A 3 × 3 Lax Form for the q -Painlev´e Equation of Type E 6

. For the q -Painlev´e equation with affine Weyl group symmetry of type E (1)6 , a 2 × 2 matrix Lax form and a second order scalar lax form were known. We give a new 3 × 3 matrix Lax form and a third order scalar equation related to it. Continuous limit is also discussed.


Introduction
The q-Painlevé equation with affine Weyl group symmetry of type E (1) 6 was first discovered in [9].The well-known form of it is as follows T : (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ; f, g) → a 1 /q, a 2 /q, a 3 /q, a 4 /q, a 5 , a 6 , a 7 , a 8 ; f , g , f g − 1 (f g − 1) f f = (g − 1/a 5 )(g − 1/a 6 )(g − 1/a 7 )(g − 1/a 8 ) (g − a 3 )(g − a 4 ) , f − a 1 /q f − a 2 /q , q = a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 , where f and g are dependent variables, a 1 , a 2 , . . ., a 8 are parameters, q is a constant and the overline symbol "¯" denotes the discrete time evolution.In previous works, the following Lax forms for the q-Painlevé equation for type E 6 has been obtained.In [12], a 2 × 2 matrix Lax pair was first derived as a reduction of the q-Garnier system.The other approach gives a 2 × 2 matrix Lax pair [2,7].In [16], a second order scalar Lax form was obtained as a reduction from the q-Painlevé equation of type E 8 .The relation between these Lax forms was given in [15].
In this article, we give a new Lax form with 3 × 3 matrix Lax pair.We derive such a Lax form as a special case of the system investigated in our previous work [8].As a result, we derive an equation which is equivalent to the q-Painlevé equation of type E (1) 6 [4,10].In the previous work [8], we defined a nonlinear q-difference system as a connection preserving deformation of the following linear equation where the exponents are ε i = ±1 (1 ≤ i ≤ M ), u j,i (1 ≤ j ≤ N ) are dependent variables and c i , d j are parameters which satisfy N j=1 u j,i = c i .
Since one can exchange the order of matrices X ±1 i by suitable rational transformations of variables u j,i , the equation (1.1) essentially depends on M + , M − , where M ± = #{ε i |ε i = ±1}.
The contents of this paper is as follows.In Section 2, we set up a linear q-difference equation (2.1), which is a case of (M + , M − , N ) = (3, 0, 3) for the equation (1.1).And we discuss about its two deformations.One deformation gives rise to a well known form of the q-Painlevé equation of type E (1) 6 , and the other deformation gives an equation for a non-standard direction.Namely, it does not give a q-shift deformation for parameters.In Section 3, we derive a scalar equation from the 3 × 3 matrix equation (2.1) and consider its characteristic properties.In Section 4, we study continuous limit of our constructions and its relation to the Boalch's Lax pair [1].In Appendix A, we give deformations considered in Section 2 on root variables.In Appendix B, we give explicit forms of coefficients of a single linear q-difference equation derived in Section 3.

A 3 × 3 matrix Lax form
In this section, we consider two types of deformations for the linear q-difference equation (1.1) in a case (M + , M − , N ) = (3, 0, 3).
We consider the connection preserving deformation for the following q-difference equation for an unknown function Ψ where the matrices D and X i (z) (1 ≤ i ≤ 3) stand for and u j,i (1 ≤ i, j ≤ 3) are dependent variables and c i , d j are parameters which satisfy The first equation in (2.2) is equivalent to that the characteristic exponents at z = 0 of the equation (2.1) are b j d j .The second equation in (2.2) is equivalent to the following condition: Through a gauge transformation by a 3 × 3 diagonal matrix we can take two components in (2.1) as 1.In the following, we use this kind of gauge fixings in case by case.By the condition (2.3), two of the remaining four components are determined by other components and parameters b j , c i , d j .
In this article, we will consider two deformations T 1 and T 2 for the equation (2.1) which act on parameters b j , c i , d j as 2.1 The deformation T 1 We will show that the deformation T 1 gives rise to the standard form of the q-E (1) 6 .We consider the following q-difference linear equation: ) Although there are many ways of gauge fixings, fixing as the equation (2.4) makes relatively easier to find a new change of variables from indeterminate points of time evolution equations v i ( * = T 1 ( * )).As a connection preserving deformation for the equation (2.4), (2.5) we take the following deformation for parameters Then, there is a matrix B(z) which satisfies the following deformation equation: We derive and show the matrix B(z).First, the matrix B(z) is a rational function for z [5].
In fact, by the argument [5] to determine a coefficient matrix of a deformation equation, the matrix B(z) is of degree one through the following: (i) The parameters c i which satisfy |A(c i )| = 0 are constant by the deformation T 1 (2.6).Therefore, the matrix B(z) does not have poles at z = −q k c i (k ∈ Z).Namely, the matrix B(z) has poles only at z = 0 or z = ∞.
(ii) The deformation T 1 (2.6) shifts characteristic exponents of the equation (2.4) at z = 0 as follows: Therefore, the matrix B(z) behaves nearby z = 0 as (iii) The deformation T 1 (2.6) shifts characteristic exponents of the equation (2.4) at z = ∞ as follows: Therefore, the matrix B(z) behaves nearby z = ∞ as From the above (i)-(iii), the matrix B(z) is a polynomial in z of degree one.At last, we derive an explicit form of the matrix B(z).We express the matrix B(z) as follows: Comparing coefficients of z for a compatibility condition equation for the matrices A(z) (2.4) and B(z) (2.7), we have the following three equations: where matrices A i denote coefficients of z i for the matrix A(z) (2.4).Solving the first and the third equations of (2.8), forms of the matrices B 0 and B 1 are as follows: From the second equation of (2.8), we obtain explicit forms of time evolutions v i (1 ≤ i ≤ 4), the components β 0 2,3 and β 1 2,1 of the matrix B(z) (2.7).Explicit forms of the remain components β 0 2,3 and β 1 2,1 are as follows: .
The equation (2.12) is the well known form of the q-Painlevé equation of type E 6 [4,10] (see also [6]).
Proof .The result is obtained by a direct computation of the compatibility condition of (2.11).Since the computation is rather heavy, we will give a comment how to do it efficiently.Though the 2 variables v 2 , v 3 can be represented by the rational functions of the remaining two variables v 1 , v 4 by the relation (2.5), it is more efficient to do this elimination after the calculation of the compatibility condition (2.11) in 4 variables, and then reduce it to 2 variables.In this way, we get the following time evolutions for v 1 , v 4 as rational functions of v 1 , v 4 : where The remaining task is to rewrite the equation (2.14) to (2.12).A useful way to solve it is to look at the singularities of the equations [3].Namely, we investigate the points at which the right hand side of the equations (2.14) are indeterminate.We focus on the equation Investigating common zero points of the equation (2.15) and the other polynomials , we find 4 indeterminate points as follows: (2.16) The other 4 points are the following: In view of the form of the points in (2.16), we define the variables f , g as follows: By the transformation (2.17), the equation Through the correspondence (2.17) and the time evolution equations for the variables v 1 , v 4 (2.14), we have time evolution equations for f , g (2.12).■

The deformation T 2
In this subsection, we take a deformation equation which is one of that considered in the previous work [8].It corresponded to the permutations of the matrices X i (z) ±1 .We consider the following Lax pair: where we define variables x, y and gauge freedom w 1 , w 2 with the variables u j,i in (2.18) as follows: we obtain the following equations: where and * stands for T 2 ( * ).
Proof .Solving a compatibility condition (2.21) with (2.20) for the variables x, y, we obtain the following equations where ) and H k (x, y) are of degree (2, 2).Using a method [6] for finding point configuration, a configuration of points for the equation (2.22), (2.23) is as follows:

25). ■
In Appendix A, we give pictures of a point configuration of the equation (2.12) and the configuration (2.24), root basis associated with them, and three deformations T 1 , T 2 and T 3  2 on root variables attached with root basis.
Before deriving, we state about a characterization of a linear q-difference equation.Linear differential equations are characterized by its singular points and characteristic exponents.Similarly, linear q-difference equations are also characterized by its singular points and characteristic exponents.We consider the following n-th order q-difference equation In the q-difference equation (3.1), singular points are at z = 0 and z = ∞.And characteristic exponents of solutions at z = 0 and z = ∞ are given as solutions of the following characteristic equations respectively To characterize the q-difference equation (3.1) is namely to determine coefficients p k,m .The number of coefficients p k,m in the equation (3.1) is (n+1)(n+2l+2)

2
. Through the following, total 3n + 2l − 1 coefficients p k,m are determined: (i) We express parameters a i and b j as zeroes of the coefficients P n (z) and P 0 (z): From the above (i) and (ii), the number of remain coefficients p k,m is 1 2 (n−1)(n+2l−2), which is the number of accessary parameters.If z = a 1 is an apparent singularity for the equation (3.1), namely all solutions of the equation (3.1) are regular at z = a 1 , we have the following relations: P 0 (a 1 /q) = 0, and f := P 0 (a 1 ) P 1 (a 1 /q) = P 1 (a 1 ) where f is a parameter.The above equations (3.6) correspond to a non-logarithmic condition via a Laplace transformation z ↔ T z .The relations (3.6) determine n coefficients p k,m .
From now on, we derive a scalar q-difference equation from the matrix q-difference equation (2.18), (2.20) for an unknown function Ψ(z) = [Ψ 1 (z), Ψ 2 (z), Ψ 3 (z)] and its properties.Eliminating functions Ψ 2 (z) and Ψ 3 (z) in the equation (2.18), (2.20), we obtain the following third linear q-difference equation for Φ(z) := Ψ 1 (z): where Here, the coefficients p k,l (1 ≤ k ≤ 3, 0 ≤ l ≤ 3) in the polynomials P k (z) (3.8) depend on parameters b j , c i , d j , and a variable u defined as the zero of P 3 (z).The variable u is expressed in terms of x, y as follows where Explicit forms of the polynomials P j (z) (0 ≤ j ≤ 3) (3.8) are given in appendix.
Then we have Lemma 3.2.The equation L(z) = 0 (3.7) has the following properties: (i) it is a linear four term equation between Φ q j z (0 ≤ j ≤ 3) and its coefficients P j (z) are polynomials for z of degree 4 − j, (ii) a polynomial P 0 (z) has four zero points at z = −c i (1 (iv) a point z = u such that P 3 (z) = 0 is an apparent singularity, namely we have Conversely, the equation L(z) = 0 (3.7) is uniquely characterized by these properties (i)-(iv) up to normalization.
The polynomial P ′ (u, v) has 14 coefficients.From the property (ii), 10 coefficients are described in terms of parameters b j , c j , d j and the coefficient c ′ 0 as follows: 0 by the property (iii).Namely, the property (iii) gives 3 linear inhomogeneous equa- . Though these relations are apparently q-difference equations, we can solve them algebraically.For example, in the equation (3.12), we solve c ′ 12 when j = 2. Then when j = 0, we solve c ′ 21 | z→qz,Φ(q k z)→Φ(q k+1 z) .And finally solving c ′ 11 when j = 1, they algebraically can be solved.■ Explicit forms of the coefficients c i,j (z) of the polynomial P (u, v) are in Appendix B.

Continuous limit
In this section, we describe a relation between our result and the result of Boalch [1].In [1], a Lax pair for the additional-difference Painlevé equation with affine Weyl symmetry group of type E 6 was described.The linear differential equation of the Lax pair is as follows where the matrices A b i (1 ≤ i ≤ 3) are 3 × 3 matrices with different eigenvalues.We show that the linear q-difference equation (2.1) reduces to the equation (4.1) via a continuous limit q → 1.The equation (2.1) takes the following form after a scale transformation z → −z and gauge transformations where k j (j = 1, 2) are constants.We put q = e h and consider the limit h → 0. We set where l j are constants.By using Taylor's expansion for (4.2), ( we find the following limit as h → 0: where eigenvalues of the matrix A 2 are γ i (1 ≤ i ≤ 3) by the condition of the determinant of the matrix A(z) (4.2).Therefore, the linear q-difference equation (2.1) reduces to the equation (4.1) via a continuous limit q → 1.
In the following, we consider a continuous limit q → 1 of the result in Section 2.1.In Section 2.1, through a compatibility condition of the equations (2.11), we derived a standard q-Painlevé equation of type E 6 .We take the following equation as a deformation equation for the differential equation (4.4) which is rewritten version of (2.10): Solving a compatibility condition for the equation (4.4), (4.5), we obtain the following additionaldifference Painlevé equation of type E 6 [6,10]: where and * stands for T ( * ).From the above, we derive the additional-difference Painlevé equation of type E 6 solving a compatibility condition of the Lax pair via a continuous limit q → 1.
A Deformations T 1 , T 2 and T 3 2 on root variables In this appendix, we show that how the deformations T 1 , T 2 and T 3 2 act on root variables.We consider a pair of root basis of {α i } (i = 0, 1, . . ., 6) and {δ j } (j = 0, 1, 2) as symmetry type E (1) 6 and surface type A  and A Pictures of a point configuration of the equation (2.12) and the configuration (2.24) are presented in Figures 2 and 3, respectively.From these pictures, we take a pair of root basis {α i } and {δ j } as symmetry type E where we stand for δ as a null root.The choice of the above root basis is the same as in [6].y = 0

A.1 A deformation T 1 on root variables
We take variables a i (i = 0, 1, . . ., 6) as root variables attached to the root α i (A.1) associated with a point configuration in coordinate (f, g) (see Figure 2) which satisfy q = a 0 a 1 a 2 2 a 3 3 a 2 4 a 5 a 2 6 .Then we have the following statement.
Proposition A.1.The action T 1 (2.10) on the root variables a i in (A.2) is given by the translation T 1 (a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) = (a 0 , a 1 , qa 2 , a 3 , a 4 , a 5 , a 6 /q).(A.gives a translation on them We show that how the deformations T 2 (2.19) and T 3 2 (A.4) on root variables.We take variables a ′ i (i = 0, 1, . . ., 6) as root variables attached to the root α i (A.1) associated with a point configuration in coordinate (x, y) (see Figure 3) 2 .Then we have the following statement.
Proposition A.2.The actions T 2 (2.19) and T 3 2 on the root variables a ′ i in (A.5) are given as follows: Proof .Applying (2.19) and (A.4), we obtain the desired results (A.6).■

B Explicit forms of coefficients in Section 3
In this appendix, we give explicit forms of P j (z) (3.8) and the coefficients c ij (0 ≤ j ≤ 3, 0 ≤ i ≤ 4 − j) of the polynomial P (u, v) in variables u and v. Explicit forms of P j (z) (3.8) are as follows:  + vq + v q 2 + v q 2 (q + 1)b 3 d 3 − v − qu v − q 2 ud 2 v − q 2 ud 3 + v d 2 u q 2 ud 3 − v + b 2 q 2 (q + 1)b 3 d 3 − v q 2 + v v − q 2 (u + b 3 )d 3 , p 10 = q 3 u 2 We give also explicit forms of the coefficients c ij (0 ≤ j ≤ 3, 0 ≤ i + j ≤ 4) of the polynomial P (u, v) in variables u and v: .25)Proof .Comparing the coefficient matrix A(z) of the equation (2.4), (2.13) to the coefficient matrix A(z) of the equation (2.18), (2.20), we can solve for the variables x, y and gauge freedom w 1 , w 2 in terms of variables f , g and parameters b j , c i , d j (1 ≤ i, j ≤ 3).Then we have the desired relation between pairs of variables (f, g) and (x, y) (2.
The time evolution equations u j,i are derived by solving the compatibility condition (2.21).For another derivation using the transformations which correspond to permutations of the matrices X ε i i (z), see [8, Proposition 2.1].Before ending this subsection, we show a relation between pairs of the variables (f, g) in (2.12) and (x, y) in (2.22).The equations (2.4), (2.13) and (2.18), (2.20) are equivalent with each other if the variables f , g and x, y are related as Deformations T 2 and T 3 2 on root variables The deformation T 2 (2.19) is not a translation on parameters b i , c i , d j but a deformation T 3 2 3)Proof .Applying (2.10), we obtain the desired result (A.3).■A.2