A Variation of the $q$-Painlev\'e System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$

Recently a certain $q$-Painlev\'e type system has been obtained from a reduction of the $q$-Garnier system. In this paper it is shown that the $q$-Painlev\'e type system is associated with another realization of the affine Weyl group symmetry of type $E_7^{(1)}$ and is different from the well-known $q$-Painlev\'e system of type $E_7^{(1)}$ from the point of view of evolution directions. We also study a connection between the $q$-Painlev\'e type system and the $q$-Painlev\'e system of type $E_7^{(1)}$. Furthermore determinant formulas of particular solutions for the $q$-Painlev\'e type system are constructed in terms of the terminating $q$-hypergeometric function.


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 P 1 ×P 1 (resp. P 2 ) by blowing up at 8 (resp. 9) singular points. In view of the configuration of 8 (resp. 9) singular points in P 1 × P 1 (resp. P 2 ), 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 P 1 × P 1 . The second order continuous and discrete Painlevé equations are classified into the 22 cases 2 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) 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 10 E 9 [3]. In the case of q-E (1) 7 , the particular solutions are expressed in [24] in terms of the terminating q-hypergeometric function 4 ϕ 3 , 4 where the function k ϕ l [3] is defined by Here the standard q-Pochhammer symbol 5 is defined by 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 P 1 × P 1 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-P VI (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 solutions 7 for the q-Garnier system have been simply expressed by applying a certain method of Padé approximation and 3 P D (1) i III symbolizes PIII having the surface connected to the affine root system of type D (1) i . 4 The 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]. 5 Actually Pochhammer himself used the symbol (a)n not as a rising shifted factorial but as a binomial coefficient [17]. 6 For the related works, see [1,2,35] (additive Garnier system), [36,50] (elliptic Garnier system). 7 These solutions have been constructed in terms of the q-Appell Lauricella function (resp. the generalized q-hypergeometric function) in [28,40] (resp. [28]). dependent variables. Define T a : a → qa by a q-shift operator of parameter a. Then we consider a q-shift operator T 1 11 Here for any object X the corresponding shifts are denoted as X := T 1 (X) and X := T −1 1 (X). The operator T 1 plays the role of the evolution of the system. The system is described by the following transformation T −1 1 (g) = g(f, g) and T 1 (f ) = f (f, g) in P 1 × P 1 : Here Then we have Proposition 2.1. The q-Painlevé type system (2.2) has the following properties: It is associated with a novel realization (2.5) of the symmetry/surface of type E (1) Proof . (i) It is easy to see that the first equation of (2.2) is a rational transformation T −1 1 (g) = g(f, g). The second equation of (2.2) is rewritten as Then the numerator and denominator of (2.4) are alternating with respect to x 1 ↔ x 2 = e 1 c 1 x 1 , and Laurent polynomials h i x i 1 where h i is depending on f , a i , b j , c 1 and d 1 . Accordingly the numerator and denominator are expressed by Laurent is given as a rational polynomial of bi-degree (1, 3) in (f, g). Therefore the property (i) is proved. (ii) Eight singular points (f s , g s ) ∈ P 1 × P 1 (s = 1, . . . , 8) in coordinates (f, g) are on one line g = ∞ of bi-degree (0, 1) and one parabolic curve e 1 f 2 + e 1 f g + c 1 = 0 of bi-degree (2, 1) as follows: Hence the property (ii) is confirmed since the configuration (2.5) is the realization of the surface type A 1 .
According to Remark 1.1, the system (2.2) is regarded as a variation of the q-E (1) 7 system. 11 The operator T1 is generally selected as T −1

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).

Case of the q-Garnier system
The scalar Lax equations for the q-Garnier system are Here the deformation direction is T 1 (2.1) and f 0 , . . . , f N , g 0 , . . . , g N −1 ∈ P 1 are variables de- Remark 3.1. The scalar Lax pair L 1 = 0 and L 2 = 0 (or L 3 = 0) is equivalent to the pair of the deformation equations L 2 = 0 and L 3 = 0.
The equation L 1 = 0 (we call it the L 1 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 N −1 are the dependent variables. Then we have the following fact 12 .

Reduction to the q-Painlevé type system
We impose a reduction condition by a constraint of the parameters and specialize the dependent variables as where w 1 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 where ϕ is given by (2.3) and Then we have Proposition 3.3. The compatibility condition of the L 1 and L 2 equations (3.6) is equivalent to the q-Painlevé type system (2.2).
The pair of L 1 and L 2 equations (3.6) is regard as the Lax pair for the system (2.2).

Characterization of the L 1 equation
In [48] Y. Yamada has formulated a Lax form for the q-Painlevé equation of type E 7 as the linear equation (say L 1 = 0) and its deformed equation. Our direction T 1 (2.1) is different from Yamada's one. In general the L 1 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 L 1 equation (3.6) and Yamada's L 1 equation.

Case of our L 1 equation
We consider characterizing our L 1 equation (3.6) in the coordinates (f, g) ∈ P 1 × P 1 . The compatibility of L 2 and L 3 equations (3.6) gives the first equation of (2.2) and two relations The relations (4.1) and (4.2) give the second equation of (2.2). Eliminating f , w 1 and w 1 from the expression L 1 (3.6) by using the first equation of (2.2) and the relation (4.1), the expression L 1 (3.6) is rewritten in terms of variables f and x 1 as the expression Then we have: has the following characterization 14 : (i) The expression L * 1 (f, g) is a polynomial of bi-degree (3,2) in the coordinates (f, g) ∈ P 1 ×P 1 . (ii) As a polynomial, the expression L * 1 (f, g) vanishes at the following 12 points (f s , g s ) ∈ P 1 × P 1 (s = 1, . . . , 12): where the first 8 points are as in (2.5) and g u is given by Conversely the L * 1 equation is uniquely characterized by these properties (i) and (ii).
Proof . By the expression L 1 (4.3), the expression L * 1 is rewritten in terms of variables f and x 1 as follows where , and x 1 , x 2 = e 1 c 1 x 1 are as in Section 3.2. Similarly to the proof of Proposition 2.1, the expres- The expression Q(x) has zeros at x = qx 1 , qx 2 which are solutions of the equation ϕ x q = 0. Therefore, due to the relation x 1 + x 2 = − e 1 g c 1 , 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 completely proved. Next the property (ii) can be easily confirmed by substituting the 12 points (4.5) into the expression L * 1 (4.6).
(4.9) 15 The direction T2 is given by a composition of the fundamental ones such as T1 (2.1) and T −1 (1) 7 9 The following scalar Lax equations 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 w 2 is a gauge freedom (as mentioned in Section 3.2). Then the compatibility of the L 1 and L 2 equations (4.10) is equivalent to the system (4.8). Next setting the expression L * 1 by 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) ∈ P 1 × P 1 . (ii) As a polynomial, the expression L * 1 vanishes at the following 12 points (λ s , µ s ) ∈ P 1 × P 1 (s = 1, . . . , 12): where the first 8 points are as in (4.9) and µ u is given by 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.  Proof . Comparing the last two points of (4.5) in Proposition 4.1 with ones of (4.12) in Proposition 4.2, we obtain the transformation (4.13) as a necessary condition to change the L * 1 equation (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) respectively. 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

Correspondence between two L 1 equations
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 g). Hence, the relation (4.13) is proved to be the sufficient condition that the L 1 equation (4.10) corresponds with the L 1 equation (4.3).
We note that the system (2.2) has the Lax pair whose L 1 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).

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.

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 where a 1 , . . . , a N , b 1 , . . . , b N , c ∈ C × and m, n ∈ Z ≥0 . Correspondingly we replace the notations A, B etc. (3.2) by We also replace the evolution direction (2.1) by We show particular solutions in terms of the τ function where the generalized q-hypergeometric function N +1 ϕ N 16 [3] is defined by (1.1). Then we have the following fact 17 .

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 and the specialization (3.5). Then the tau function τ m,n (5.4) is reduced to the following function where the generalized q-hypergeometric function 4 ϕ 3 is defined by (1.1). As the case of a reduction of Proposition 5.1, we have the following 18 .
The particular values of f and g determined by , i, j = 1, 2, 3, give particular solutions of the following overdetermined bi-rational equation: Here α , β and γ are given by , and the evolution direction is as in T 1 (5.3) and x = x 1 , x 2 are solutions of an equation ϕ = 0: where κ = a 2 a 3 b 1 q n . Proof . Substituting the conditions (5.6) and (3.5) into the particular solutions (5.5), we obtain (5.7).

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 . 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 particular case N = 3 of the q-Garnier system (3.3), and clarified the connection between the system (2.2) and the q-E 7 system (4.8) by comparing their L 1 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 4 ϕ 3 through the similar reduction of the particular case N = 3.

13
Extending the results of this paper, we naturally have the following open problems. One may consider several variations of the q-E 7 system according to several deformation direction such as T 1 and T 2 , 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), a 1 , a 2 , a 3 , b 1 , b 2 and b 3 ∈ C × with the constraint (5.6). Then we consider a function 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: The common normalizations of the polynomials P and Q in x are fixed as P (0) = 1. The parameter shift operator is given by T 1 (5.3). Consider two linear relations: L 2 = 0 among y(x), y(qx), y(x) and L 3 = 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 L 2 and L 3 19 can be expressed as follows where ϕ is given by (5.9) and A 1 , B 1 are the same as the case N = 3 of (5.2). Here f, g, C 0 , C 1 ∈ P 1 are constants depending on parameters a i , b j ∈ C × , m, n ∈ Z ≥0 .
Proof . By the definition of the relations L 2 = 0 and L 3 = 0, they can be written as  Taking note of the relations where A and B are the same as the case N = 3 of (5.2), we rewrite the Casorati determinants (A.5) into the following determinants where 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 qhypergeometric function 4 ϕ 3 in Section 5.2. We derive the explicit forms (5.7) of variables {f, g} appearing in the Casorati determinants D 1 and D 3 (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 ψ s , the polynomials P (x) and Q(x) of degree m and n satisfying a Padé interpolation problem Proof . This proof follows from the formula (A.11) and the sequence ψ s = ψ(q s ) = Proof . From the first equation of (A.7), we have i, j = 1, 2, 3, (A.14) where A and B are as in Appendix A.1. From the second and third equations of (A.7), we have where A 1 and B 1 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).