Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾

Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾

We consider q-Painlevé equations arising from birational representations of the extended affine Weyl groups of A⁽¹⁾₄- and (A₁+A₁)⁽¹⁾-types. We study their hypergeometric solutions on the level of τ functions.

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2016
Main Author: Nakazono, N.
Format: Article
Language:English
Published: Інститут математики НАН України 2016
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/147747
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Cite this:Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾ / N. Nakazono // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 59 назв. — англ.

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spelling Nakazono, N.
2019-02-15T19:07:09Z
2019-02-15T19:07:09Z
2016
Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾ / N. Nakazono // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 59 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 33D05; 33D15; 33E17; 39A13
DOI:10.3842/SIGMA.2016.051
https://nasplib.isofts.kiev.ua/handle/123456789/147747
We consider q-Painlevé equations arising from birational representations of the extended affine Weyl groups of A⁽¹⁾₄- and (A₁+A₁)⁽¹⁾-types. We study their hypergeometric solutions on the level of τ functions.
The author would like to express his sincere thanks to Dr. Milena Radnovic for her valuable comments. This research was supported by grant # DP130100967 from the Australian Research Council.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾
spellingShingle Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾
Nakazono, N.
title_short Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾
title_full Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾
title_fullStr Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾
title_full_unstemmed Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾
title_sort hypergeometric τ functions of the q-painlevé systems of types a⁽¹⁾₄ and (a₁+a′₁)⁽¹⁾
author Nakazono, N.
author_facet Nakazono, N.
publishDate 2016
language English
container_title Symmetry, Integrability and Geometry: Methods and Applications
publisher Інститут математики НАН України
format Article
description We consider q-Painlevé equations arising from birational representations of the extended affine Weyl groups of A⁽¹⁾₄- and (A₁+A₁)⁽¹⁾-types. We study their hypergeometric solutions on the level of τ functions.
issn 1815-0659
url https://nasplib.isofts.kiev.ua/handle/123456789/147747
citation_txt Hypergeometric τ Functions of the q-Painlevé Systems of Types A⁽¹⁾₄ and (A₁+A′₁)⁽¹⁾ / N. Nakazono // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 59 назв. — англ.
work_keys_str_mv AT nakazonon hypergeometricτfunctionsoftheqpainlevesystemsoftypesa14anda1a11
first_indexed 2025-11-26T00:08:36Z
last_indexed 2025-11-26T00:08:36Z
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 051, 23 pages Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′ 1) (1) Nobutaka NAKAZONO School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia E-mail: nobua.n1222@gmail.com URL: http://researchmap.jp/nakazono/ Received February 01, 2016, in final form May 16, 2016; Published online May 20, 2016 http://dx.doi.org/10.3842/SIGMA.2016.051 Abstract. We consider q-Painlevé equations arising from birational representations of the extended affine Weyl groups of A (1) 4 - and (A1+A1)(1)-types. We study their hypergeometric solutions on the level of τ functions. Key words: q-Painlevé equation; basic hypergeometric function; affine Weyl group; τ func- tion 2010 Mathematics Subject Classification: 33D05; 33D15; 33E17; 39A13 1 Introduction 1.1 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. This work is motivated by the project to construct all possible hypergeometric τ functions associated with the multiplicative surface types in the Sakai’s classification [56], that is, A (1) 0 -, A (1) 1 -, A (1) 2 -, A (1) 3 -, A (1) 4 -, A (1) 5 - and A (1) 6 -surface types. The corresponding symmetry groups are W ( E (1) 8 ) , W̃ ( E (1) 7 ) , W̃ ( E (1) 6 ) , W̃ ( D (1) 5 ) , W̃ ( A (1) 4 ) , W̃ ( (A2 + A1) (1) ) and W̃ ( (A1 + A′1) (1) ) , respectively. The works for W ( E (1) 8 ) -type [41], W̃ ( E (1) 7 ) -type [40] and W̃ ( (A2+A1) (1) ) -type [43] have been done. In this paper, we consider the hypergeometric τ functions of W̃ ( A (1) 4 ) - and W̃ ( (A1 +A′1) (1) ) -types. 1.2 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 hyper- geometric 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. mailto:nobua.n1222@gmail.com http://researchmap.jp/nakazono/ http://dx.doi.org/10.3842/SIGMA.2016.051 2 N. Nakazono 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: Z(2) n = ∫ ∞ −∞ · · · ∫ ∞ −∞ ∆(t1, . . . , tn)2 n∏ i=1 e−g1ti 2−g2ti4dti, where g2 > 0 and ∆(t1, . . . , tn) is Vandermonde’s determinant. Letting Rn = Z (2) n+1Z (2) n−1( Z (2) n )2 , we obtain the following difference equation [11, 13, 15, 53] Rn+1 +Rn +Rn−1 = n 4g2 1 Rn − g1 2g2 . (1.1) Equation (1.1) is known as a discrete analogue of the Painlevé I equation and also as a Bäcklund transformation of the Painlevé IV equation. The partition function Z (2) n corresponds to hyper- geometric 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 W̃ ( A (1) 4 ) and W̃ ( (A1 +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]. 1.3 Plan of the paper This paper is organized as follows: in Section 2, we first introduce τ functions with the repre- sentation of the affine Weyl group W̃ ( A (1) 4 ) . Next, we construct the hypergeometric τ functions Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 3 of W̃ ( 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 W̃ ( (A1 +A′1) (1) ) -type (or, A (1) 6 -surface type). 1.4 q-Special functions We use the following conventions of q-analysis with |p|, |q| < 1 throughout this paper [12]. • q-Shifted factorials: (a; q)n = n−1∏ i=0 ( 1− qia ) , n = 1, 2, . . . , (a; q)∞ = ∞∏ i=0 ( 1− qia ) , (a; p, q)∞ = ∞∏ i,j=0 ( 1− qipja ) . • Modified 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)∞ . • 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∏ i=1 (ai; q)n. We note that the following formulae hold (qna; q)∞ (a; q)∞ = n−1∏ i=0 1 1− qia , Θ(qna; q) Θ(a; q) = (−1)n n−1∏ i=0 1 qia , (qna; p, q)∞ (a; p, q)∞ = n−1∏ i=0 1 (qia; p)∞ , (pna; p, q)∞ (a; p, q)∞ = n−1∏ i=0 1 (pia; q)∞ , Γ(qna; p, q) Γ(a; p, q) = n−1∏ i=0 Θ ( qia; p ) , Γ(pna; p, q) Γ(a; p, q) = n−1∏ i=0 Θ ( pia; q ) , where n ∈ Z>0. 2 Hypergeometric τ functions of W̃ ( A (1) 4 ) -type In this section, we construct the hypergeometric τ functions of W̃ ( A (1) 4 ) -type. 4 N. Nakazono 2.1 τ functions Let us consider ten variables: τ (j) i (i = 1, 2, j = 1, . . . , 5) and six parameters: a0, . . . , a4, q ∈ C∗ with the following three relations for the variables τ (1) 2 = a0a1 ( a3τ (3) 1 τ (5) 1 + a0τ (4) 1 τ (3) 2 ) a2a32τ (5) 2 , (2.1a) τ (2) 2 = a1a2 ( a4τ (1) 1 τ (4) 1 + a1τ (5) 1 τ (4) 2 ) a3a42τ (1) 2 , (2.1b) τ (4) 2 = a3a4 ( a1τ (1) 1 τ (3) 1 + a3τ (2) 1 τ (1) 2 ) a0a12τ (3) 2 , (2.1c) and the following condition for the parameters a0a1a2a3a4 = q. The action of the transformation group 〈s0, s1, s2, s3, s4, σ, ι〉 on the parameters is given by si(aj) = ajai −aij , σ(ai) = ai+1, ι : (a0, a1, a2, a3, a4) 7→ ( a0 −1, a4 −1, a3 −1, a2 −1, a1 −1), where i, j ∈ Z/5Z and the symmetric 5× 5 matrix (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 ( τ (i+5) 1 ) = τ (i+4) 2 , si ( τ (i+3) 2 ) = ai+3ai+4 ( aiai+1τ (i+1) 1 τ (i+3) 1 + ai+3τ (i+2) 1 τ (i+1) 2 ) ai+1 2τ (i+5) 1 , (2.2a) si ( τ (i+4) 2 ) = τ (i+5) 1 , si ( τ (i+5) 2 ) = ai+4 ( ai+2τ (i+2) 1 τ (i+4) 1 + aiai+4τ (i+3) 1 τ (i+2) 2 ) aiai+1ai+2 2τ (i+5) 1 , (2.2b) σ ( τ (i) 1 ) = τ (i+1) 1 , σ ( τ (i) 2 ) = τ (i+1) 2 , (2.2c) ι : ( τ (1) 1 , τ (2) 1 , τ (3) 1 , τ (4) 1 , τ (1) 2 , τ (2) 2 , τ (3) 2 , τ (5) 2 ) 7→ ( τ (4) 1 , τ (3) 1 , τ (2) 1 , τ (1) 1 , τ (2) 2 , τ (1) 2 , τ (5) 2 , τ (3) 2 ) , (2.2d) where i ∈ Z/5Z. In general, for a function F = F ( ai, τ (k) j ) , we let an element w ∈ W̃ ( A (1) 4 ) act as w.F = F ( w.ai, w.τ (k) j ) , that is, w acts on the arguments from the left. Note that q = a0a1a2a3a4 is invariant under the action of 〈s0, s1, s2, s3, s4, σ〉. Proposition 2.1 ([26, 58]). The group of birational transformations 〈s0, s1, s2, s3, s4, σ, ι〉, de- noted by W̃ ( A (1) 4 ) , forms the extended affine Weyl group of type A (1) 4 . Namely, the transforma- tions satisfy the fundamental relations si 2 = 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 τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 5 To iterate each variable τ (j) i , we need the translations Ti, i = 0, . . . , 4, defined by T0 = σs4s3s2s1, T1 = σs0s4s3s2, T2 = σs1s0s4s3, T3 = σs2s1s0s4, (2.3a) T4 = σs3s2s1s0. (2.3b) The action of translations on the parameters is given by Ti(ai) = qai, Ti(ai+1) = q−1ai+1, where i ∈ Z/5Z. Note that Ti, i = 0, . . . , 4, commute with each other and T0T1T2T3T4 = 1. We define τ functions by τ l0,l2,l3l1 = T0 l0T1 l1T2 l2T3 l3 ( τ (3) 2 ) , (2.4) where li ∈ Z. We note that τ (1) 1 = τ1,0,10 , τ (2) 1 = τ1,0,11 , τ (3) 1 = τ1,1,11 , τ (4) 1 = τ1,1,21 , τ (5) 1 = τ0,0,10 , (2.5a) τ (1) 2 = τ1,1,10 , τ (2) 2 = τ1,0,21 , τ (3) 2 = τ0,0,00 , τ (4) 2 = τ2,1,21 , τ (5) 2 = τ0,0,11 . (2.5b) 2.2 Hypergeometric τ functions The aim of this section is to construct the hypergeometric τ functions of W̃ ( A (1) 4 ) -type. Hereinafter, we consider the τ functions τ l0,l2,l3l1 satisfying the following conditions: (i) τ l0,l2,l3l1 satisfy the action of the translation subgroup of W̃ ( A (1) 4 ) , 〈T0, T1, T2, T3, T4〉; (ii) τ l0,l2,l3l1 are functions in a0, a2 and a4 consistent with the action of 〈T0, T2, T3〉, i.e., τ l0,l2,l3l1 = τl1 ( ql0a0, q l2a2, q −l3a4 ) ; (iii) τ l0,l2,l3l1 satisfy the following boundary conditions: τ l0,l2,l3l1 = 0, (2.6) for l1 < 0; under the conditions of parameters a0a1 = q. (2.7) We here call such functions τ l0,l2,l3l1 hypergeometric τ functions of W̃ ( A (1) 4 ) -type. From the condition (i), every τ l0,l2,l3l1 can be given by a rational function of ten variables τ (j) i (or, { τ l0,l2,l30 } li∈Z and { τ l0,l2,l31 } li∈Z ). Therefore, our purpose in this section is to obtain the explicit formulae for { τ l0,l2,l30 } li∈Z and { τ l0,l2,l31 } li∈Z , satisfying the condition (ii) under the condition (iii) and construct the closed-form expressions of { τ l0,l2,l3l1 } li∈Z, l1≥2 . Step 1. Begin by preparing the equations necessary for the construction of the hypergeo- metric τ functions of W̃ ( A (1) 4 ) -type. From the actions (2.2) and the definitions (2.3), the actions of T0, T2 and T3 and their inverses on ten variables τ (j) i are given by the following T0 ( τ (4) 1 ) = τ (4) 2 , T0 ( τ (5) 1 ) = τ (1) 1 , T0 ( τ (5) 2 ) = τ (2) 1 , 6 N. Nakazono T2 ( τ (1) 1 ) = τ (1) 2 , T2 ( τ (2) 1 ) = τ (3) 1 , T2 ( τ (2) 2 ) = τ (4) 1 , T3 ( τ (2) 1 ) = τ (2) 2 , T3 ( τ (3) 1 ) = τ (4) 1 , T3 ( τ (3) 2 ) = τ (5) 1 , T0 ( τ (1) 1 ) = qa0 2a4 ( a3τ (1) 1 T0 ( τ (3) 1 ) + a0a1τ (4) 2 T0 ( τ (3) 2 )) a3τ (3) 1 , (2.8a) T0 ( τ (2) 1 ) = a0a1 ( qa0τ (4) 2 T0 ( τ (3) 2 ) + a2a3τ (1) 1 T0 ( τ (3) 1 )) a32τ (1) 2 , (2.8b) T0 ( τ (3) 1 ) = a3a4 ( a0a1τ (1) 1 τ (3) 1 + a3τ (2) 1 τ (1) 2 ) a12τ (5) 1 , (2.8c) T0 ( τ (1) 2 ) = a0a1 ( qa0τ (4) 2 T0 ( τ (3) 2 ) + a3τ (1) 1 T0 ( τ (3) 1 )) a2a32τ (2) 1 , (2.8d) T0 ( τ (2) 2 ) = a1a2 ( q−1a1τ (1) 1 T0 ( τ (4) 2 ) + a4τ (4) 2 T0 ( τ (1) 1 )) qa3a42T0 ( τ (1) 2 ) , (2.8e) T0 ( τ (3) 2 ) = a3 ( a1τ (1) 1 τ (3) 1 + a3a4τ (2) 1 τ (1) 2 ) a0a12a4τ (4) 1 , (2.8f) T0 ( τ (4) 2 ) = a3a4 ( a1T0 ( τ (1) 1 ) T0 ( τ (3) 1 ) + qa3T0 ( τ (2) 1 ) T0 ( τ (1) 2 )) a0a12T0 ( τ (3) 2 ) , (2.8g) T0 −1(τ (3)1 ) = a0 ( a3τ (3) 1 τ (5) 1 + a0a1τ (4) 1 τ (3) 2 ) a1a2a32τ (1) 1 , (2.8h) T0 −1(τ (4)1 ) = a3 ( qa1τ (5) 1 T0 −1(τ (3)1 ) + a3a4τ (5) 2 T0 −1(τ (1)2 )) qa0a12a4τ (3) 2 , (2.8i) T0 −1(τ (5)1 ) = a3a4 ( a0a1τ (5) 1 T0 −1(τ (3)1 ) + a3τ (5) 2 T0 −1(τ (1)2 )) q2a12τ (3) 1 , (2.8j) T0 −1(τ (1)2 ) = a0a1 ( a2a3τ (3) 1 τ (5) 1 + a0τ (4) 1 τ (3) 2 ) a32τ (2) 1 , (2.8k) T0 −1(τ (2)2 ) = qa1a2 ( qa1τ (4) 1 T0 −1(τ (5)1 ) + a4τ (5) 1 T0 −1(τ (4)1 )) a3a42T0 −1(τ (1)2 ) , (2.8l) T0 −1(τ (3)2 ) = a3a4 ( qa1τ (5) 1 T0 −1(τ (3)1 ) + a3τ (5) 2 T0 −1(τ (1)2 )) qa0a12τ (4) 1 , (2.8m) T0 −1(τ (5)2 ) = a0a1 ( q−1a0T0 −1(τ (4)1 ) T0 −1(τ (3)2 ) + a3T0 −1(τ (3)1 ) T0 −1(τ (5)1 )) a2a32T0 −1(τ (1)2 ) , (2.8n) T2 ( τ (3) 1 ) = qa2 2a1 ( a0τ (3) 1 T2 ( τ (5) 1 ) + a2a3τ (1) 2 T2 ( τ (5) 2 )) a0τ (5) 1 , (2.9a) T2 ( τ (4) 1 ) = a2a3 ( qa2τ (1) 2 T2 ( τ (5) 2 ) + a4a0τ (3) 1 T2 ( τ (5) 1 )) a02τ (3) 2 , (2.9b) T2 ( τ (5) 1 ) = a0a1 ( a2a3τ (3) 1 τ (5) 1 + a0τ (4) 1 τ (3) 2 ) a32τ (2) 1 , (2.9c) Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 7 T2 ( τ (3) 2 ) = a2a3 ( qa2τ (1) 2 T2 ( τ (5) 2 ) + a0τ (3) 1 T2 ( τ (5) 1 )) a4a02τ (4) 1 , (2.9d) T2 ( τ (4) 2 ) = a3a4 ( q−1a3τ (3) 1 T2 ( τ (1) 2 ) + a1τ (1) 2 T2 ( τ (3) 1 )) qa0a12T2 ( τ (3) 2 ) , (2.9e) T2 ( τ (5) 2 ) = a0 ( a3τ (3) 1 τ (5) 1 + a0a1τ (4) 1 τ (3) 2 ) a2a32a1τ (1) 1 , (2.9f) T2 ( τ (1) 2 ) = a0a1 ( a3T2 ( τ (3) 1 ) T2 ( τ (5) 1 ) + qa0T2 ( τ (4) 1 ) T2 ( τ (3) 2 )) a2a32T2 ( τ (5) 2 ) , (2.9g) T2 −1(τ (5)1 ) = a2 ( a0τ (5) 1 τ (2) 1 + a2a3τ (1) 1 τ (5) 2 ) a3a4a02τ (3) 1 , (2.9h) T2 −1(τ (1)1 ) = a0 ( qa3τ (2) 1 T2 −1(τ (5)1 ) + a0a1τ (2) 2 T2 −1(τ (3)2 )) qa2a32a1τ (5) 2 , (2.9i) T2 −1(τ (2)1 ) = a0a1 ( a2a3τ (2) 1 T2 −1(τ (5)1 ) + a0τ (2) 2 T2 −1(τ (3)2 )) q2a32τ (5) 1 , (2.9j) T2 −1(τ (3)2 ) = a2a3 ( a4a0τ (5) 1 τ (2) 1 + a2τ (1) 1 τ (5) 2 ) a02τ (4) 1 , (2.9k) T2 −1(τ (4)2 ) = qa3a4 ( qa3τ (1) 1 T2 −1(τ (2)1 ) + a1τ (2) 1 T2 −1(τ (1)1 )) a0a12T2 −1(τ (3)2 ) , (2.9l) T2 −1(τ (5)2 ) = a0a1 ( qa3τ (2) 1 T2 −1(τ (5)1 ) + a0τ (2) 2 T2 −1(τ (3)2 )) qa2a32τ (1) 1 , (2.9m) T2 −1(τ (2) 2 ) = a2a3 ( q−1a2T2 −1(τ (1)1 ) T2 −1(τ (5)2 ) + a0T2 −1(τ (5)1 ) T2 −1(τ (2)1 )) a4a02T2 −1(τ (3)2 ) , (2.9n) T3 ( τ (4) 1 ) = qa3 2a2 ( a1τ (4) 1 T3 ( τ (1) 1 ) + a3a4τ (2) 2 T3 ( τ (1) 2 )) a1τ (1) 1 , (2.10a) T3 ( τ (5) 1 ) = a3a4 ( qa3τ (2) 2 T3 ( τ (1) 2 ) + a0a1τ (4) 1 T3 ( τ (1) 1 )) a12τ (4) 2 , (2.10b) T3 ( τ (1) 1 ) = a1a2 ( a3a4τ (4) 1 τ (1) 1 + a1τ (5) 1 τ (4) 2 ) a42τ (3) 1 , (2.10c) T3 ( τ (4) 2 ) = a3a4 ( qa3τ (2) 2 T3 ( τ (1) 2 ) + a1τ (4) 1 T3 ( τ (1) 1 )) a0a12τ (5) 1 , (2.10d) T3 ( τ (5) 2 ) = a4a0 ( q−1a4τ (4) 1 T3(τ (2) 2 ) + a2τ (2) 2 T3 ( τ (4) 1 )) qa1a22T3 ( τ (4) 2 ) , (2.10e) T3 ( τ (1) 2 ) = a1 ( a4τ (4) 1 τ (1) 1 + a1a2τ (5) 1 τ (4) 2 ) a3a42a2τ (2) 1 , (2.10f) T3(τ (2) 2 ) = a1a2 ( a4T3 ( τ (4) 1 ) T3 ( τ (1) 1 ) + qa1T3 ( τ (5) 1 ) T3 ( τ (4) 2 )) a3a42T3 ( τ (1) 2 ) , (2.10g) 8 N. Nakazono T3 −1(τ (1)1 ) = a3 ( a1τ (1) 1 τ (3) 1 + a3a4τ (2) 1 τ (1) 2 ) a4a0a12τ (4) 1 , (2.10h) T3 −1(τ (2)1 ) = a1 ( qa4τ (3) 1 T3 −1(τ (1)1 ) + a1a2τ (3) 2 T3 −1(τ (4)2 )) qa3a42a2τ (1) 2 , (2.10i) T3 −1(τ (3)1 ) = a1a2 ( a3a4τ (3) 1 T3 −1(τ (1)1 ) + a1τ (3) 2 T3 −1(τ (4)2 )) q2a42τ (1) 1 , (2.10j) T3 −1(τ (4)2 ) = a3a4 ( a0a1τ (1) 1 τ (3) 1 + a3τ (2) 1 τ (1) 2 ) a12τ (5) 1 , (2.10k) T3 −1(τ (5)2 ) = qa4a0 ( qa4τ (2) 1 T3 −1(τ (3)1 ) + a2τ (3) 1 T3 −1(τ (2)1 )) a1a22T3 −1(τ (4)2 ) , (2.10l) T3 −1(τ (1)2 ) = a1a2 ( qa4τ (3) 1 T3 −1(τ (1)1 ) + a1τ (3) 2 T3 −1(τ (4)2 )) qa3a42τ (2) 1 , (2.10m) T3 −1(τ (3)2 ) = a3a4 ( q−1a3T3 −1(τ (2)1 ) T3 −1(τ (1)2 ) + a1T3 −1(τ (1)1 ) T3 −1(τ (3)1 )) a0a12T3 −1(τ (4)2 ) . (2.10n) Moreover, by using the action of T1, we obtain the following lemma. Lemma 2.2. The following discrete Toda type bilinear equations hold τ l0,l2,l3l1+1 τ l0,l2,l3l1−1 = q3l1−l2−l3 a0a1 a22a3 ( −1 + q−l0+l1a1 )( τ l0,l2,l3l1 )2 + q4(−l0+l1)a1 4τ l0+1,l2,l3 l1 τ l0−1,l2,l3l1 , (2.11a) τ l0,l2,l3l1+1 τ l0,l2,l3l1−1 = q−l0+4l1−l2−l3 a0a1 2 a22a3 ( 1− q−l1+l2a2 )( τ l0,l2,l3l1 )2 + q4(l1−l2)a2 −4τ l0,l2+1,l3 l1 τ l0,l2−1,l3l1 , (2.11b) τ l0,l2,l3l1+1 τ l0,l2,l3l1−1 = q−l0+3l1−l2 a1 a22a3a4 ( −1 + ql1−l3a0a1a4 )( τ l0,l2,l3l1 )2 + q4(l1−l3)a0 4a1 4a4 4τ l0,l2,l3+1 l1 τ l0,l2,l3−1l1 . (2.11c) Proof. The actions of T0, T1 −1 and T2 −1 on τ (1) 1 are given by T0 ( τ (1) 1 ) = qa0 2a3a4 2τ (1) 1 ( a0a1τ (1) 1 τ (3) 1 + a3τ (2) 1 τ (1) 2 ) a12τ (3) 1 τ (5) 1 + qa0 2τ (4) 2 ( a1τ (1) 1 τ (3) 1 + a3a4τ (2) 1 τ (1) 2 ) a1τ (3) 1 τ (4) 1 , (2.12) T1 −1(τ (1)1 ) = τ (1) 2 ( a3a4τ (1) 1 τ (4) 1 + a1τ (5) 1 τ (4) 2 ) qa22a3a4τ (3) 1 τ (4) 1 + a1τ (1) 1 ( a4τ (1) 1 τ (4) 1 + a1a2τ (5) 1 τ (4) 2 ) qa23a32a42τ (2) 1 τ (4) 1 , (2.13) T2 −1(τ (1)1 ) = a2 2τ (2) 1 ( a3a4τ (1) 1 τ (4) 1 + a1τ (5) 1 τ (4) 2 ) qa3a4τ (3) 1 τ (4) 1 + a1a2 2τ (1) 1 ( a4τ (1) 1 τ (4) 1 + a1τ (5) 1 τ (4) 2 ) qa32a42τ (1) 2 τ (4) 1 , (2.14) Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 9 respectively. Eliminating the terms τ (3) 1 , τ (4) 1 , τ (1) 2 and τ (4) 2 from equations (2.12) and (2.13), we obtain τ (2) 1 T1 −1(τ (1)1 ) = q−1 a0a1 a22a3 ( −1 + q−1a1 )( τ (1) 1 )2 + q−4a1 4T0 ( τ (1) 1 ) τ (5) 1 , (2.15) which is equivalent to equation (2.11a). Furthermore, eliminating the terms τ (3) 1 , τ (4) 1 , τ (5) 1 and τ (4) 2 from equations (2.13) and (2.14), we obtain τ (2) 1 T1 −1(τ (1)1 ) = q−2 a0a1 2 a22a3 (1− a2) ( τ (1) 1 )2 + a2 −4τ (1) 2 T2 −1(τ (1)1 ) , (2.16) which is equivalent to equation (2.11b). Eliminating the term τ (2) 1 T1 −1(τ (1)1 ) from equations (2.15) and (2.16), we obtain T0 ( τ (1) 1 ) τ (5) 1 = q2 a0 a12a2a3 (−1 + a4a0a3) ( τ (1) 1 )2 + a4 4a0 4a3 4τ (1) 2 T2 −1(τ (1)1 ) . (2.17) Applying the transformation σ on equation (2.17), we obtain T1 ( τ (2) 1 ) τ (1) 1 = q2 a1 a22a3a4 (−1 + a0a1a4) ( τ (2) 1 )2 + a0 4a1 4a4 4τ (2) 2 T3 −1τ (2) 1 , which is equivalent to equation (2.11c). Therefore, we have completed the proof. � Step 2. In this step, we get the explicit formulae for τ l0,l2,l30 and τ l0,l2,l31 . Letting τ l0,l2,l31 = τ l0,l2,l30 Hl0,l2,l3 , (2.18) where Hl0,l2,l3 = H ( ql0a0, q l2a2, q −l3a4 ) , we obtain the following lemma. Lemma 2.3. A solution of the system of the equations (2.1) and (2.8)–(2.10) are given by the solution of the following system under the condition (2.7): τ0,0,00 τ0,1,10 + a0a4 qa2 τ0,1,00 τ0,0,10 = 0, (2.19a) τ0,0,00 τ1,1,10 − qa42τ0,0,10 τ1,1,00 = 0, (2.19b) τ0,0,00 τ1,1,10 − 1 qa22 τ1,0,10 τ0,1,00 = 0, (2.19c) τ0,0,00 τ1,1,10 − q a02 τ0,1,10 τ1,0,00 = 0, (2.19d) τ1,0,00 τ−1,0,00 − a0 4a4(1− qa0−1) q3a2 ( τ0,0,00 )2 = 0, (2.19e) τ0,1,00 τ0,−1,00 + q2a2 3a4(1− a2) a0 ( τ0,0,00 )2 = 0, (2.19f) τ0,0,10 τ0,0,−10 − 1− qa4 q3a0a2a44 ( τ0,0,00 )2 = 0, (2.19g) H0,0,0 = q2a4H1,1,0 + q(1− qa4)H1,1,1, (H01) H0,0,0 = −q2a2a4H0,1,0 + q2a4 ( 1− q−1a0 ) H1,1,0, (H02) H0,0,0 = −q3a0−1a2a4H0,1,0 − q2a0−1 ( 1− q−1a0 ) (1− qa4)H1,1,1, (H03) 10 N. Nakazono H0,0,0 = −a0a2−1H1,0,0 − a2−1a4−1(1− qa4)H1,0,1, (H04) H0,0,0 = −q2a0−1a2H0,1,0 − qa0−1a4−1(1− qa4)H0,1,1, (H05) H0,0,0 = −qa0a4H1,0,0 + q2a4(1− a2)H1,1,0, (H06) H0,0,0 = −qa0a2−1a4H1,0,0 − qa2−1(1− a2)(1− qa4)H1,1,1, (H07) H0,0,0 = −a2−1a4−1H0,0,1 + qa2 −1(1− q−1a0)H1,0,1, (H08) H0,0,0 = q2a0 −1H0,1,1 − q2a0−1 ( 1− q−1a0 ) H1,1,1, (H09) a4(1− a0)H2,1,1H0,0,0 = qa4H1,1,0H1,0,1 − a0a4H1,0,0H1,1,1. (H10) Proof. By using notation (2.4) and relation (2.18), equations (2.1) and (2.8)–(2.10) can be classified by the type of contiguity relations of function Hl0,l2,l3 (see Figs. 1–4) as the following table: Type Equation number Type 1 (2.1a), (2.8d), (2.8n), (2.9g), (2.9m) Type 2 (2.1b), (2.8c), (2.8e), (2.8j), (2.8l), (2.10b), (2.10g), (2.10k), (2.10m) Type 3 (2.1c), (2.8g), (2.8m), (2.9e), (2.9l), (2.10d), (2.10n) Type 4 (2.8a), (2.8h), (2.9d), (2.9f), (2.9i), (2.9n) Type 5 (2.8b), (2.8f), (2.8i), (2.8k), (2.9c), (2.9j), (2.10a), (2.10h) Type 6 (2.9a), (2.9h) Type 7 (2.9b), (2.9k) Type 8 (2.10c), (2.10j) Type 9 (2.10f), (2.10i) Type 10 (2.10e), (2.10l) Under the condition (2.7), comparing the coefficients of Hl0,l2,l3 in the same types, for example (2.1a)≡ (2.8d), and substituting the boundary condition (2.6) in equations (2.11) with l1 = 0, we obtain equations (2.19). Moreover, by using the relations (2.19), the equations of Type 1, . . . , Type 10 are given by equations (H01)–(H10), respectively. Therefore, we have completed the proof. � We can easily verify the following lemma by the direct calculation. Lemma 2.4. A solution of system (2.19) is given by τ l0,l2,l30 = ( ql0a0; q, q ) ∞ ( ql2+1a2; q, q ) ∞ ( ql3a4 −1; q, q ) ∞Kl0,l2,l3 , where Kl0,l2,l3 (2.20) = ( Γ ( ql0+l2+1a0a2; q, q ) Γ ( ql2+l3+1a2a4 −1; q, q ) Γ ( ql3+l0a4 −1a0; q, q ))2 Γ ( ql0+l2+l3+1a0a2a4−1; q, q )( Γ ( ql0+1/3a0; q, q ) Γ ( ql2+4/3a2; q, q ) Γ ( ql3+1/3a4−1; q, q ))6 . We consider a solution of system of the equations (H01)–(H10). First, we get the essential relations for the function Hl0,l2,l3 . Lemma 2.5. If the function Hl0,l2,l3 satisfies equations (H02), (H06) and (H08) and the fol- lowing three-term relation qa0a4(a0 − q)H1,0,0 + (a0 − qa2 + qa0a2a4)H0,0,0 + qa2H−1,0,0 = 0, (H11) then it also satisfies equations (H01), (H03), (H04), (H05), (H07), (H09) and (H10). Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 11 Figure 1. Left: Type 1, center: Type 2, right: directions. Figure 2. Left: Type 3, center: Type 4, right: Type 5. Figure 3. Left: Type 6, center: Type 7, right: Type 8. Figure 4. Left: Type 9, right: Type 10. Proof. Equation (H04) can be obtained by using equations (H08) and (H11) as follows. Erasing the term H1,0,1 from equations (H08) and (H11)3, we obtain a0a4H0,0,0 + (q − a0a4)H0,0,1 − qH−1,0,1 = 0. (H12) We note that a subscript i of equation number means Ti-shifted corresponding equation. More- over, erasing the term H0,0,1 from equations (H12)0 and (H08), we obtain equation (H04). This procedure is described in Fig. 5. 12 N. Nakazono In a similar manner, we can derive equations (H01), (H03), (H07), (H05) and (H09) as shown in Figs. 6–10, respectively. On the other hand, we can prove equation (H10) by reducing it to equation (H06) with equations (H02), (H04) and (H07) as shown in Fig. 11. Therefore, we have completed the proof. � Figure 5. Derivation of equation (H04). The black points are removed. Figure 6. Derivation of equation (H01). Figure 7. Derivation of equation (H03). Figure 8. Derivation of equation (H07). Figure 9. Derivation of equation (H05). Figure 10. Derivation of equation (H09). Figure 11. Reduction from equation (H10) to equation (H06). Next, we solve the essential relations for the function Hl0,l2,l3 , that is, equations (H02), (H06), (H08) and (H11). Lemma 2.6. The solution of the system of equations (H02), (H06), (H08) and (H11) are given by Hl0,l2,l3 = q2/3 ( q−l0−l2−l3+2 a4 a0a2 )1/2 ( ql0−1a0 −1; q ) ∞Gl0,l2,l3 , where Gl0,l2,l3 = C1q (−2l2+l3)/2a2 −1a4 −1/2 ( q−l2+l3a2 −1a4 −1; q ) ∞( q−l2+1a2−1; q ) ∞ Θ ( q−l0+l2−1/2a0 −1a2; q ) Θ ( q−l0a0−1; q ) Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 13 × 1ϕ1 ( q−l2+1a2 −1 q−l2+l3a2 −1a4 −1; q, q −l0+l3a0 −1a4 −1 ) + C2q (l2−2l3)/2a2 1/2a4 ( ql2−l3+2a2a4; q ) ∞( q−l3+2a4; q ) ∞ Θ ( q−l0+l3−3/2a0 −1a4 −1; q ) Θ ( q−l0a0−1; q ) × 1ϕ1 ( q−l3+2a4 ql2−l3+2a2a4 ; q, q−l0+l2+1a0 −1a2 ) . (2.21) Here, Ci = Ci(l0, l2, l3), i = 1, 2, are periodic functions of period one for l0, l2, l3 ∈ Z, i.e., Ci(l0 + 1, l2, l3) = Ci(l0, l2 + 1, l3) = Ci(l0, l2, l3 + 1) = Ci(l0, l2, l3). Proof. By letting Hl0,l2,l3 = q2/3 ( q−l0−l2−l3+2 t αβ )1/2 ( ql0−1t−1; q ) ∞G ( q−l0t, ql2α, ql3β ) , where t = a0 −1, α = a2, β = a4 −1, equations (H02), (H06), (H08) and (H11) can be rewritten as the following βG(qt, α, β)− qG(t, qα, β) + q3/2αG(qt, qα, β) = 0, (2.22) β ( 1− q2t ) G(qt, α, β) + q3(1− α)tG(t, qα, β)− q3/2G(t, α, β) = 0, (2.23) q1/2αG(qt, α, β)− q1/2G(t, α, qβ) + βG(qt, α, qβ) = 0, (2.24) αβ ( q3t− 1 ) G ( q2t, α, β ) − ( q5/2αβt− q1/2(qα+ β) ) G(qt, α, β)− q2G(t, α, β) = 0, (2.25) respectively. Substituting G(t, α, β) = ∞∑ n=0 cnt n+ρ, where cn = cn(α, β), in equation (2.25), we obtain G(t, α, β) = A(α, β) Θ ( q−1/2αt; q ) Θ(t; q) 1ϕ1 ( qα−1 α−1β ; q, βt ) +B(α, β) Θ ( q−3/2βt; q ) Θ(t; q) 1ϕ1 ( q2β−1 q2αβ−1 ; q, qαt ) , (2.26) where A(α, β) and B(α, β) are arbitrary functions. Moreover, substituting (2.26) in equations (2.22), (2.23) and (2.24), we obtain the following relations A(qα, β) = 1− q−1α−1β q ( 1− α−1 ) A(α, β), A(α, qβ) = q1/2 1− α−1β A(α, β), (2.27a) B(qα, β) = q1/2 1− q2αβ−1 B(α, β), B(α, qβ) = 1− qαβ−1 q ( 1− qβ−1 )B(α, β), (2.27b) which can be solved by A(α, β) = α−1β1/2 ( α−1β; q ) ∞( qα−1; q ) ∞ , B(α, β) = α1/2β−1 ( q2αβ−1; q ) ∞( q2β−1; q ) ∞ . 14 N. Nakazono To obtain the relations (2.27), we used the following recurrence relations of hypergeometric series 1ϕ1: 1ϕ1 ( a b ; q, z ) − q + b− q2z q 1ϕ1 ( a b ; q, qz ) − q2az − b q 1ϕ1 ( a b ; q, q2z ) = 0, 1ϕ1 ( a b ; q, z ) = 1ϕ1 ( a b ; q, qz ) + (a− 1)z 1− b 1ϕ1 ( qa qb ; q, qz ) , 1ϕ1 ( a b ; q, z ) = 1 1− b 1ϕ1 ( a qb ; q, z ) − b 1− b 1ϕ1 ( a qb ; q, qz ) , which can be verified by the direct calculation. Therefore, we have completed the proof. � Step 3. In this final step, we give the hypergeometric τ functions of W̃ ( A (1) 4 ) -type. Substituting τ l0,l2,l3l1 = q2l1 3/3 ( q−l0−l2−l3+2 a4 a0a2 )l12/2 (ql0−l1a0 −1; q, q)∞ (ql0a0−1; q, q)∞ τ l0,l2,l30 Φl0,l2,l3 l1 , in equation (2.11a), we obtain the following bilinear equation Φl0,l2,l3 l1+1 Φl0,l2,l3 l1−1 = (Φl0,l2,l3 l1 )2 − Φl0+1,l2,l3 l1 Φl0−1,l2,l3 l1 . (2.28) In general, equation (2.28) admits a solution expressed in terms of Jacobi–Trudi type determi- nant Φl0,l2,l3 l1 = det(cl0+i−j,l2,l3)i,j=1,...,l1 , where l1 ∈ Z>1, under the boundary conditions Φl0,l2,l3 l1 = 0 (l1 < 0), Φl0,l2,l3 0 = 1, Φl0,l2,l3 1 = cl0,l2,l3 , where cl0,l2,l3 is an arbitrary function. Therefore, we obtain the following theorem. Theorem 2.7. The hypergeometric τ functions of W̃ (A (1) 4 )-type are given by the following τ l0,l2,l30 = ( ql0a0; q, q ) ∞ ( ql2+1a2; q, q ) ∞ ( ql3a4 −1; q, q ) ∞Kl0,l2,l3 , τ l0,l2,l3l1 = q2l1 3/3 ( q−l0−l2−l3+2 a4 a0a2 )l12/2 (ql0−l1a0−1; q, q)∞( ql0a0−1; q, q ) ∞ τ l0,l2,l30 Φl0,l2,l3 l1 , where l0, l2, l3 ∈ Z, l1 ∈ Z>0 and the functions { Φl0,l2,l3 l1 } li∈Z, l1>0 are given by the following l1 × l1 determinants Φl0,l2,l3 l1 = ∣∣∣∣∣∣∣∣∣ Gl0,l2,l3 Gl0+1,l2,l3 · · · Gl0+l1−1,l2,l3 Gl0−1,l2,l3 Gl0,l2,l3 · · · Gl0+l1−2,l2,l3 ... ... · · · ... Gl0−l1+1,l2,l3 Gl0−l1+2,l2,l3 · · · Gl0,l2,l3 ∣∣∣∣∣∣∣∣∣ . (2.29) Here, the functions Kl0,l2,l3 and Gl0,l2,l3 are given in equations (2.20) and (2.21), respectively. Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 15 2.3 Discrete Painlevé equations Let us define the ten f -variables by f (j) 1 = τ (j+1) 1 τ (j) 2 τ (j) 1 τ (j+2) 1 , f (j) 2 = ajaj+1 aj+3 2 τ (j+1) 1 ( aj+2aj+3τ (j) 1 τ (j+3) 1 + ajτ (j+4) 1 τ (j+3) 2 ) τ (j) 1 τ (j+2) 1 τ (j+1) 2 , (2.30) where j ∈ Z/5Z. From the definition above and conditions (2.1), the following eight relations hold f (j) 2 = ajaj+1 ( aj+2aj+3 + ajf (j+3) 1 ) aj+3 2f (j+1) 1 , j ∈ Z/5Z, a4a0 2f (2) 1 f (3) 1 = a2a3 ( a0 + a2f (5) 1 ) , a0a1 2f (3) 1 f (4) 1 = a3a4 ( a1 + a3f (1) 1 ) , a2a3 2f (5) 1 f (1) 1 = a0a1 ( a3 + a0f (3) 1 ) . Therefore, the f -variables are essentially two. The action of W̃ ( A (1) 4 ) on these variables f (j) i is given by the following lemma, which follows from the actions (2.2). Lemma 2.8. The action of W̃ ( A (1) 4 ) on variables f (j) i is given by sj ( f (j+3) 1 ) = f (j+3) 2 , sj ( f (j+3) 2 ) = f (j+3) 1 , sj ( f (j) 1 ) = aj+4 ajaj+1aj+2 2 aj+2 + ajaj+4f (j+2) 1 f (j+4) 1 , sj ( f (j+2) 2 ) = ajaj+3aj+4 aj+1 aj+2 + ajaj+4f (j+2) 1 + ajaj+1aj+2f (j+4) 1 f (j+4) 1 f (j+3) 2 , sj ( f (j+4) 2 ) = ajaj+1aj+2 2 aj+4 f (j+4) 1 f (j) 1 f (j+4) 2 aj+2 + ajaj+4f (j+2) 1 , sj ( f (j) 2 ) = ajaj+1aj+4 + aj+3aj+4f (j+1) 1 + ajaj+1 2aj+2f (j+4) 1 ajaj+1aj+3f (j+1) 1 f (j+4) 1 , π ( f (j) i ) = f (j+1) i , ι ( f (j) 1 ) = f (3−j) 1 , ι ( f (j) 2 ) = a2−j ( a5−j + a2−ja3−jf (5−j) 1 ) a3−ja4−ja5−j2f (2−j) 1 , where i = 1, 2 and j ∈ Z/5Z. It is well known that the translation part of W̃ ( A (1) 4 ) give discrete Painlevé equations [56]. Let X (i) l1 = T1 l1 ( f (i) 1 ) , Y (i) l1 = T1 l1 ( f (i) 2 ) , α (i) l1 = T1 l1(ai) =  ql1a1 if i = 1, q−l1a2 if i = 2, ai otherwise. (2.31) The action of Ti: Ti : (ai, ai+1) 7→ ( qai, q −1ai+1 ) , Ti ( f (i+3) 1 ) f (i+3) 1 = ai+3 ai2ai+1 2ai+4 ( ai+1 + ai+3ai+4f (i+1) 1 )( ai+1 + ai+3f (i+1) 1 ) aiai+1 + ai+3f (i+1) 1 , 16 N. Nakazono Ti −1(f (i+1) 1 ) f (i+1) 1 = aiai+1 3 ai+3 2 ( ai+2ai+3 + aif (i+3) 1 )( ai+3 + aif (i+3) 1 ) ai+3 + aiai+1f (i+3) 1 , where i ∈ Z/5Z, lead a q-discrete analogue of Painlevé V equation [56] Ti ( X (i+3) l1 ) X (i+3) l1 = α (i+3) l1( α (i) l1 )2( α (i+1) l1 )2 α (i+4) l1 × ( α (i+1) l1 + α (i+3) l1 α (i+4) l1 X (i+1) l1 )( α (i+1) l1 + α (i+3) l1 X (i+1) l1 ) α (i) l1 α (i+1) l1 + α (i+3) l1 X (i+1) l1 , Ti −1(X(i+1) l1 ) X (i+1) l1 = α (i) l1 ( α (i+1) l1 )3( α (i+3) l1 )2 ( α (i+2) l1 α (i+3) l1 + α (i) l1 X (i+3) l1 )( α (i+3) l1 + α (i) l1 X (i+3) l1 ) α (i+3) l1 + α (i) l1 α (i+1) l1 X (i+3) l1 . (2.32) Moreover, the action of T (i) 23 = Ti+2Ti+3: T (i) 23 : (ai+2, ai+4) 7→ ( qai+2, q −1ai+4 ) ,( T (i) 23 ( f (i+2) 2 ) f (i+3) 1 − ai+2ai+3ai+4 ai )( f (i+2) 2 f (i+3) 1 − ai+2ai+3ai+4 ai ) = ai+2 3ai+3ai+4 ai2 ( ai+3 + aif (i+3) 1 )( ai+3 + aiai+1f (i+3) 1 ) ai+2ai+3 + aif (i+3) 1 ,( f (i+2) 2 f (i+3) 1 − ai+2ai+3ai+4 ai )( f (i+2) 2 T (i) 23 −1( f (i+3) 1 ) − ai+2ai+3ai+4 ai ) = ai+2ai+3 ai2ai+1ai+4 ( ai+1ai+2ai+4 + f (i+2) 2 )( ai+2ai+4 + f (i+2) 2 ) ai+2 + aif (i+2) 2 , where i ∈ Z/5Z, and that of T (i) 13 = Ti+1Ti+3: T (i) 13 : (ai+1, ai+2, ai+3, ai+4) 7→ ( qai+1, q −1ai+2, qai+3, q −1ai+4 ) ,( T (i) 13 ( f (i+1) 1 ) f (i+2) 1 − ai+1ai+2 ai+3ai+4 )( f (i+1) 1 f (i+2) 1 − ai+1ai+2 ai+3ai+4 ) = ai+1 3ai+2 ai+3ai+4 2 ( ai+2 + aiai+4f (i+2) 1 )( ai+2 + ai+4f (i+2) 1 ) ai+1ai+2 + ai+4f (i+2) 1 ,( f (i+1) 1 f (i+2) 1 − ai+1ai+2 ai+3ai+4 )( f (i+1) 1 T (i) 13 −1( f (i+2) 1 ) − ai+1ai+2 ai+3ai+4 ) = ai+1ai+2 aiai+3 2ai+4 2 ( ai+1 + ai+3f (i+1) 1 )( aiai+1 + ai+3f (i+1) 1 ) ai+1 + ai+3ai+4f (i+1) 1 , where i ∈ Z/5Z, respectively give the systems( T (i) 23 ( Y (i+2) l1 ) X (i+3) l1 − α (i+2) l1 α (i+3) l1 α (i+4) l1 α (i) l1 )( Y (i+2) l1 X (i+3) l1 − α (i+2) l1 α (i+3) l1 α (i+4) l1 α (i) l1 ) = α (i+2) l1 3 α (i+3) l1 α (i+4) l1( α (i) l1 )2 ( α (i+3) l1 + α (i) l1 X (i+3) l1 )( α (i+3) l1 + α (i) l1 α (i+1) l1 X (i+3) l1 ) α (i+2) l1 α (i+3) l1 + α (i) l1 X (i+3) l1 , (2.33a) Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 17( Y (i+2) l1 X (i+3) l1 − α (i+2) l1 α (i+3) l1 α (i+4) l1 α (i) l1 )( Y (i+2) l1 T (i) 23 −1( X (i+3) l1 ) − α (i+2) l1 α (i+3) l1 α (i+4) l1 α (i) l1 ) = α (i+2) l1 α (i+3) l1( α (i) l1 )2 α (i+1) l1 α (i+4) l1 ( α (i+1) l1 α (i+2) l1 α (i+4) l1 + Y (i+2) l1 )( α (i+2) l1 α (i+4) l1 + Y (i+2) l1 ) α (i+2) l1 + α (i) l1 Y (i+2) l1 , (2.33b) and ( T (i) 13 ( X (i+1) l1 ) X (i+2) l1 − α (i+1) l1 α (i+2) l1 α (i+3) l1 α (i+4) l1 )( X (i+1) l1 X (i+2) l1 − α (i+1) l1 α (i+2) l1 α (i+3) l1 α (i+4) l1 ) = ( α (i+1) l1 )3 α (i+2) l1 α (i+3) l1 ( α (i+4) l1 )2 ( α (i+2) l1 + α (i) l1 α (i+4) l1 X (i+2) l1 )( α (i+2) l1 + α (i+4) l1 X (i+2) l1 ) α (i+1) l1 α (i+2) l1 + α (i+4) l1 X (i+2) l1 , (2.34a) ( X (i+1) l1 X (i+2) l1 − α (i+1) l1 α (i+2) l1 α (i+3) l1 α (i+4) l1 )( X (i+1) l1 T (i) 13 −1( X (i+2) l1 ) − α (i+1) l1 α (i+2) l1 α (i+3) l1 α (i+4) l1 ) = α (i+1) l1 α (i+2) l1 α (i) l1 ( α (i+3) l1 )2( α (i+4) l1 )2 ( α (i+1) l1 + α (i+3) l1 X (i+1) l1 )( α (i) l1 α (i+1) l1 + α (i+3) l1 X (i+1) l1 ) α (i+1) l1 + α (i+3) l1 α (i+4) l1 X (i+1) l1 . (2.34b) Systems (2.33) and (2.34) are also known as q-discrete analogues of Painlevé V equation [57]. From equation (2.5), definitions (2.30) and (2.31) and Theorem 2.7, we obtain the following corollary. Corollary 2.9. Under the condition (2.7), the hypergeometric solutions of q-Painlevé equations (2.32), (2.33) and (2.34) are given by X (1) l1 = ql1+1/2 Φ1,1,1 l1 Φ1,0,1 l1+1 Φ1,0,1 l1 Φ1,1,1 l1+1 , X (2) l1 = − qa2 a0a4 Φ1,0,2 l1+1Φ 1,1,1 l1+1 Φ1,0,1 l1+1Φ 1,1,2 l1+1 , X (3) l1 = 1− a4 ql1+1/2a0a2a42 Φ0,0,0 l1 Φ1,1,2 l1+1 Φ0,0,1 l1 Φ1,1,1 l1+1 , X (4) l1 = − a0a4 ql1+1/2a2 Φ0,0,1 l1 Φ2,1,2 l1+1 Φ1,0,1 l1 Φ1,1,2 l1+1 , X (5) l1 = ql1+1 − a0 q1/2 Φ1,0,1 l1 Φ0,0,1 l1+1 Φ0,0,1 l1 Φ1,0,1 l1+1 , Y (1) l1 = 1 q1/2a2 Φ1,0,1 l1+1Φ 2,1,2 l1+1 Φ1,0,2 l1+1Φ 1,1,1 l1+1 ( Φ0,0,1 l1 Φ1,0,1 l1 − q1/2 a4 Φ1,1,2 l1+1 Φ2,1,2 l1+1 ) , Y (2) l1 = q1/2a2a4 2 1− a4 Φ1,0,1 l1 Φ1,1,1 l1+1 Φ0,0,0 l1 Φ1,1,2 l1+1 ( Φ0,0,1 l1 Φ1,0,1 l1 + a2 ( ql1+1 − a0 ) ql1+1/2a0a4 Φ0,0,1 l1+1 Φ1,0,1 l1+1 ) , Y (3) l1 = −q 1/2a0 a4 Φ1,1,1 l1 Φ1,1,2 l1+1 Φ0,0,1 l1 Φ2,1,2 l1+1 ( Φ1,0,1 l1 Φ1,1,1 l1 + 1 q1/2a2a4 Φ1,0,1 l1+1 Φ1,1,1 l1+1 ) , Y (4) l1 = q2l1+1/2a4 a2 ( ql1+1 − a0 ) Φ0,0,1 l1 Φ1,0,2 l1+1 Φ1,0,1 l1 Φ0,0,1 l1+1 ( Φ1,0,1 l1+1 Φ1,0,2 l1+1 − Φ1,1,1 l1+1 Φ1,1,2 l1+1 ) , Y (5) l1 = a2(1− a4) ql1 Φ1,0,1 l1 Φ1,1,2 l1+1 Φ1,1,1 l1 Φ1,0,1 l1+1 ( Φ0,0,0 l1 Φ0,0,1 l1 + q1/2a2a4 1− a4 Φ1,1,1 l1+1 Φ1,1,2 l1+1 ) , where the functions { Φl0,l2,l3 l1 } l1∈Z≥0 are def ined by (2.29). Note that the actions of translations Ti, i = 0, . . . , 4, on these solutions are given by the following T0 : ( a0, a2, a4, l1, q,Φ l0,l2,l3 l1 ) 7→ ( qa0, a2, a4, l1, q,Φ l0+1,l2,l3 l1 ) , 18 N. Nakazono T1 : ( a0, a2, a4, l1, q,Φ l0,l2,l3 l1 ) 7→ ( a0, q −1a2, a4, l1 + 1, q,Φl0,l2,l3 l1+1 ) , T2 : ( a0, a2, a4, l1, q,Φ l0,l2,l3 l1 ) 7→ ( a0, qa2, a4, l1, q,Φ l0,l2+1,l3 l1 ) , T3 : ( a0, a2, a4, l1, q,Φ l0,l2,l3 l1 ) 7→ ( a0, a2, q −1a4, l1, q,Φ l0,l2,l3+1 l1 ) , T4 : ( a0, a2, a4, l1, q,Φ l0,l2,l3 l1 ) 7→ ( q−1a0, a2, qa4, l1 − 1, q,Φl0−1,l2−1,l3−1 l1−1 ) . 3 Hypergeometric τ functions of W̃ ( (A1 +A′ 1) (1) ) -type In this section, we construct the hypergeometric τ functions of W̃ ( (A1 +A′1) (1) ) -type. 3.1 τ functions The action of the transformation group W̃ ( (A1 + A′1) (1) ) = 〈s0, s1, w0, w1, π〉 on the parame- ters a0, a1 and b are given by s0 : (a0, a1, b) 7→ ( 1 a0 , a0 2a1, b a0 ) , s1 : (a0, a1, b) 7→ ( a0a1 2, 1 a1 , a1b ) , w0 : (a0, a1, b) 7→ ( 1 a0 , 1 a1 , b a0 ) , w1 : (a0, a1, b) 7→ ( 1 a0 , 1 a1 , b a02a1 ) , π : (a0, a1, b) 7→ ( 1 a1 , 1 a0 , b a0a1 ) , while its actions on the variables τi, i = −3, . . . , 3, are given by s0 : (τ−3, τ−1, τ1) 7→ ( a0τ1τ−2 2 + τ−1τ0τ−2 + τ−3τ0 2 a0τ−1τ1 , a0τ0 2 + bτ−2τ2 a0τ1 , bτ−2τ2 + τ0 2 τ−1 ) , s1 : (τ−2, τ0) 7→ ( a0a1τ−1 2 + bτ−3τ1 a0a1τ0 , a0τ−1 2 + bτ−3τ1 a0τ−2 ) , w0 : (τ−3, τ−2, τ−1, τ1) 7→ (τ3, τ2, τ1, τ−1) , w1 : (τ−3, τ−2, τ0, τ1) 7→ (τ1, τ0, τ−2, τ−3) , π : (τ−3, τ−2, τ−1, τ0, τ1) 7→ (τ2, τ1, τ0, τ−1, τ−2) , where τ2 = a0 (τ−1τ0 + τ−2τ1) bτ−3 , τ3 = τ0τ1 + τ−1τ2 bτ−2 . For each element w ∈ W̃ ( (A1 + A′1) (1) ) and function F = F (ai, b, τj), we use the notation w.F to mean w.F = F (w.ai, w.b, w.τj), that is, w acts on the arguments from the left. We note that the group of transformations W̃ ( (A1 + A′1) (1) ) forms the extended affine Weyl group of type (A1 +A1) (1) [25]. Namely, the transformations satisfy the fundamental relations s0 2 = s1 2 = (s0s1) ∞ = 1, w0 2 = w1 2 = (w0w1) ∞ = 1, π2 = 1, πs0 = s1π, πw0 = w1π, and the action of W ( A (1) 1 ) = 〈s0, s1〉 and that of W ( A (1) 1 ′) = 〈w0, w1〉 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. To iterate each variable τi, we need the translations Ti, i = 1, 2, 3, defined by T1 = w0w1, T2 = πs1w0, T3 = πs0w0. Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 19 Note that Ti, i = 1, 2, 3, commute with each other and T1T2T3 = 1. The actions of these on the parameters are given by T1 : (a0, a1, b) 7→ (a0, a1, qb), T2 : (a0, a1, b) 7→ ( qa0, q −1a1, b ) , T3 : (a0, a1, b) 7→ ( q−1a0, qa1, q −1b ) , where the parameter q = a0a1 is invariant under the action of translations. We define τ functions by τ l1l2 = T1 l1T2 l2(τ−3), where l1, l2 ∈ Z. We note that τ−3 = τ00 , τ−2 = τ11 , τ−1 = τ10 , τ0 = τ21 , τ1 = τ20 , τ2 = τ31 , τ3 = τ30 . 3.2 Discrete Painlevé equations Let f0 = τ−2τ1 τ−1τ0 , f1 = τ−3τ0 τ−2τ−1 , f2 = (τ−1) 2 τ−3τ1 , where f0f1f2 = 1. The action of W̃ ( (A1 +A′1) (1) ) on the variables fi, i = 0, 1, 2, is given by s0 : (f0, f1, f2) 7→ ( f0(a0f0 + a0 + f1) f0 + f1 + 1 , f1(a0f0 + f1 + 1) a0(f0 + f1 + 1) , a0f2(f0 + f1 + 1)2 (a0f0 + a0 + f1)(a0f0 + f1 + 1) ) , s1 : (f0, f1) 7→ ( f0(a0a1 + bf0f1) a1(a0 + bf0f1) , a1f1(a0 + bf0f1) a0a1 + bf0f1 ) , w0 : (f0, f1, f2) 7→ ( a0(f0 + 1) bf0f1 , a0f0 + a0 + bf0f1 a0bf0(f0 + 1) , b2f0 f2(a0f0 + a0 + bf0f1) ) , w1 : (f0, f1) 7→ (f1, f0) , π : (f1, f2) 7→ ( a0(f0 + 1) bf0f1 , bf1 a0(f0 + 1) ) . By letting f (0) l2 = T2 l2(f0), f (1) l2 = T2 l2(f1), f (2) l2 = T2 l2(f2), the actions of Ti, i = 1, 2, 3: T1(f1)f1 = a0(f0 + 1) bf0 , T1(f0)f0 = T1(f1) + 1 bT1(f1) , T2(f2)f2 = b qf1(f1 + 1) , T2(f1)f1 = a0(b+ qT2(f2)) T2(f2)(qa0T2(f2) + b) , T3(f0)f0 = a1b+ qf2 f2(b+ qf2) , T3(f2)f2 = a1b qT3(f0)(T3(f0) + 1) , 20 N. Nakazono lead the following q-Painlevé equations T1 ( f (1) l2 ) f (1) l2 = ql2a0 ( f (0) l2 + 1 ) bf (0) l2 , T1 ( f (0) l2 ) f (0) l2 = T1 ( f (1) l2 ) + 1 bT1 ( f (1) l2 ) , (3.1) T2 ( f (2) l2 ) f (2) l2 = b qf (1) l2 ( f (1) l2 + 1 ) , T2 ( f (1) l2 ) f (1) l2 = ql2a0 ( b+ qT2 ( f (2) l2 )) T2 ( f (2) l2 )( ql2+1a0T2 ( f (2) l2 ) + b ) , (3.2) T3 ( f (0) l2 ) f (0) l2 = a1b+ ql2+1f (2) l2 ql2f (2) l2 ( b+ qf (2) l2 ) , T3 ( f (2) l2 ) f (2) l2 = a1b ql2+1T3 ( f (0) l2 )( T3 ( f (0) l2 ) + 1 ) . (3.3) We note that equation (3.1) is known as a q-discrete analogue of Painlevé II equation [39] and can be rewritten as the following single second-order ordinary difference equation [52, 54, 56]:( T1 ( f (0) l2 ) f (0) l2 − 1 b )( T1 −1(f (0)l2 ) f (0) l2 − q b ) = a1 ql2b f (0) l2 1 + f (0) l2 . (3.4) 3.3 Hypergeometric τ functions We here define hypergeometric τ functions of W̃ ( (A1+A′1) (1) ) -type by τ l1l2 satisfying the following conditions: (i) τ l1l2 satisfy the action of the translation subgroup of W̃ ( (A1 +A′1) (1) ) , 〈T1, T2, T3〉; (ii) τ l1l2 are functions in b consistent with the action of T1, i.e., τ l1l2 = τl2(ql1b); (iii) τ l1l2 satisfy the following boundary conditions: τ l1l2 = 0, for l2 < 0; under the conditions of parameters a0 = 1, a1 = q. (3.5) In a similar manner as Section 2.2, we obtain the following theorem. Theorem 3.1. The hypergeometric τ functions of W̃ ( (A1 + A′1) (1) ) -type are given by the fol- lowing τ l10 = Γ ( ql1b; q, q ) , τ l1l2 = Γ ( ql1b; q, q ) Θ ( ql1b; q )l2 ψl1l2 , where l1 ∈ Z, l2 ∈ Z>0 and the functions { ψl1l2 } l1∈Z, l2∈Z>0 are given by the following l2 × l2 determinants ψl1l2 = ∣∣∣∣∣∣∣∣∣ Fl1 Fl1+1 · · · Fl1+l2−1 Fl1−1 Fl1 · · · Fl1+l2−2 ... ... . . . ... Fl1−l2+1 Fl1−l2+2 · · · Fl1 ∣∣∣∣∣∣∣∣∣ . Here, the function Fn is given by Fn = Θ ( −q(2n−3)/4b1/2; q1/2 ) qnb ( An 1ϕ1 ( 0 −q1/2; q 1/2,−q(2n−5)/4b1/2 ) +Bne πi log b/ log q 1ϕ1 ( 0 −q1/2; q 1/2, q(2n−5)/4b1/2 )) , where An and Bn are periodic functions of period one with respect to n, that is, An+1 = An, Bn+1 = Bn. Hypergeometric τ Functions of the q-Painlevé Systems of Types A (1) 4 and (A1 +A′1) (1) 21 Moreover, Theorem 3.1 leads the following corollary. Corollary 3.2. Under the condition (3.5), the hypergeometric solutions of q-Painlevé equations (3.1)–(3.4) are given by f (0) l2 = − 1 qb ψ1 l2+1ψ 2 l2 ψ1 l2 ψ2 l2+1 , f (1) l2 = ql2+1 ψ0 l2 ψ2 l2+1 ψ1 l2+1ψ 1 l2 , f (2) l2 = − b ql2 (ψ1 l2 )2 ψ0 l2 ψ2 l2 . Note that the actions of translations Ti, i = 1, 2, 3, on these solutions are given by the following T1 : ( b, q, ψl1l2 ) 7→ ( qb, q, ψl1+1 l2 ) , T2 : ( b, q, ψl1l2 ) 7→ ( b, q, ψl1l2+1 ) , T3 : ( b, q, ψl1l2 ) 7→ ( q−1b, q, ψl1−1l2−1 ) . 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Phys. 262 (2006), 595–609. http://dx.doi.org/10.1111/1467-9590.00167 http://arxiv.org/abs/solv-int/9812011 http://dx.doi.org/10.1143/JPSJ.61.4295 http://dx.doi.org/10.1143/JPSJ.61.4295 http://dx.doi.org/10.1007/BF01762370 http://dx.doi.org/10.1007/BF01458459 http://dx.doi.org/10.1103/PhysRevLett.64.1326 http://dx.doi.org/10.1016/0378-4371(95)00439-4 http://arxiv.org/abs/solv-int/9510011 http://dx.doi.org/10.1103/PhysRevLett.67.1829 http://dx.doi.org/10.1016/S0898-1221(01)00180-8 http://dx.doi.org/10.1088/0951-7715/11/4/004 http://dx.doi.org/10.1007/s002200100446 http://dx.doi.org/10.1070/RD2004v009n01ABEH000260 http://dx.doi.org/10.1007/s11005-005-0037-3 http://dx.doi.org/10.1007/s00220-005-1461-z 1 Introduction 1.1 Purpose 1.2 Background 1.3 Plan of the paper 1.4 q-Special functions 2 Hypergeometric functions of W"0365W(to.A4(1))to.-type 2.1 functions 2.2 Hypergeometric functions 2.3 Discrete Painlevé equations 3 Hypergeometric functions of W"0365W(to.(A1+A1')(1))to.-type 3.1 functions 3.2 Discrete Painlevé equations 3.3 Hypergeometric functions References

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