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

We consider a q-Painlevé III equation and a q-Painlevé 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 of τ functions.

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Дата:	2010
Автор:	Nakazono, N.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Hypergeometric τ Functions of the q-Painlevé Systems of Type ((A₂+A₁)⁽¹⁾ / N. Nakazono // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 34 назв. — англ.

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spelling	irk-123456789-1465182019-02-10T01:25:09Z Hypergeometric τ Functions of the q-Painlevé Systems of Type (A₂+A₁)⁽¹⁾ Nakazono, N. We consider a q-Painlevé III equation and a q-Painlevé 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 of τ functions. 2010 Article Hypergeometric τ Functions of the q-Painlevé Systems of Type ((A₂+A₁)⁽¹⁾ / N. Nakazono // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D05; 33D15; 33E17; 39A13 DOI:10.3842/SIGMA.2010.084 http://dspace.nbuv.gov.ua/handle/123456789/146518 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We consider a q-Painlevé III equation and a q-Painlevé 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 of τ functions.
format	Article
author	Nakazono, N.
spellingShingle	Nakazono, N. Hypergeometric τ Functions of the q-Painlevé Systems of Type (A₂+A₁)⁽¹⁾ Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Nakazono, N.
author_sort	Nakazono, N.
title	Hypergeometric τ Functions of the q-Painlevé Systems of Type (A₂+A₁)⁽¹⁾
title_short	Hypergeometric τ Functions of the q-Painlevé Systems of Type (A₂+A₁)⁽¹⁾
title_full	Hypergeometric τ Functions of the q-Painlevé Systems of Type (A₂+A₁)⁽¹⁾
title_fullStr	Hypergeometric τ Functions of the q-Painlevé Systems of Type (A₂+A₁)⁽¹⁾
title_full_unstemmed	Hypergeometric τ Functions of the q-Painlevé Systems of Type (A₂+A₁)⁽¹⁾
title_sort	hypergeometric τ functions of the q-painlevé systems of type (a₂+a₁)⁽¹⁾
publisher	Інститут математики НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/146518
citation_txt	Hypergeometric τ Functions of the q-Painlevé Systems of Type ((A₂+A₁)⁽¹⁾ / N. Nakazono // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 34 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT nakazonon hypergeometrictfunctionsoftheqpainlevesystemsoftypea2a11
first_indexed	2025-07-11T00:10:16Z
last_indexed	2025-07-11T00:10:16Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 084, 16 pages Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2 + A1) (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 a q-Painlevé III equation and a q-Painlevé 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 of τ functions. Key words: q-Painlevé 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 Painlevé III equation (q-PIII) [8, 12, 13, 32] gn+1 = q2N+1c2 fngn 1 + a0q nfn a0qn + fn , fn+1 = q2N+1c2 fngn+1 1 + a2a0q n−mgn+1 a2a0qn−m + gn+1 , (1.1) and that of the Painlevé II equation (q-PII) [12, 30, 20] Xk+1 = q2N+1c2 XkXk−1 1 + a0q k/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 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 expressib- le in the form of ratio of determinants whose entries are given by hypergeometric type functions. mailto:n-nakazono@math.kyushu-u.ac.jp http://researchmap.jp/nakazono/ http://dx.doi.org/10.3842/SIGMA.2010.084 2 N. Nakazono 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 (1) 7 and E (1) 8 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 (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 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-PIII and then those of q-PII. We next introduce a representation of the affine Weyl group of type (A2 +A1)(1). In Section 3, we construct the hypergeometric τ functions of q-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 series 2ϕ1. We use the following conventions of q-analysis throughout this paper [1]. q-Shifted factorials: (a; q)k = k∏ i=1 ( 1− aqi−1 ) . 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. 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 ) . It holds that Θ(qa; q) = −a−1Θ(a; q), Γ(qa; q, q) = Θ(a; q)Γ(a; q, q). Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2 +A1)(1) 3 2 q-PIII and q-PII 2.1 Hypergeometric solutions of q-PIII and q-PII First, we review the hypergeometric solutions of q-PIII and q-PII. The hypergeometric solutions of q-PIII have been constructed as follows: Proposition 2.1 ([8]). The hypergeometric solutions of q-PIII, (1.1), with c = 1 are given by fn = −a0q n ψn,m−1 N+1 ψn,m N ψn,m N+1ψ n,m−1 N , gn = a0 −1a2q −n−m+1 ψn,m N+1ψ n−1,m−1 N ψn−1,m−1 N+1 ψn,m N , where ψn,m N (N ∈ Z≥0) is an N ×N determinant defined by ψn,m N = ∣∣∣∣∣∣∣∣∣ Fn,m Fn+1,m · · · Fn+N−1,m Fn−1,m Fn,m · · · Fn+N−2,m ... ... . . . ... Fn−N+1,m Fn−N+2,m · · · Fn,m ∣∣∣∣∣∣∣∣∣ , ψn,m 0 = 1, and Fn,m is an arbitrary solution of the systems Fn+1,m − Fn,m = −a0 2q2nFn,m−1, (2.1) Fn,m+1 − Fn,m = −a2 −2q2m+2Fn−1,m. (2.2) The general solution of (2.1) and (2.2) is given by Fn,m = An,m (a2 −2q2m+2; q2)∞ 1ϕ1 ( 0 a2 2q−2m; q2, a2 2a0 2q2n−2m ) +Bn,m Θ(a0 2a2 2q2n−2m−2; q2) (a2 2q−2m−2; q2)∞Θ(a0 2q2n; q2) 1ϕ1 ( 0 a2 −2q2m+4; q 2, a0 2q2n+2 ) , (2.3) where An,m and Bn,m are periodic functions of period one with respect to n and 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 of q-PII are given as follows: Proposition 2.2 ([20]). The hypergeometric solutions of q-PII, (1.2), with c = 1 are given by Xk = −a0q k/2+N φk N+1φ k−1 N φk−1 N+1φ k N , (2.4) where φk N (N ∈ Z≥0) is an N ×N determinant defined by φk N = ∣∣∣∣∣∣∣∣∣ Gk Gk−1 · · · Gk−N+1 Gk+2 Gk+1 · · · Gk−N+3 ... ... . . . ... Gk+2N−2 Gk+2N−3 · · · Gk+N−1 ∣∣∣∣∣∣∣∣∣ , φk 0 = 1, (2.5) and Gk is an arbitrary solution of the system Gk+1 −Gk + a0 −2q−kGk−1 = 0. (2.6) 4 N. Nakazono The general solution of (2.6) is given by Gk = AkΘ(ia0q (2k+1)/4; q1/2) 1ϕ1 ( 0 −q1/2; q 1/2,−ia0q (3+2k)/4 ) +BkΘ(−ia0q (2k+1)/4; q1/2) 1ϕ1 ( 0 −q1/2; q 1/2, 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 Bäcklund 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 Painlevé 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, for i, 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- tions wj (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, for i ∈ Z/3Z. Proposition 2.3 ([13]). The group of birational transformations 〈s0, s1, s2, π, w0, w1, r〉 forms the affine Weyl group of type (A2 + A1)(1), denoted by W̃ ((A2 + A1)(1)). Namely, the transfor- mations satisfy the fundamental relations si 2 = (sisi+1)3 = π3 = 1, πsi = si+1π (i ∈ Z/3Z), w0 2 = w1 2 = r2 = 1, rw0 = w1r, and the action of W̃ (A(1) 2 ) = 〈s0, s1, s2, π〉 and that of W̃ (A(1) 1 ) = 〈w0, w1, r〉 commute with each other. Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2 +A1)(1) 5 In general, for a function F = F (ai, fj), we let an element w ∈ W̃ ((A2 + A1)(1)) act as w.F (ai, fj) = F (ai.w, fj .w), that is, w acts on the arguments from the right. Note that a0a1a2 = q and f0f1f2 = qc2 are invariant under the action of W̃ ((A2 +A1)(1)) and W̃ (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 c is 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 and T1T2T3 = 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, applying T1 nT2 mT4 N (n,m,N ∈ Z) on (2.9) and putting fn,m i,N = T1 nT2 mT4 N (fi) (i = 0, 1, 2), we obtain fn+1,m 1,N = q2N+1c2 fn,m 1,N fn,m 0,N 1 + a0q nfn,m 0,N a0qn + fn,m 0,N , fn+1,m 0,N = q2N+1c2 fn,m 0,N fn+1,m 1,N 1 + a2a0q n−mfn+1,m 1,N a2a0qn−m + fn+1,m 1,N , which is equivalent to q-PIII. Then T1 and Ti (i = 2, 4) are regarded as the time evolution and Bäcklund 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 to q-PII, it is crucial to introduce the transforma- tion R1 defined by R1 = π2s1, (2.10) which satisfies R1 2 = 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 R1 kT4 N on (2.11) and putting fk i,N = R1 kT4 N (fi) (i = 0, 1, 2), we have fk+1 0,N = q2N+1c2 fk 0,Nf k−1 0,N 1 + a0q k/2fk 0,N a0qk/2 + fk 0,N , 6 N. Nakazono which is equivalent to q-PII. Then R1 and T4 are regarded as the time evolution and a Bäcklund transformation of q-PII, 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. 2.3 Birational representation of W̃ ((A2 + A1) (1)) on the τ function We introduce the new variables τi and τ i (i ∈ Z/3Z) by letting fi = q1/3c2/3 τ i+1τi−1 τi+1τ i−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+1τ i−1 + τ i+1τi−1 ui 1/2τ i , si(τj) = τj (i 6= j), si(τ i) = viτ i+1τi−1 + τi+1τ i−1 vi 1/2τi , si(τ j) = τ j (i 6= j), π(τi) = τi+1, π(τ i) = τ i+1, w0(τ i) = ai+1 1/3(τ iτi+1τi+2+ ui−1τiτ i+1τi+2+ ui+1 −1τiτi+1τ i+2) ai+2 1/3τ i+1τ i+2 , w0(τi) = τi, w1(τi) = ai+1 1/3(τiτ i+1τ i+2+ vi−1τ iτi+1τ i+2+ vi+1 −1τ iτ i+1τi+2) ai+2 1/3τi+1τi+2 , w1(τ i) = τ i, r(τi) = τ i, r(τ i) = τi, with ui = q−1/3c−2/3ai, vi = q1/3c2/3ai, where i, j ∈ Z/3Z. Then, 〈s0, s1, s2, π, w0, w1, r〉 forms the affine Weyl group W̃ ((A2 +A1)(1)). We define the τ function τn,m N (n,m,N ∈ Z) by τn,m N = T1 nT2 mT4 N (τ1). We note that τ0 = τ−1,0 0 , τ1 = τ0,0 0 , τ2 = τ0,1 0 , τ0 = τ−1,0 1 , τ1 = τ0,0 1 , τ2 = τ0,1 1 , (2.12) and fn,m 0,N = q(2N+1)/3c2/3 τn,m N+1τ n,m+1 N τn,m N τn,m+1 N+1 , fn,m 1,N = q(2N+1)/3c2/3 τn,m+1 N+1 τn−1,m N τn,m+1 N τn−1,m N+1 , fn,m 2,N = q(2N+1)/3c2/3 τn−1,m N+1 τn,m N τn−1,m N τn,m N+1 . Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2 +A1)(1) 7 Let us consider the τ functions for q-PII. We set τk N = R1 kT4 N (τ1). Note that τ0 = τ−2 0 , τ1 = τ0 0 , τ2 = τ−1 0 , τ0 = τ−2 1 , τ1 = τ0 1 , τ2 = τ−1 1 , (2.13) and fk 0,N = q(2N+1)/3c2/3 τ k N+1τ k−1 N τk Nτ k−1 N+1 . In general, it follows that τn,0 N = τ2n N , τn,1 N = τ2n−1 N . For convenience, we introduce αi, γ, and Q by αi 6 = ai, γ6 = c, Q6 = q. 3 Hypergeometric τ functions of the q-Painlevé systems of type (A2 + A1) (1) In this section, we construct the hypergeometric τ functions of q-PIII and q-PII. We define the hypergeometric τ functions of q-PIII by τn,m N consistent with the action of 〈T1, T2, T3, T4〉. We also define the hypergeometric τ functions of q-PII by τk N consistent with the action of 〈R1, T4〉. Here, we mean τ(α) consistent with a action of transformation r as r.τ(α) = τ(α.r). We then regard τn,m N as function in α0 and α2, i.e., τn,m N = τ0,0 N (Qnα0, Q −mα2). We also regard τk N as function in α0, i.e., τk N = τ0 N (Qk/2α0). 3.1 Hypergeometric τ functions of q-PIII We construct the hypergeometric τ functions of q-PIII. By the action of the affine Weyl group, τn,m N is determined as a rational function in τn,m 0 and τn,m 1 (or τi and τ i). Thus, our purpose is determining τn,m 0 and τn,m 1 consistent with the action of 〈T1, T2, T3, T4〉 and constructing τn,m N under the condition γ = 1, (3.1) and the boundary condition τn,m N = 0 (N < 0). (3.2) First we consider the condition for τn,m 0 which follows from the boundary condition (3.2). We use the bilinear equations obtained in [12]: 8 N. Nakazono Proposition 3.1. The following bilinear equations hold: τn,m N+1τ n,m N−1 +Q4n−8m+4α1 −4α2 4 ( τn,m N )2 −Qn−2m+1α1 −1α2τ n,m+1 N τn,m−1 N = 0, (3.3) τn,m N+1τ n,m N−1 +Q4n+4mα0 4α2 −4 ( τn,m N )2 −Qn+mα0α2 −1τn+1,m+1 N τn−1,m−1 N = 0, (3.4) τn,m N+1τ n,m N−1 +Q−8n+4m−4α0 −4α1 4 ( τn,m N )2 −Q−2n+m−1α0 −1α1τ n+1,m N τn−1,m N = 0. (3.5) By putting N = 0 in (3.3)–(3.5), we get Q4n−8m+4α1 −4α2 4 (τn,m 0 )2 −Qn−2m+1α1 −1α2τ n,m+1 0 τn,m−1 0 = 0, (3.6) Q4n+4mα0 4α2 −4 (τn,m 0 )2 −Qn+mα0α2 −1τn+1,m+1 0 τn−1,m−1 0 = 0, (3.7) Q−8n+4m−4α0 −4α1 4 (τn,m 0 )2 −Q−2n+m−1α0 −1α1τ n+1,m 0 τn−1,m 0 = 0. (3.8) We set τn,m 0 = Γ(Q2n−m+1α0 2α2;Q,Q)Γ(Q−n+2m−1α1 2α0;Q,Q)Γ(Q−n−mα2 2α1;Q,Q)An,m 0 . (3.9) From (3.6)–(3.8), the following equations hold: (An,m 0 )2 = An,m+1 0 An,m−1 0 , (3.10) (An,m 0 )2 = An+1,m+1 0 An−1,m−1 0 , (3.11) (An,m 0 )2 = An+1,m 0 An−1,m 0 . (3.12) We next determine τn,m 0 and τn,m 1 . From (2.8) and Proposition 2.4, we see that the action of T1, T2, and T3 are given by Ti(τi−1) = τi, (3.13) Ti(τ i−1) = τ i, (3.14) Ti(τi+1) = αi−1 6τiτ i+1 +Q2τ iτi+1 Qαi−1 3τ i−1 , (3.15) Ti(τ i+1) = Q2αi−1 6τi+1τ i + τ i+1τi Qαi−1 3τi−1 , (3.16) Ti(τi) = 1 αi+1 3αi−1 6 τi 2 τi−1 + αi+1αi−1 4 αi 2 τiτ i τ i−1 + αi 2αi−1 2 αi+1 τ iτiτi+1 τ i+1τi−1 + αi+1 3 τ i 2τi+1 τ i+1τ i−1 , (3.17) Ti(τ i) = 1 αi+1 3αi−1 6 τ i 2 τ i−1 + αi 2αi+1 5αi−1 8 τiτ i τi−1 + 1 αi 2αi+1 5αi−1 2 τiτ iτ i+1 τi+1τ i−1 + αi+1 3 τi 2τ i+1 τi+1τi−1 , (3.18) where i = 1, 2, 3. Lemma 3.1. If τi and τ i are consistent with (3.13)–(3.16), then they are also consistent with (3.17) and (3.18). Proof. Applying Ti−1 on (3.16) and using (3.13) and (3.14), we have Ti(τi) = Q αi+1 3αi−1 3 τi τ i+1 Ti(τ i+1) + αi+1 2αi−1 2 αi τ i τ i+1 Ti(τi+1). (3.19) By using (3.15) and (3.16) for (3.19), we get (3.17). Similarly, applying Ti−1 on (3.15) and using (3.13) and (3.14), we have Ti(τ i) = 1 Qαi+1 3αi−1 3 τ i τi+1 Ti(τi+1) +Qαi−1 3αi+1 3 τi τi+1 Ti(τ i+1). (3.20) By using (3.15) and (3.16) for (3.20), we get (3.18). � Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2 +A1)(1) 9 From (2.12), we rewrite (3.15) and (3.16) as follows: τ−1,0 0 τ0,1 1 −Q−1α1 3τ−1,1 0 τ0,0 1 +Q−2α1 6τ0,1 0 τ−1,0 1 = 0, (3.21) τ0,0 0 τ−1,0 1 −Q−1α2 3τ−1,−1 0 τ0,1 1 +Q−2α2 6τ−1,0 0 τ0,0 1 = 0, (3.22) τ0,1 0 τ0,0 1 −Q−1α0 3τ1,1 0 τ−1,0 1 +Q−2α0 6τ0,0 0 τ0,1 1 = 0, (3.23) τ0,1 0 τ−1,0 1 −Qα1 3τ0,0 0 τ−1,1 1 +Q2α1 6τ−1,0 0 τ0,1 1 = 0, (3.24) τ−1,0 0 τ0,0 1 −Qα2 3τ0,1 0 τ−1,−1 1 +Q2α2 6τ0,0 0 τ−1,0 1 = 0, (3.25) τ0,0 0 τ0,1 1 −Qα0 3τ−1,0 0 τ1,1 1 +Q2α0 6τ0,1 0 τ0,0 1 = 0. (3.26) We set τn,m 1 = −Q2n+2mα0 2α2 −2 Θ(−Q−6nα0 −6;Q6)Θ(−Q6mα2 −6;Q6) Θ(Q−6(n−m)α0 −6α2 −6;Q6) τn,m 0 Fn,m−1. (3.27) Here, Fn,m is equivalent to (2.3) because we obtain (2.1) and (2.2) from (3.24)–(3.26) and (3.21)–(3.23), respectively. If we assume An,m 0 is an arbitrary constant, it does not contradict (3.10)–(3.18). Therefore, we may set An,m 0 = 1. Finally we construct τn,m N . Theorem 3.1. Under the assumption (3.1) and (3.2), the hypergeometric τ functions of q-PIII are given as the follows: τn,m N = (−1)N(N+1)/2Q−2(2n−m)N2+6nNα0 −4N2+6Nα2 −2N2 × ( Θ(−Q−6nα0 −6;Q6)Θ(−Q6mα2 −6;Q6) Θ(Q−6(n−m)α0 −6α2 −6;Q6) )N × Γ(Q2n−m+1α0 2α2;Q,Q)Γ(Q−n+2m−1α1 2α0;Q,Q) × Γ(Q−n−mα2 2α1;Q,Q)ψn,m−1 N , (3.28) where ψn,m N = ∣∣∣∣∣∣∣∣∣ Fn,m Fn+1,m · · · Fn+N−1,m Fn−1,m Fn,m · · · Fn+N−2,m ... ... . . . ... Fn−N+1,m Fn−N+2,m · · · Fn,m ∣∣∣∣∣∣∣∣∣ , ψn,m 0 = 1, ψn,m −N = 0 (N > 0), and Fn,m = An,m (a2 −2q2m+2; q2)∞ 1ϕ1 ( 0 a2 2q−2m; q2, a2 2a0 2q2n−2m ) +Bn,m Θ(a0 2a2 2q2n−2m−2; q2) (a2 2q−2m−2; q2)∞Θ(a0 2q2n; q2) 1ϕ1 ( 0 a2 −2q2m+4; q 2, a0 2q2n+2 ) . (3.29) Here, An,m and Bn,m are periodic functions of period one with respect to n and m. Proof. We set τn,m N = (−1)N(N+1)/2Q−2(2n−m)N2+6nNα0 −4N2+6Nα2 −2N2 × ( Θ(−Q−6nα0 −6;Q6)Θ(−Q6mα2 −6;Q6) Θ(Q−6(n−m)α0 −6α2 −6;Q6) )N 10 N. Nakazono × Γ(Q2n−m+1α0 2α2;Q,Q)Γ(Q−n+2m−1α1 2α0;Q,Q)Γ(Q−n−mα2 2α1;Q,Q)ψn,m−1 N . From (3.2), (3.9), and (3.27), we find ψn,m N = 0 (N < 0), ψn,m 0 = 1, ψn,m 1 = Fn,m. Furthermore, it is easily verified that ψn,m N satisfy ψn,m N+1ψ n,m N−1 − ( ψn,m N )2 + ψn+1,m N ψn−1,m N = 0, (3.30) from (3.5). In general, (3.30) admits a solution expressed in terms of the Toeplitz type deter- minant ψn,m N = det (cn−i+j,m)i,j=1,...,N (N > 0), under the boundary conditions ψn,m N = 0 (N < 0), ψn,m 0 = 1, ψn,m 1 = 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 of q-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 that τn,m N consistent with the action of 〈s2, T1, T2, T3, T4〉 is also consistent with the action of R1 because R1 = s2T2 −1. Therefore, we construct τn,m N consistent with the action of 〈s2, T1, T2, T3, T4〉. The action of s2 on τn,m N is s2(τ n,m N ) = τn−m,−m N . (3.31) We consider only τn,m 0 and τn,m 1 because τn,m N is determined as a rational function in τn,m 0 and τn,m 1 . It easily verified that τn,m 0 , (3.28) (or (3.9)), is consistent with the action of s2. 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 (α2 12Q12m;Q12)∞ 1ϕ1 ( 0 α2 −12Q−12m+12;Q 12, α0 12Q12n−12m+12 ) = (An,m − s2(Bn,m))Θ(α0 12Q12n−12m;Q12) (α2 −12Q−12m;Q12)∞Θ(α0 12α2 12Q12n;Q12) 1ϕ1 ( 0 α2 12Q12m+12;Q 12, α0 12α2 12Q12n+12 ) , which implies that τn,m 1 is also consistent with the action of s2 when s2(An,m) = Bn,m. (3.33) Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2 +A1)(1) 11 Lemma 3.2. Under the assumption (3.33), the hypergeometric τ functions (3.28) are consistent with the action of 〈s2, T1, T2, T3, T4〉. Therefore we easily obtain the following theorem: Theorem 3.2. Setting R1(An,m) = Bn,m, (3.34) α2 = Q1/2, and putting τ2n N = τn,0 N , τ2n−1 N = τn,1 N , we obtain the hypergeometric τ functions of q-PII. Here τn,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. How- ever the hypergeometric τ functions of q-PII 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-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, τk N is determined as a rational function of τk 0 and τk 1 (or τi and τ i). Thus, our purpose is determining τk 0 and τk 1 consistent with the action of 〈R1, T4〉 and constructing τk N under the conditions α2 = Q1/2, γ = 1, (3.35) and the boundary condition τk N = 0 (N < 0). (3.36) First we consider the condition for τk 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: τk N+1τ k+1 N−1 −Q(k−4N+1)/2γ−2α0τ k+2 N τk−1 N −Q−k+4N−1γ4α0 −2τk+1 N τk N = 0. (3.37) By putting N = 0 in (3.37), we get Q3(k+1)/2α0 3τk+2 0 τk−1 0 + τk+1 0 τk 0 = 0. (3.38) We set τk 0 = Γ ( Q(2k+3)/2α0 2;Q,Q ) Γ ( Q−k/2α0 −1;Q,Q ) Γ ( Q(−k+3)/2α0 −1;Q,Q ) Ak 1. (3.39) From (3.38), Ak 1 satisfies Ak+2 1 Ak−1 1 = Ak+1 1 Ak 1. (3.40) 12 N. Nakazono We next determine τk 0 and τk 1 . From (2.10) and Proposition 2.4, we see that the action of R1 on τk 0 and τk 1 is given by R1(τ0) = τ2, (3.41) R1(τ1) = Q−2α0 6τ1τ2 + τ1τ2 Q−1α0 3τ0 , (3.42) R1(τ2) = τ1, (3.43) R1(τ0) = τ2, (3.44) R1(τ1) = Q2α0 6τ1τ2 + τ1τ2 Qα0 3τ0 , (3.45) R1(τ2) = τ1. (3.46) From (2.13), we rewrite (3.42) and (3.45) as Qα0 −3τ−2 1 τ1 0 −Q2α0 −6τ0 1 τ −1 0 − τ0 0 τ −1 1 = 0, (3.47) Q−1α0 −3τ−2 0 τ1 1 −Q−2α0 −6τ0 0 τ −1 1 − τ0 1 τ −1 0 = 0, (3.48) respectively. Setting τk 1 = τk 0 Θ(Q3k+1α0 6;Q3) Gk, (3.49) then the systems (3.47) and (3.48) reduce to (2.6). Therefore Gk is equivalent to (2.7). If we assume Ak 1 is an arbitrary constant, it does not contradict (3.40)–(3.46). Therefore, we may put Ak 1 = 1. Finally we present an explicit formula for τk N . Theorem 3.3. Under the assumption (3.35) and (3.36), the hypergeometric τ functions of q-PII are given as the follows: τk N = (−1)N(N−1)/2QN(N−1)(k+N)α0 2N(N−1) × Γ(Q(2k+3)/2α0 2;Q,Q)Γ(Q−k/2α0 −1;Q,Q)Γ(Q(−k+3)/2α0 −1;Q,Q) Θ(Q3k+1α0 6;Q3)N φk N , where φk N = ∣∣∣∣∣∣∣∣∣ Gk Gk−1 · · · Gk−N+1 Gk+2 Gk+1 · · · Gk−N+3 ... ... . . . ... Gk+2N−2 Gk+2N−3 · · · Gk+N−1 ∣∣∣∣∣∣∣∣∣ , φk 0 = 1, φk −N = 0 (N > 0), and Gk = AkΘ ( ia0q (2k+1)/4; q1/2 ) 1ϕ1 ( 0 −q1/2; q 1/2,−ia0q (3+2k)/4 ) +BkΘ ( −ia0q (2k+1)/4; q1/2 ) 1ϕ1 ( 0 −q1/2; q 1/2, ia0q (3+2k)/4 ) . Here, Ak and Bk are periodic functions of period one. Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2 +A1)(1) 13 Proof. We set τk N = (−1)N(N−1)/2QN(N−1)(k+N)α0 2N(N−1) × Γ(Q(2k+3)/2α0 2;Q,Q)Γ(Q−k/2α0 −1;Q,Q)Γ(Q(−k+3)/2α0 −1;Q,Q) Θ(Q3k+1α0 6;Q3)N φk N . From (3.36), (3.39), and (3.49), we find that φk N = 0 (N < 0), φk 0 = 1, φk 1 = Gk. From (3.37), φk N satisfies φk N+1φ k+1 N−1 − φk Nφ k+1 N + φk+2 N φk−1 N = 0, (3.50) which is a variant of the discrete Toda equation. Under the conditions φk N = 0 (N < 0), φk 0 = 1, φk 1 = ck, where ck is an arbitrary function. Equation (3.50) admits a solution expressed by φk N = 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 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 10ϕ9 transformation formula [17]. We here show that the consistency of s0 to the hypergeometric τ functions of q-PIII give rise to a transformation of 1ϕ1 which is obtained by Heine’s transform for 2ϕ1. The action of s0 on τn,m N is s0(τ n,m N ) = τ−n,m−n N . (3.51) We consider only τn,m 0 and τn,m 1 because τn,m N is determined as a rational function in τn,m 0 and τn,m 1 . It easily verified that τn,m 0 , (3.28) (or (3.9)), is consistent with the action of s0. 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 α0 12α2 12Q−12m+12;Q 12, α2 12Q12n−12m+12 ) −An,m (α2 12Q12n−12m+12;Q12)∞ (α0 12α2 12Q−12m+12;Q12)∞ 1ϕ1 ( 0 α2 12Q12n−12m+12;Q 12, α2 12α0 12Q−12m+12 ) −Bn,mα0 12Q−12n (α0 −12α2 −12Q12m, α2 −12Q−12n+12m+12;Q12)∞ Θ(α0 12Q−12n;Q12) 14 N. Nakazono × 1ϕ1 ( 0 α2 −12Q−12n+12m+12;Q 12, α0 12Q−12n+12 ) + s0(Bn,m) (α0 −12α2 −12Q12m;Q12)∞Θ(α2 12Q12n−12m;Q12) (α0 12α2 12Q−12m;Q12)∞Θ(α0 −12Q12n;Q12) × 1ϕ1 ( 0 α0 −12α2 −12Q12m+12;Q 12, α0 −12Q12n+12 ) = 0. (3.53) In particular, setting s0(An,m) = An,m, Bn,m = 0, in (3.53), we obtain 1ϕ1 ( 0 α0 12α2 12Q−12m+12;Q 12, α2 12Q12n−12m+12 ) (3.54) = (α2 12Q12n−12m+12;Q12)∞ (α0 12α2 12Q−12m+12;Q12)∞ 1ϕ1 ( 0 α2 12Q12n−12m+12;Q 12, α2 12α0 12Q−12m+12 ) . Equation (3.54) corresponds to a specialization of Heine’s transform. 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[34] Wadim Z., Heine’s basic transform and a permutation group for q-harmonic series, Acta Arith. 111 (2004), 153–164. http://dx.doi.org/10.1016/0167-2789(89)90233-9 http://dx.doi.org/10.1016/0378-4371(95)00439-4 http://arxiv.org/abs/solv-int/9510011 http://dx.doi.org/10.1088/0951-7715/11/4/004 http://dx.doi.org/10.1007/s002200100446 http://dx.doi.org/10.1007/s11005-005-0037-3 http://dx.doi.org/10.4064/aa111-2-4 1 Introduction 2 q-PIII and q-PII 2.1 Hypergeometric solutions of q-PIII and q-PII 2.2 Projective reduction from q-PIII to q-PII 2.3 Birational representation of W"0365W((A2+A1)(1)) on the function 3 Hypergeometric functions of the q-Painlevé systems of type (A2+A1)(1) 3.1 Hypergeometric functions of q-PIII 3.2 Hypergeometric functions of q-PII 3.2.1 Hypergeometric functions of q-PII (I) 3.2.2 Hypergeometric functions of q-PII (II) 3.3 Relation between the hypergeometric functions of q-PIII and Heine's transform References

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

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