A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾

Recently a certain q-Painlevé type system has been obtained from a reduction of the q-Garnier system. In this paper it is shown that the q-Painlevé type system is associated with another realization of the affine Weyl group symmetry of type E₇⁽¹⁾ and is different from the well-known q-Painlevé syste...

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Дата:	2017
Автор:	Nagao, H.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2017
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/149269
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Цитувати:	A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾ / H. Nagao // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 50 назв. — англ.

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spelling	irk-123456789-1492692019-02-20T01:23:43Z A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾ Nagao, H. Recently a certain q-Painlevé type system has been obtained from a reduction of the q-Garnier system. In this paper it is shown that the q-Painlevé type system is associated with another realization of the affine Weyl group symmetry of type E₇⁽¹⁾ and is different from the well-known q-Painlevé system of type E₇⁽¹⁾ from the point of view of evolution directions. We also study a connection between the q-Painlevé type system and the q-Painlevé system of type E₇⁽¹⁾. Furthermore determinant formulas of particular solutions for the q-Painlevé type system are constructed in terms of the terminating q-hypergeometric function. 2017 Article A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾ / H. Nagao // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 50 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H70; 33D15; 33D70; 34M55; 37K20; 39A13; 41A21 DOI:10.3842/SIGMA.2017.092 http://dspace.nbuv.gov.ua/handle/123456789/149269 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	Recently a certain q-Painlevé type system has been obtained from a reduction of the q-Garnier system. In this paper it is shown that the q-Painlevé type system is associated with another realization of the affine Weyl group symmetry of type E₇⁽¹⁾ and is different from the well-known q-Painlevé system of type E₇⁽¹⁾ from the point of view of evolution directions. We also study a connection between the q-Painlevé type system and the q-Painlevé system of type E₇⁽¹⁾. Furthermore determinant formulas of particular solutions for the q-Painlevé type system are constructed in terms of the terminating q-hypergeometric function.
format	Article
author	Nagao, H.
spellingShingle	Nagao, H. A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾ Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Nagao, H.
author_sort	Nagao, H.
title	A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾
title_short	A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾
title_full	A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾
title_fullStr	A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾
title_full_unstemmed	A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾
title_sort	variation of the q-painlevé system with affine weyl group symmetry of type e₇⁽¹⁾
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/149269
citation_txt	A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾ / H. Nagao // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 50 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT nagaoh avariationoftheqpainlevesystemwithaffineweylgroupsymmetryoftypee71 AT nagaoh variationoftheqpainlevesystemwithaffineweylgroupsymmetryoftypee71
first_indexed	2025-07-12T21:12:22Z
last_indexed	2025-07-12T21:12:22Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 092, 18 pages A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E (1) 7 Hidehito NAGAO Department of Arts and Science, National Institute of Technology, Akashi College, Hyogo 674-8501, Japan E-mail: nagao@akashi.ac.jp Received July 03, 2017, in final form November 24, 2017; Published online December 10, 2017 https://doi.org/10.3842/SIGMA.2017.092 Abstract. Recently a certain q-Painlevé type system has been obtained from a reduction of the q-Garnier system. In this paper it is shown that the q-Painlevé type system is associated with another realization of the affine Weyl group symmetry of type E (1) 7 and is different from the well-known q-Painlevé system of type E (1) 7 from the point of view of evolution directions. We also study a connection between the q-Painlevé type system and the q-Painlevé system of type E (1) 7 . Furthermore determinant formulas of particular solutions for the q-Painlevé type system are constructed in terms of the terminating q-hypergeometric function. Key words: q-Painlevé system of type E (1) 7 ; q-Garnier system; Padé method; q-hypergeo- metric function 2010 Mathematics Subject Classification: 14H70; 33D15; 33D70; 34M55; 37K20; 39A13; 41A21 1 Introduction 1.1 Background In [39] H. Sakai has classified the second order continuous and discrete Painlevé equations into 22 cases by using the geometric theory of certain rational surfaces, called the“spaces of initial values”1, connected to affine root systems. The spaces of initial values are obtained from P1×P1 (resp. P2) by blowing up at 8 (resp. 9) singular points. In view of the configuration of 8 (resp. 9) singular points in P1 × P1 (resp. P2), there exist three types of discrete Painlevé equations and six continuous Painlevé equations in the classification: elliptic difference (e-), multiplicative difference (q-), additive difference (d-) and continuous (differential). Each of these Painlevé equations is constructed in a unified manner as the bi-rational action of a translation part of the corresponding affine Weyl group symmetry on a certain family of the rational surfaces. The sole e-Painlevé equation [31] having the affine Weyl group symmetry of type E (1) 8 is obtained from the most generic configuration on the unique curve of bi-degree (2, 2) called the smooth “elliptic curve”. All of the other Painlevé equations are derived from its degeneration. For instance, the q-Painlevé system with the symmetry of type E (1) 7 is well known to be obtained from a configuration of eight singular points on two curves of bi-degree (1, 1) in P1 × P1. The second order continuous and discrete Painlevé equations are classified into the 22 cases2 according to the degeneration diagram of affine Weyl group symmetries (see Fig. 1), where the symbol A → B represents that B is obtained from A by a certain limiting procedure. The d-Painlevé equation of type D (1) 4 and its degeneration (expect for A (1) 0 ) arise as Bäcklund (Schlesinger) 1For each of the six continuous Painlevé equations, K. Okamoto has constructed certain rational surfaces, called the “spaces of initial values”, which parametrize all the solutions [34]. 2Some q-Painlevé equations, such as a second order case of the system [15] (see also [45]), does not belong to the list of discrete Painlevé equations appeared in [39]. mailto:nagao@akashi.ac.jp https://doi.org/10.3842/SIGMA.2017.092 2 H. Nagao ell.(e-) E (1) 8 �� A (1) 1 \|α\|2=8 %% mul.(q-) E (1) 8 // �� E (1) 7 // �� E (1) 6 // �� D (1) 5 // �� A (1) 4 // $$ (A2 +A1)(1) // (( �� (A1 + A1 \|α\|2=14) (1) // && 66 A (1) 1 "" �� A (1) 0 �� add.(d-) E (1) 8 //E(1) 7 //E(1) 6 // D(1) 4 (PVI) //A(1) 3 (PV) // '' 2(A1)(1) (PIII) // %% A (1) 1 \|α\|2=4 (P D (1) 7 III ) // A(1) 0 (P D (1) 8 III ) A (1) 2 (PIV) //A(1) 1 (PII) //A(1) 0 (PI) Figure 1. transformations of the six continuous Painlevé equations3 (PI, . . .,PVI). The symbol A (1) 1 \|α\|2=l means the root subsystem of type A (1) 1 whose square length of roots is l. Similarly to the differential Painlevé systems, the discrete Painlevé systems are known to have particular solutions expressed by various hypergeometric functions [12, 13, 14, 16, 21, 22, 38]. The particular solutions of the elliptic Painlevé equation are expressed in [12] in terms of the elliptic hypergeometric function 10E9 [3]. In the case of q-E (1) 7 , the particular solutions are expressed in [24] in terms of the terminating q-hypergeometric function 4ϕ3, 4 where the function kϕl [3] is defined by kϕl ( α1, . . . , αk β1, . . . , βl , x ) = ∞∑ s=0 (α1, . . . , αk)s (β1, . . . , βl, q)s [ (−1)sq( s 2) ]1+l−k xs, (1.1) with (s2) = s(s−1) 2 . Here the standard q-Pochhammer symbol5 is defined by (x)∞ := ∞∏ i=0 (1− qix), (x)s := (x)∞ (xqs)∞ , (x1, x2, . . . , xk)s := (x1)s(x2)s · · · (xk)s. It is common to nonlinear integrable systems that they arise as the compatibility condition of linear equations and their deformed equations. The pair of the linear equations is called a “Lax pair” for the nonlinear system. Similarly to the continuous Painlevé equations [8, 9, 10, 32, 33], Lax pairs for the discrete Painlevé equations have been studied from various points of view in [4, 11, 16, 23, 37, 42, 46, 48]. For instance, as a geometric approach, the Lax pair for the e-Painlevé equation has been formulated in [46] as a curve of bi-degree (3, 2) in P1 × P1 passing through 12 points. In the case of q-E (1) 7 , the Lax pair has been similarly formulated in [16, 46]. In [41] the q-Garnier system was formulated as a multivariable extension of the well-known q-PVI (i.e., q-D (1) 5 ) system [11] by H. Sakai, and has recently been studied in [28, 35, 40]6. In [28] a Lax pair, an evolution equation and two kinds of particular solutions7 for the q-Garnier system have been simply expressed by applying a certain method of Padé approximation and 3P D (1) i III symbolizes PIII having the surface connected to the affine root system of type D (1) i . 4The terminating balanced 4ϕ3 is rewritten into the terminating q-hypergeometric function 8W7 by Watson’s transformation formula [3]. For particular solutions in terms of 8W7, see [13, 14, 21]. 5Actually Pochhammer himself used the symbol (a)n not as a rising shifted factorial but as a binomial coefficient [17]. 6For the related works, see [1, 2, 35] (additive Garnier system), [36, 50] (elliptic Garnier system). 7These solutions have been constructed in terms of the q-Appell Lauricella function (resp. the generalized q-hypergeometric function) in [28, 40] (resp. [28]). A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E (1) 7 3 its analogue (i.e., Padé interpolation)8. The q-D (1) 5 (resp. q-E (1) 6 ) system appears as a reduction of case N = 1 [41] (resp. particular case of N = 2 [42]) of the q-Garnier system having 2N dependent variables. Recently the q-Painlevé type system9 [28, Section 2.5] has appeared as a particular case of N = 3. However the q-E (1) 8 system has been not obtained from a reduction of the q-Garnier system. Remark 1.1. We call a certain q-Painlevé type system “a variation of a q-Painlevé system” having a well-known direction10, when both systems satisfy the following: (i) They are associated with different realizations of the symmetry/surface of the same type in the Sakai’s classification. (ii) Their time evolutions are different from the viewpoint of shift operator on parameters. 1.2 Purpose and organization Our main subject is the q-Painlevé type system [28] regarded as a variation of the well-known q-E (1) 7 system [5]. The purpose of this paper is the following three. • We show that the q-Painlevé type system is a bi-rational transformation and is related to a novel realization of the symmetry/surface of type E (1) 7 /A (1) 1 . Then it is clarified to be a variation of the q-E (1) 7 system. • The Lax pair for the q-Painlevé type system is obtained from a certain reduction of the q-Garnier system and we study a connection between the q-Painlevé type system and the q-E (1) 7 system by comparing their Lax equations. • Particular solutions for the q-Painlevé type system are given as a reduction of the q-Garnier system. This paper is organized as follows. In Section 2 we prove that the q-Painlevé type system is a bi-rational transformation and we investigate its configuration of 8 singular points on a curve of bi-degree (2, 2) in coordinates (f, g) ∈ P1 × P1. In Section 3 we briefly recall Lax equations for the q-Garnier system [28, Section 2.1], and study a reduction of particular case N = 3 of the q-Garnier system. Consequently, we obtain Lax equations for the q-Painlevé type system. In Section 4 the Lax equation for the q-Painlevé type system is uniquely determined by a char- acterization, and recall the characterization of Yamada’s Lax equation for the q-E (1) 7 system. Then we investigate a connection between these systems by comparing characterizations of their Lax equations. In Section 5 we recall the particular solutions of the q-Garnier system and we construct particular solutions of the q-Painlevé type system by applying a reduction. In Ap- pendix A we derive the q-Painlevé type system, its Lax pair and its particular solutions by using a Padé interpolation. 2 q-Painlevé type system In this section we first recall the q-Painlevé type system [28, Section 2.5]. Then we prove that the system is a bi-rational transformation and confirm that the system has the symmetry/surface of type E (1) 7 /A (1) 1 by its configuration of eight singular points. Let q (\|q\| < 1), a1, . . . , a4, b1, . . . , b4, c1 and d1 ∈ C× be complex parameters with a constraint 4∏ i=1 ai bi = q c21 d21 , and let (f, g) ∈ P1×P1 be 8The Padé method has been also applied to the continuous/discrete Painlevé systems in [6, 24, 25, 26, 27, 30, 47, 49]. For the case of q-E (1) 7 [24], see Section A. For the works related to the differential Garnier system, see [18, 47] (Padé approximation), [19, 20] (Hermite–Padé approximation). 9As another derivation of the equations (2.2) and (3.6), see Appendix A.1 (Padé interpolation method). 10In case of q-E (1) 7 , T2 (4.7) is the well-known direction and T1 (2.1) is a variation direction. 4 H. Nagao dependent variables. Define Ta : a→ qa by a q-shift operator of parameter a. Then we consider a q-shift operator T1 11 T1 = T−1a1 T −1 b1 . (2.1) Here for any object X the corresponding shifts are denoted as X := T1(X) and X := T−11 (X). The operator T1 plays the role of the evolution of the system. The system is described by the following transformation T−11 (g) = g(f, g) and T1(f) = f(f, g) in P1 × P1: ( e1f 2 + e1fg + c1 )(e1 q f2 + e1 q gf + c1 ) = c21 4∏ i=2 (1− aif)(1− bif) (1− a1f)(1− b1f) , x21(1− fx1)(1− fx1) x22(1− fx2)(1− fx2) = 4∏ i=2 (x1 − ai)(x1 − bi) (x2 − ai)(x2 − bi) . (2.2) Here e1 = d1a2a3a4b −1 1 , and x = x1, x2 ( = e1 c1x1 ) are solutions of an equation ϕ = 0 where ϕ(x) = e1 + e1gx+ c1x 2. (2.3) Then we have Proposition 2.1. The q-Painlevé type system (2.2) has the following properties: (i) It is a bi-rational transformation T−11 (g) = g(f, g) and T1(f) = f(f, g) ∈ P1 × P1. (ii) It is associated with a novel realization (2.5) of the symmetry/surface of type E (1) 7 /A (1) 1 . Proof. (i) It is easy to see that the first equation of (2.2) is a rational transformation T−11 (g) = g(f, g). The second equation of (2.2) is rewritten as f = x21(1− fx1)A(x2)− x22(1− fx2)A(x1) x31(1− fx1)A(x2)− x32(1− fx2)A(x1) , (2.4) where A(x) = 4∏ i=2 (x− ai)(x− bi). Then the numerator and denominator of (2.4) are alternating with respect to x1 ↔ x2 = e1 c1x1 , and Laurent polynomials 4∑ i=−4 hix i 1 where hi is depending on f , ai, bj , c1 and d1. Accordingly the numerator and denominator are expressed by Laurent polynomials (x1 − x2) 3∑ i=0 h̃i(x1 + x2) i, where h̃i is depending on f , ai, bj , c1 and d1. Due to the relation x1 + x2 = − e1g c1 , the transformation T1(f) = f(f, g) is given as a rational polynomial of bi-degree (1, 3) in (f, g). Therefore the property (i) is proved. (ii) Eight singular points (fs, gs) ∈ P1 × P1 (s = 1, . . . , 8) in coordinates (f, g) are on one line g = ∞ of bi-degree (0, 1) and one parabolic curve e1f 2 + e1fg + c1 = 0 of bi-degree (2, 1) as follows:( 1 a1 ,∞ ) , ( 1 b1 ,∞ ) , ( 1 ai ,− 1 ai − aic1 e1 ) i=2,3,4 , ( 1 bi ,− 1 bi − bic1 e1 ) i=2,3,4 . (2.5) Hence the property (ii) is confirmed since the configuration (2.5) is the realization of the surface type A (1) 1 . � According to Remark 1.1, the system (2.2) is regarded as a variation of the q-E (1) 7 system. 11The operator T1 is generally selected as T−1 ai T−1 bj . The directions such as T−1 ai T−1 bj , T−1 ai T−1 aj T−1 ck and T−1 ai T−1 aj Tdk are fundamental ones. A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E (1) 7 5 3 Lax equations In this section we recall Lax equations for the q-Garnier system [28, Section 2.1] and investigate a reduction of particular case N = 3 of them. As a result, we obtain the Lax equations for the q-Painlevé type system (2.2). 3.1 Case of the q-Garnier system The scalar Lax equations for the q-Garnier system are L1(x) = A(x)F ( x q ) y(qx) + qc1c2B ( x q ) F (x)y ( x q ) − { (x− a1)(x− b1)F ( x q ) G(x) + F (x) G ( x q )V (x q )} y(x), L2(x) = F (x)y(x)−A1(x)y(qx) + (x− b1)G(x)y(x), L3(x) = F ( x q ) y(x) + (x− a1)G ( x q ) y(x)− qc1c2B1 ( x q ) y ( x q ) , (3.1) where A(x) = N+1∏ i=1 (x− ai), B(x) = N+1∏ i=1 (x− bi), A1(x) = A(x) x− a1 , B1(x) = B(x) x− b1 , F (x) = N∑ i=0 fix i, G(x) = N−1∑ i=0 giz i, V (x) = qc1c2A1(x)B1(x)− F (x)F (x). (3.2) Here the deformation direction is T1 (2.1) and f0, . . . , fN , g0, . . . , gN−1 ∈ P1 are variables de- pending on parameters ai, bi, ci, di with a constraint N+1∏ i=1 ai bi = q c1c2d1d2 . Remark 3.1. The scalar Lax pair L1 = 0 and L2 = 0 (or L3 = 0) is equivalent to the pair of the deformation equations L2 = 0 and L3 = 0. The equation L1 = 0 (we call it the L1 equation) is equivalent to one for Sakai’s system given in [41] and the deformation direction is opposite to one for Sakai’s system. The q-Garnier system is G(x)G(x) = c1c2 A1(x)B1(x) (x− a1)(x− b1) for F (x) = 0, F (x)F (x) = qc1c2A1(x)B1(x) for G(x) = 0, (3.3) fNfN = q(gN−1 − c1)(gN−1 − c2), f0f0 = a1b1 ( g0 − d1 a1b1 A(0) )( g0 − d2 a1b1 A(0) ) , where 2N variables f1 f0 , . . . , fNf0 , g0, . . . , gN−1 are the dependent variables. Then we have the following fact12. Proposition 3.2. The compatibility condition of the Lax pair L1 = 0 and L2 = 0 (3.1) is equivalent to the q-Garnier system (3.3). 12For the proof of Proposition 3.2, see [28, Section 2.3]. 6 H. Nagao 3.2 Reduction to the q-Painlevé type system We impose a reduction condition by a constraint of the parameters c1 = c2, d1 = d2 (3.4) and specialize the dependent variables as f0 = f3 = 0, f1 = w1, f2 = −fw1, g0 = e1, g1 = e1g, g2 = c1, (3.5) where w1 13 is a “gauge freedom”. Applying the conditions (3.4) and (3.5) into (3.1) and (3.3), we obtain the following linear equations L1(x) = A(x) ( 1− fx q ) y(qx) + q2c21B ( x q ) (1− fx)y ( x q ) − { (x− a1)(x− b1) ( 1− fx q ) ϕ(x) + q(1− fx) ϕ ( x q ) V ( x q )} y(x), L2(x) = w1x(1− fx)y(x)−A1(x)y(qx) + (x− b1)ϕ(x)y(x), L3(x) = w1 x q ( 1− f x q ) y(x) + (x− a1)ϕ ( x q ) y(x)− qc21B1 ( x q ) y ( x q ) , (3.6) where ϕ is given by (2.3) and A(x) = 4∏ i=1 (x− ai), B(x) = 4∏ i=1 (x− bi), V (x) = qc21A1(x)B1(x)− w1w1x 2(1− fx)(1− fx). (3.7) Then we have Proposition 3.3. The compatibility condition of the L1 and L2 equations (3.6) is equivalent to the q-Painlevé type system (2.2). Proof. Thanks to Proposition 3.2 and the conditions (3.4), (3.5). � The pair of L1 and L2 equations (3.6) is regard as the Lax pair for the system (2.2). 4 Characterization of the L1 equation In [48] Y. Yamada has formulated a Lax form for the q-Painlevé equation of type E (1) 7 as the linear equation (say L1 = 0) and its deformed equation. Our direction T1 (2.1) is different from Yamada’s one. In general the L1 equation is expressed in terms of different dependent variables according to several deformation directions. In this section, from the viewpoint of coordinates of dependent variables, we study a connection between our L1 equation (3.6) and Yamada’s L1 equation. 13For convenience, f1 is replaced by a different symbol w1 since f1 looks like f . A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E (1) 7 7 4.1 Case of our L1 equation We consider characterizing our L1 equation (3.6) in the coordinates (f, g) ∈ P1 × P1. The compatibility of L2 and L3 equations (3.6) gives the first equation of (2.2) and two relations w1w1x 2 1(1− fx1)(1− fx1) = qc21A1(x1)B1(x1), (4.1) w1w1x 2 2(1− fx2)(1− fx2) = qc21A1(x2)B1(x2). (4.2) The relations (4.1) and (4.2) give the second equation of (2.2). Eliminating f , w1 and w1 from the expression L1 (3.6) by using the first equation of (2.2) and the relation (4.1), the expression L1 (3.6) is rewritten in terms of variables f and x1 as the expression L1(x) = A(x) ( 1− f x q )[ y(qx)− (x− b1)ϕ(x) A1(x) y(x) ] + q2c21B ( x q ) (1− fx) [ y( x q )− A1 ( x q )( x q − b1 ) ϕ ( x q )] (4.3) + c21x 2(1− fx) ( 1− f xq ) (x1 − x2)ϕ ( x q ) [( x q − x2 ) A1(x1)B1(x1) x21(1− fx1) − ( x q − x1 ) A1(x2)B1(x2) x22(1− fx2) ] y(x), where x2 = e1 c1x1 . Next when we set the expression L∗1 by L∗1(x) = (1− fx1)(1− fx2)L1(x). (4.4) Then we have: Proposition 4.1. The L∗1 equation (4.4) has the following characterization14: (i) The expression L∗1(f, g) is a polynomial of bi-degree (3, 2) in the coordinates (f, g) ∈ P1×P1. (ii) As a polynomial, the expression L∗1(f, g) vanishes at the following 12 points (fs, gs) ∈ P1 × P1 (s = 1, . . . , 12):( 1 a1 ,∞ ) , ( 1 b1 ,∞ ) , ( 1 ai ,− 1 ai − aic1 e1 ) i=2,3,4 , ( 1 bi ,− 1 bi − bic1 e1 ) i=2,3,4 ,( q x ,∞ ) , ( 1 x ,−1 x − c1x e1 ) , ( 1 x , g 1 x ) , ( q x , g q x ) , (4.5) where the first 8 points are as in (2.5) and gu is given by y ( q u ) y ( 1 u ) =: ( 1 u − b1 )( e1 + e1gu u + c1 u2 ) A1 ( 1 u ) , u = 1 x , q x . Conversely the L∗1 equation is uniquely characterized by these properties (i) and (ii). Proof. By the expression L1 (4.3), the expression L∗1 is rewritten in terms of variables f and x1 as follows L∗1(x) = A(x)(1− fx1)(1− fx2) ( 1− f x q ) y(qx) + q2c21(1− fx1)(1− fx2)(1− fx)y ( x q ) + P (x)y(x), (4.6) 14For other cases, see [16, 46] (case e-E (1) 8 ), [16, 48] (case q-E (1) 8 ). 8 H. Nagao where P (x) = q2c21(1− fx) ( 1− f xq )( x q )2 (x1 − x2)ϕ ( x q ) Q(x) − (x− a1)(x− b1)(1− fx1)(1− fx2) ( 1− f x q ) ϕ(x), Q(x) = (1− fx2) ( x q − x2 ) A1(x1)B1(x1) x21 − (1− fx1) ( x q − x1 ) A1(x2)B1(x2) x22 − (x1 − x2)(1− fx1)(1− fx2)A1 ( x q ) B1 ( x q )( 1− f xq )( x q )2 , and x1, x2 ( = e1 c1x1 ) are as in Section 3.2. Similarly to the proof of Proposition 2.1, the expres- sion Q(x) is given as a Laurent polynomial (x1 − x2) 3∑ i=0 ki(x1 + x2) i where ki depends on f , ai, bi, c1 and d1. The expression Q(x) has zeros at x = qx1, qx2 which are solutions of the equation ϕ ( x q ) = 0. Therefore, due to the relation x1 +x2 = − e1g c1 , the expression P (x) (i.e., the coefficient of y(x)) is a polynomial of bi-degree (3, 2) in (f, g). It is obvious that the coefficients of y(qx) and y ( x q ) are polynomials of bi-degree (3, 1) in (f, g). Hence the property (i) is com- pletely proved. Next the property (ii) can be easily confirmed by substituting the 12 points (4.5) into the expression L∗1 (4.6). � 4.2 Case of Yamada’s L1 equation Firstly we recall the well-known q-E (1) 7 system [5] and the corresponding Yamada’s Lax form [48]. Next we characterize Yamada’s L1 equation in terms of dependent variables. The complex parameters ai, bi, c1 and d1 be as in Section 2 and let (λ, µ) ∈ P1 × P1 be dependent variables. Then we consider a q-shift operator T2 15 as T2 = T−1a1 T −1 a2 T −1 b1 T−1b2 . (4.7) The operator T2 plays the role of the evolution of the q-E (1) 7 system. The system is well-known as the bi-rational transformation T−12 (µ) = µ(λ, µ) and T2(λ) = λ(λ, µ) in P1 × P1 as follows [5, 13, 16, 39] ( λµ− e2 c1 )( λµ− e2 qc1 ) (λµ− 1)(λµ− 1) = e22 qc21 ∏ i=1,2 (1− aiλ)(1− biλ)∏ i=3,4 (1− aiλ)(1− biλ) , ( λµ− e2 c1 )( λµ− qe2 c1 ) (λµ− 1)(λµ− 1) = ∏ i=1,2 ( µ− aie2 c1 )( µ− bie2 c1 ) ∏ i=3,4 (µ− ai)(µ− bi) , (4.8) where e2 = a3a4d1 b1b2 . Here eight singular points (λs, µs) (s = 1, . . . , 8) in coordinates (λ, µ) are on two curves λµ = 1 and λµ = e2 c1 as follows( 1 ai , aie2 c1 ) i=1,2 , ( 1 bi , bie2 c1 ) i=1,2 , ( 1 ai , ai ) i=3,4 , ( 1 bi , bi ) i=3,4 . (4.9) 15The direction T2 is given by a composition of the fundamental ones such as T1 (2.1) and T−1 a2 T−1 b2 . A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E (1) 7 9 The following scalar Lax equations L1(x) = A(x) 1− λx y(qx)− e2 ∏ i=1,2 (x− bi)(µ− x)∏ i=3,4 (x− ai) ( µ− e2x c1 )y(x)  + q2c21B ( x q ) 1− λxq y(x q ) − ∏ i=3,4 ( x q − ai )( µ− e2x qc1 ) e2 ∏ i=1,2 ( x q − bi )( µ− x q )y(x)  + c1 ( 1− c1 e2 ) x2 µ  ∏ i=3,4 (µ− ai)(µ− bi) (λµ− 1) ( µ− x q ) − ∏ i=1,2 ( µ− aie2 c1 )( µ− bie2 c1 ) ( λµ− e2 c1 )( µ− e2x c1 )  y(x), (4.10) L2(x) = w2x(1− λx)y(x)− 4∏ i=3 (x− ai) ( 1− e2x c1µ ) y(qx) + e2 2∏ i=1 (x− bi) ( 1− x µ ) y(z) are equivalent to those in [16, 24, 48] up to a gauge transformation of y(x). Here A and B are as in (3.7) and w2 is a gauge freedom (as mentioned in Section 3.2). Then the compatibility of the L1 and L2 equations (4.10) is equivalent to the system (4.8). Next setting the expres- sion L∗1 by L∗1(x) = (1− λx) ( 1− λx q ) (λµ− 1) ( λµ− e2 c1 ) L1(x), (4.11) we have: Proposition 4.2. The L∗1 equation (4.11) has the following characterization: (i) The expression L∗1 is a polynomial of bi-degree (3, 2) in the coordinates (λ, µ) ∈ P1 × P1. (ii) As a polynomial, the expression L∗1 vanishes at the following 12 points (λs, µs) ∈ P1 × P1 (s = 1, . . . , 12):( 1 ai , aie2 c1 ) i=1,2 , ( 1 bi , bie2 c1 ) i=1,2 , ( 1 ai , ai ) i=3,4 , ( 1 bi , bi ) i=3,4 ,( 1 x , x ) , ( q x , e2x qc1 ) , ( 1 x , µ 1 x ) , ( q x , µ q x ) , (4.12) where the first 8 points are as in (4.9) and µu is given by Y ( qu) Y ( 1 u) =: e2 ∏ i=1,2 ( 1 u − bi )( µu − 1 u ) ∏ i=3,4 ( 1 u − ai )( µu − e2 c1u ) , u = 1 x , q x . Conversely the equation L∗1 is uniquely characterized by these properties (i) and (ii). Proof. This proof is the similar as for Proposition 4.1. � 4.3 Correspondence between two L1 equations Thanks to Propositions 4.1 and 4.2, we have: 10 H. Nagao Theorem 4.3. The L1 equation (4.3) is equivalent to Yamada’s L1 equation (4.10) under a re- lation f = λ, ( λµ− e2 c1 )( e1f 2 + e1fg + c1 ) e2(1− b2λ)(λµ− 1)(1− a2f) = 1. (4.13) Proof. Comparing the last two points of (4.5) in Proposition 4.1 with ones of (4.12) in Propo- sition 4.2, we obtain the transformation (4.13) as a necessary condition to change the L∗1 equa- tion (4.11) into the L∗1 equation (4.4). Conversely, under the relation (4.13), µ(f, g) is written as a rational function with the numerator and the denominator of bi-degree (1, 1) in (f, g) re- spectively. Substituting the expression µ(f, g) (4.13) into the L∗1 equation (4.11), it is shown that the algebraic curve L∗1 = 0 (4.11) of bi-degree (3, 2) in (λ, µ) changes to the algebraic curve of bi-degree (5, 2) in (f, g). Furthermore, due to the relation (4.13), 12 points (4.12) are changed to 12 points (4.5) and 2 lines f = 1 a2 , f = 1 b2 . Namely it turns out that the algebraic curve L∗1(λ, µ) = 0 (4.11) of bi-degree (3, 2) is changed to the algebraic curve L∗1(f, g) × (1 − a2f)(1 − b2f) = 0 (4.4) of bi-degree (3, 2) × (2, 0) in (f, g). Hence, the rela- tion (4.13) is proved to be the sufficient condition that the L1 equation (4.10) corresponds with the L1 equation (4.3). � We note that the system (2.2) has the Lax pair whose L1 equation (4.3) is equivalent to that (4.10) of the q-E (1) 7 system (4.8) and clarify the relation (4.13) of the dependent variables between the systems (2.2) and (4.8). 5 Particular solutions In this section we recall the particular solutions for the system (3.3) given in [28], and derive ones for the system (2.2) by the similar reduction as in Section 3.2. 5.1 Case of the q-Garnier system The contents are extracts from [28, Section 5.2]. For convenience we change the notations in Section 3.1 as follows a1 . . . aN aN+1 b1 . . . bN bN+1 c1 c2 d1 d2  7→  1 a1 . . . 1 aN qm+n 1 b1 . . . 1 bN 1 q cqn N∏ 1 bi ai qm c 1  , (5.1) where a1, . . . , aN , b1, . . . , bN , c ∈ C× and m,n ∈ Z≥0. Correspondingly we replace the nota- tions A, B etc. (3.2) by A(x) = N∏ i=1 (aix)1, B(x) = N∏ i=1 (bix)1, A1(x) = A(x) (a1x)1 , B1(x) = B(x) (b1x)1 . (5.2) We also replace the evolution direction (2.1) by T1 = Ta1Tb1 . (5.3) We show particular solutions in terms of the τ function τm,n = det [ N+1ϕN ( b1,...,bN ,q −(m+n) a1,...,aN , cqi+j+1 )]n i,j=0 , (5.4) A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E (1) 7 11 where the generalized q-hypergeometric function N+1ϕN 16 [3] is defined by (1.1). Then we have the following fact17. Proposition 5.1. The polynomials F (x) and G(x) determined by F ( 1 ai ) F ( 1 bj ) = α Tai(τm,n)T−1ai (τm+1,n−1) T−1bj (τm,n)Tbj (τm+1,n−1) , i, j = 1, . . . , N, G ( 1 ai ) = β Tai(τm,n)T−1ai (τm+1,n−1) Ta1(τm,n)T−1a1 (τm+1,n−1) , i = 2, . . . , N, G ( 1 bi ) = γ T−1bi (τm,n)Tbi(τm+1,n−1) T−1b1 (τm,n)Tb1(τm+1,n−1) , i = 2, . . . , N, (5.5) give particular solutions of a bi-rational equation G(x)G(x) = c ( qx, x qm+n ) 1 A1(x)B1(x) (a1x, b1x)1 for F (x) = 0, F (x)F (x) = ( qx, x qm+n ) 1 A1(x)B1(x) for G(x) = 0, fNfN = qa1b1 c ( gN−1 − c N∏ i=2 (−bi) a1qm−1 )( gN−1 − N∏ i=2 (−ai) b1qn ) , f0f0 = ( g0, g0 c ) 1 . Here α, β and γ are given by α = −cqn−m (aiq m+n)1 ( ai q )n 1 ( bj q )n 1 (ai) n+1 1 (bj)n1 B ( 1 ai ) A ( 1 bj ) , β = c (b1, aiq m+n)1 ( ai q )n 1 B1 ( 1 ai ) a1qm ( b1 a1 ) 1 (ai) n+1 1 , γ = (a1)1(bi) n 1A1 ( 1 bi ) b1qn ( a1 b1 ) 1 ( bi q )n 1 . 5.2 Reduction to the q-Painlevé type system In this subsection we derive particular solutions for the Painlevé type equation (2.2). In a similar way as in Section 3.2, we consider the reduction from the particular case N = 3 of the q-Garnier system (3.3). In order to do this, we impose a constraint of parameters c = 1, qm−n 3∏ i=1 ai bi = 1, (5.6) and the specialization (3.5). Then the tau function τm,n (5.4) is reduced to the following function τm,n = det [ 4ϕ3 ( b1,b2,b3,q−(m+n) a1,a2,a3 , qi+j+1 )]n i,j=0 , where the generalized q-hypergeometric function 4ϕ3 is defined by (1.1). As the case of a reduc- tion of Proposition 5.1, we have the following18. 16In [43, 44] the particular solutions of a higher order q-Painlevé system was constructed in terms of the q-hypergeometric function N+1ϕN by T. Suzuki. 17For the proof of Proposition 5.1, see [28, Section 5]. 18As another derivation of the particular solutions (5.7), see Section A.2 (Padé interpolation method). 12 H. Nagao Proposition 5.2. The particular values of f and g determined by 1− f ai 1− f bj = α′ Tai(τm,n)T−1ai (τm+1,n−1) T−1bj (τm,n)Tbj (τm+1,n−1) , i, j = 1, 2, 3, 1 + g ai + κ a2i = β′ Tai(τm,n)T−1ai (τm+1,n−1) Ta1(τm,n)T−1a1 (τm+1,n−1) , i = 2, 3, 1 + g bi + κ b2i = γ′ T−1bi (τm,n)Tbi(τm+1,n−1) T−1b1 (τm,n)Tb1(τm+1,n−1) , i = 2, 3, (5.7) give particular solutions of the following overdetermined bi-rational equation: ( f2 + gf + κ )( f2 + gf + qκ ) = ( f − 1 qm+n ) (f − q) ∏ i=2,3 (f − ai)(f − bi) (f − a1)(f − b1) , x21(1− fx1)(1− fx1) x22(1− fx2)(1− fx2) = ( 1− x1 qm+n ) (1− qx1) ∏ i=2,3 (1− aix1)(1− bix1)( 1− x2 qm+n ) (1− qx2) ∏ i=2,3 (1− aix2)(1− bix2) . (5.8) Here α′, β′ and γ′ are given by α′ = −qn−m ai(aiq m+n)1 ( ai q , bj q )n 1 bj(ai) n+1 1 (bj)n1 3∏ s=1 ( bs ai ) 1 3∏ s=1 ( as bj ) 1 , β′ = ( b1, aiq m+n, b2ai , b3 ai ) 1 ( ai q )n 1 a1qm ( b1 a1 ) 1 (ai) n+1 1 , γ′ = ( a1, a2 bi , a3bi ) 1 (bi) n 1 b1qn ( a1 b1 ) 1 ( b2 q )n 1 , and the evolution direction is as in T1 (5.3) and x = x1, x2 are solutions of an equation ϕ = 0: ϕ(x) = 1 + gx+ κx2, (5.9) where κ = a2a3 b1qn . Proof. Substituting the conditions (5.6) and (3.5) into the particular solutions (5.5), we ob- tain (5.7). � 6 Conclusions The main results of this paper are the following. • We showed in Proposition 2.1 that the q-Painlevé type system (2.2) is the bi-rational transformation and is related to the novel realization (i.e., configuration) (2.5) of the symmetry/surface of type E (1) 7 /A (1) 1 . Then the system (2.2) turned out to be a variation of the q-E (1) 7 system (4.8). • We obtained the Lax equations (3.6) for the system (2.2) from the reduction of the par- ticular case N = 3 of the q-Garnier system (3.3), and clarified the connection between the system (2.2) and the q-E (1) 7 system (4.8) by comparing their L1 equations in Theorem 4.3. • In Proposition 5.2 the determinant formulas of the particular solutions for the system (2.2) was expressed in terms of the generalized q-hypergeometric function 4ϕ3 through the si- milar reduction of the particular case N = 3. A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E (1) 7 13 Extending the results of this paper, we naturally have the following open problems. One may consider several variations of the q-E7 system according to several deformation direction such as T1 and T2, and investigates a connection among these systems. We will carry out similar research on discrete Painlevé and Garnier systems [29]. It seems to be interesting to study reductions of cases N ≥ 4 of the q-Garnier system. A From Padé interpolation to q-Painlevé type system By using a Padé interpolation problem with q-grid, in [24] we derived the scalar Lax pair, the evolution equation and the particular solutions for the q-E (1) 7 system. In this appendix, in a similar manner as in [24], we directly derive the data of the q-Painlevé type system (2.2) given in Sections 3.2 and 5.2. A.1 Scalar Lax pair and evolution equation Suppose we have complex parameters q (\|q\| < 1), a1, a2, a3, b1, b2 and b3 ∈ C× with the constraint (5.6). Then we consider a function ψ(x) = 3∏ i=1 (aix, bi)∞ (ai, bix)∞ . (A.1) Let P (x) and Q(x) be polynomials of degree m and n ∈ Z≥0 in x. Then we assume that the polynomials P and Q satisfy the following Padé interpolation condition: ψ(xs) = P (xs) Q(xs) , xs = qs, s = 0, 1, . . . ,m+ n. (A.2) The common normalizations of the polynomials P and Q in x are fixed as P (0) = 1. The parameter shift operator is given by T1 (5.3). Consider two linear relations: L2 = 0 among y(x), y(qx), y(x) and L3 = 0 among y(x), y(x), y ( x q ) satisfied by the functions y = P and y = ψQ. Then we have: Proposition A.1. The linear relations L2 and L3 19 can be expressed as follows L2(x) = C0x(fx)1y(x)− ( x qm+n ) 1 A1(x)y(qx) + (b1x)1ϕ(x)y(x) = 0, L3(z) = C1 x q ( fx q ) 1 y(x) + (a1x)1ϕ ( x q ) y(x)− (x)1B1 ( x q ) y ( x q ) = 0, (A.3) where ϕ is given by (5.9) and A1, B1 are the same as the case N = 3 of (5.2). Here f, g, C0, C1 ∈ P1 are constants depending on parameters ai, bj ∈ C×, m,n ∈ Z≥0. Proof. By the definition of the relations L2 = 0 and L3 = 0, they can be written as L2(x) ∝ ∣∣∣∣y(x) y(qx) y(x) y(x) y(qx) y(x) ∣∣∣∣ = D1(x)y(x)−D2(x)y(qx) +D3(x)y(x) = 0, L3(x) ∝ ∣∣∣∣∣y(x) y(x) y ( x q ) y(x) y(x) y ( x q )∣∣∣∣∣ = D1 ( x q ) y(x) +D3 ( x q ) y(x)−D2(x)y ( x q ) = 0, (A.4) 19L2 = 0 and L3 = 0 (A.3) can be derived by substituting (5.1), (5.6) and w1 → − q2mC0 a2a3 , w1 → − q2mC1 b2b3 into L2 = 0 and L3 = 0 (3.6). 14 H. Nagao where y(x) = [ P (x) ψ(x)Q(x) ] and Casorati determinants D1(x) = \|y(x),y(qx)\|, D2(x) = \|y(x),y(x)\|, D3(x) = \|y(qx),y(x)\|. (A.5) Taking note of the relations ψ(qx) ψ(x) = B(x) A(x) , ψ(x) ψ(x) = (a1, b1x)1 (a1x, b1)1 , (A.6) where A and B are the same as the case N = 3 of (5.2), we rewrite the Casorati determi- nants (A.5) into the following determinants D1(x) = ψ(x) A(x) R1(x) =: ψ(x) A(x) m+n−1∏ i=0 ( x qi ) 1 c0x(fx)1, D2(x) = ψ(x) (a1x, b1)1 R2(x) =: ψ(x) (a1x, b1)1 m+n∏ i=0 ( x qi ) 1 c′0, D3(x) = ψ(x) A(x) R3(x) =: ψ(x) A(x)(b1)1 m+n−1∏ i=0 ( x qi ) 1 c′0(b1x)1ϕ(x), (A.7) where R1(x) = B(x)P (x)Q(qx)−A(x)P (qx)Q(x), R2(x) = (a1, b1x)1P (x)Q(x)− (a1x, b1)1P (x)Q(x), R3(x) = (a1, b1x)1A1(x)P (qx)Q(x)− (b1)1B(x)P (x)Q(qx). (A.8) Here c0 and c′0 are some constants depending on the parameters ai, bj , m and n. Computing Taylor expansions at x = 0 and x =∞ in the expressions R2(x) and R3(x)(A.8), we determine ϕ by (5.9). As a result, we obtain the desired relations L2 and L3 (A.3) where C0 = c0 c′0 and C1 = (a1)1c0 c′0 . � Next we have: Proposition A.2. The constants f and g satisfy the q-Painlevé type system (5.8), and they play the role of dependent variables for (5.8). Proof. The compatibility of the relations (A.3) gives the system (5.8). � A.2 Particular solution We construct particular solutions of the q-Painlevé type system (2.2) given in terms of the q- hypergeometric function 4ϕ3 in Section 5.2. We derive the explicit forms (5.7) of variables {f, g} appearing in the Casorati determinants D1 and D3 (A.7). They are interpreted as the particular solutions for the system (5.8), due to Proposition A.2. Proposition A.3 ([7], see also [6, 24, 28]). For a given sequence ψs, the polynomials P (x) and Q(x) of degree m and n satisfying a Padé interpolation problem ψs = P (xs) Q(xs) , s = 0, 1, . . . ,m+ n, (A.9) A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E (1) 7 15 is given as the following determinant expressions: P (x) = F(x) det [ m+n∑ s=0 us xi+js x− xs ]n i,j=0 , Q(x) = det [ m+n∑ s=0 usx i+j s (x− xs) ]n−1 i,j=0 , (A.10) where us = ψs F ′(xs) and F(x) = m+n∏ i=0 (x− xi). Proposition A.4 ([6, 24, 28]). In the q-grid case of the problem (A.9) (i.e., interpolation points xs = qs), the formula (A.10) takes the following form: P (x) = F(x) (q)n+1 m+n det [ m+n∑ s=0 ψs (q−(m+n))s (q)s qs(i+j+1) x− qs ]n i,j=0 , Q(x) = 1 (q)nm+n det [ m+n∑ s=0 ψs (q−(m+n))s (q)s qs(i+j+1)(x− qs) ]n−1 i,j=0 . (A.11) Proof. Substituting the expressions F(x) = m+n∏ s=0 ( x− qs ) , F ′(xs) = (q)s(q)m+n qs(q−(m+n))s , (A.12) into the formula (A.10), then we obtain the desired form (A.11). � Remark A.5. The normalization of the polynomials P (x) and Q(x) expressed in the formu- las (A.10) and (A.11) differ from the convention P (0) = 1 as fixed beneath the interpolation condition (A.2). This difference does not influence the result in Proposition A.7, because the common normalization factors of P and Q cancel in (A.14) and (A.15). Proposition A.6. The polynomials P (x) and Q(x) defined in Section A.1 have the following particular values: P ( 1 as ) = (as)m+n+1 ams (as) n+1 1 (q)n+1 m+n Tas(τm,n), Q ( q as ) = qn ( as q )n 1 ans (q)nm+n T−1as (τm+1,n−1), P ( q bs ) = qm ( bs q ) m+n+1 bms ( bs q )n+1 1 (q)n+1 m+n T−1bs (τm,n), Q ( 1 bs ) = (bs) n 1 bns (q)nm+n Tbs(τm+1,n−1), (A.13) for s = 1, 2, 3. Here τm,n is defined by (5.4). Proof. This proof follows from the formula (A.11) and the sequence ψs = ψ(qs) = 3∏ i=1 (bi)s (ai)s . � Proposition A.7. The particular values of f and g determined by (5.7) give particular solutions of the system (5.8). Proof. From the first equation of (A.7), we have 1− f ai 1− f bj = −ai bj m+n−1∏ s=0 ( 1 bjqs ) 1( 1 aiqs ) 1 B ( 1 ai ) A ( 1 bj ) P ( 1 ai ) Q ( q ai ) P ( q bj ) Q ( 1 bj ) , i, j = 1, 2, 3, (A.14) 16 H. Nagao where A and B are as in Appendix A.1. From the second and third equations of (A.7), we have 1 + g ai + κ a2i = − m+n∏ s=0 ( 1 a1qs ) 1 m+n−1∏ s=0 ( 1 aiqs ) 1 (b1)1B1 ( 1 ai )( a1, b1 a1 ) 1 P ( 1 ai ) Q ( q ai ) P ( 1 a1 ) Q ( 1 a1 ) , i = 2, 3, 1 + g bi + κ b2i = − m+n∏ s=0 ( 1 b1qs ) 1 m+n−1∏ s=0 ( 1 biqs ) 1 (a1)1A1 ( 1 bi )( a1 b1 , b1 ) 1 P ( q bi ) Q ( 1 bi ) P ( 1 b1 ) Q ( 1 b1 ) , i = 2, 3, (A.15) where A1 and B1 are as in Appendix A.1. Substituting the particular values (A.13) into the expressions (A.14) and (A.15) respectively, we obtain the desired particular solutions (5.7). � Acknowledgements The author shall be thankful to Professor Yasuhiko Yamada for valuable discussions. The author is also grateful to the referees for stimulating comments. 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[50] Yamada Y., An elliptic Garnier system from interpolation, SIGMA 13 (2017), 069, 8 pages, arXiv:1706.05155. https://doi.org/10.1090/conm/651/13037 https://arxiv.org/abs/1105.4240 https://doi.org/10.1093/integr/xyw017 https://doi.org/10.1093/integr/xyw017 https://arxiv.org/abs/1602.01573 https://doi.org/10.1619/fesi.46.173 https://doi.org/10.3842/SIGMA.2009.042 https://arxiv.org/abs/0811.1796 https://doi.org/10.1619/fesi.52.83 https://doi.org/10.1093/imrn/rnq232 https://doi.org/10.1093/imrn/rnq232 https://arxiv.org/abs/1004.1687 https://doi.org/10.3842/SIGMA.2017.069 https://arxiv.org/abs/1706.05155 1 Introduction 1.1 Background 1.2 Purpose and organization 2 q-Painlevé type system 3 Lax equations 3.1 Case of the q-Garnier system 3.2 Reduction to the q-Painlevé type system 4 Characterization of the L1 equation 4.1 Case of our L1 equation 4.2 Case of Yamada's L1 equation 4.3 Correspondence between two L1 equations 5 Particular solutions 5.1 Case of the q-Garnier system 5.2 Reduction to the q-Painlevé type system 6 Conclusions A From Padé interpolation to q-Painlevé type system A.1 Scalar Lax pair and evolution equation A.2 Particular solution References

A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E₇⁽¹⁾

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