solution expressed in terms of Jacobi-Trudi type determinant where l 1 ∈ Z >1 , under the boundary conditions where c l 0 ,l 2 ,l 3 is an arbitrary function.Therefore, we obtain the following theorem.
on variables f (j) i is given by , where i = 1, 2 and j ∈ Z/5Z.

τ functions
The action of the transformation group W (A 1 + A 1 ) (1) = s 0 , s 1 , w 0 , w 1 , π on the parameters a 0 , a 1 and b are given by while its actions on the variables τ i , i = −3, . . ., 3, are given by where For each element w ∈ W (A 1 + A 1 ) (1) and function F = F (a i , b, τ j ), we use the notation w.F to mean w.F = F (w.a i , w.b, w.τ j ), that is, w acts on the arguments from the left.We note that the group of transformations W (A 1 + A 1 ) (1) forms the extended affine Weyl group of type (A 1 + A 1 ) (1) [25].Namely, the transformations satisfy the fundamental relations and the action of W A (1) 1 = s 0 , s 1 and that of W A (1) 1 = w 0 , w 1 commute.We note that the relation (ww ) ∞ = 1 for transformations w and w means that there is no positive integer N such that (ww ) N = 1.