An Elliptic Garnier System from Interpolation

Considering a certain interpolation problem, we derive a series of elliptic difference isomonodromic systems together with their Lax forms. These systems give a multivariate extension of the elliptic Painlev\'e equation.


Introduction
There is a simple way to derive isomonodromic equations by studying suitable Padé approximation or interpretation problem. It has been applied various examples both continuous and discrete (see [2,14] and references therein). The aim of this paper is to apply this method to certain elliptic interpolation problems and derive a multivariate extension of the elliptic-difference 1 Painlevé equation [5,11]. This work is a natural generalization of [4].
Recently, there have been some progress in multivariate elliptic isomonodromic systems. In [7,8], an elliptic analog of the Garnier system is constructed. In [2], an elliptic deformation of q-Garnier system is suggested from a geometric points of view. In [3], certain elliptic analog of Garnier system is obtained from viewpoint of lattice equations. Moreover, a general framework of elliptic isomonodromic systems is established in [9]. For the equations obtained in this paper, the proper isomonodromic interpretation and the relation to the constructions mentioned above are not clear so far. However, since the equations obtained in this paper are quite explicit, we expect that they will give a clue to elucidate the multivariate elliptic isomonodromic systems.
The paper is organized as follows. In Section 2, we set up our interpolation problem (2.2): ψ(z) ∼ P (z) Q(z) . In Section 3, we derive two contiguous relations satisfied by the interpolants P (z) and ψ(z)Q(z) (Theorem 3.3). These relations play the role of the Lax pair for the isomonodromic system. In Section 4, we analyze the Lax equations and derive the isomonodromic system as the necessary and sufficient conditions for the compatibility (Theorem 4.2). The proof becomes quite simple due to the use of the contiguous type Lax pair.

Set up of the interpolation problem
Fix p, q ∈ C such that |p|, |q| < 1. The elliptic Gamma function Γ p,q (z) [10] and the theta function [z] (of base p) are defined as This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications. The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html 1 The q-difference limit of the obtained system is expected to be the one considered in [6].
They satisfy the following fundamental relations: We also use the following notations: Fix N ∈ Z ≥2 . Let k, u 1 , . . . , u 2N be complex parameters satisfying a constraint and define a function ψ(z) as 2 We also define a shift T : x → x of parameters x = k, u i as This action is naturally extended to any functions f = f (k, u i ) of parameters by f = f (k, u i ).
We put These functions are p-periodic: f (pz) = f (z), and satisfy Let f (z) be an elliptic function of degree 2d such that p-periodic and h-symmetric: f (h/z) = f (z). Any such function can be written as f (z) = θnum(z) θ den (z) , where θ * (z) ( * = num, den) are hsymmetric entire function with common quasi periodicity: θ * (pz) = (h/pz 2 ) d θ * (z). The totality of such functions f (z) form a linear space of dimension d + 1.
For m, n ∈ Z ≥0 , consider the interpolation problem 3 where P (z) (resp. Q(z)) are k/q-symmetric and p-periodic elliptic functions of order 2m (resp. 2n), with specified denominators P den (z) (resp. Q den (z)). For convenience, we will choose them as

Derivation of the contiguous relations
Let P (z), Q(z) be solutions for the interpolation problem (2.2). We will compute the contiguous relations satisfied by the functions w(z) = P (z), ψ(z)Q(z): The coefficients are determined by the Casorati determinants as .
Certain explicit formulas for P (z), Q(z) are available (see Remark 3.4), however, we do not need them for the computations here.
where X 1,den (z) is given in equation (3.3) below, and F (z) is a k-symmetric p-quasi periodic entire function of degree 2N − 4. Explicitly, we have where C, λ 1 , . . . , λ N −2 are some constants independent of z.
From (i)-(iii), one obtain the desired result.
Lemma 3.2. We have where X 2,den (z), X 3,den (z) is given in equation (3.5) below, C is a constant, and G(z) is a pquasi periodic function of degree N − 1 which can be written as Proof . We put (i) Obviously X 2 (z), X 3 (z) are p-periodic elliptic functions. The denominators can be written as (3.5) Hence X 2 (z), X 3 (z) are both of degree N + 2m + 2n. We note that X 3,den (k/qz) = X 2,den (z).
From (i)-(iii), we obtain the desired results.
Theorem 3.3. By a suitable gauge transformation y(z) = g(z)w(z), the L 2 , L 3 equations take the following forms L 2 : F (z) k/z 2 y(z/q) − G(z)A(k/z)y(z/q) + G(k/z)A(z)y(z) = 0, where F (z), G(z) are given by equations (3.2), (3.4), and Proof . First, using Lemmas 3.1 and 3.2, we rewrite the equations (3.1) as Then, by the gauge transformation y(z) = [u 1 /z, u 1 qz/k] m w(z), we obtain The additional factors in front of F (z), F (z) can be absorbed into the normalization of F (z), F (z) by a z-independent gauge transformation of y(z). Hence, we arrive at the desired results (3.6).
Remark 3.4. An explicit expression of the Padé interpolants P (z), Q(z) is given by the determinant as follows where c is a constant and and n V n−1 is the elliptic hypergeometric series [12,15] defined by n+5 V n+4 (a 0 ; a 1 , . . . , a n ; z) = ∞ s=0 a 0 q 2s The proof is completely the same as the case N = 3 [4]. Application of the explicit formulae to the special solution of the isomonodromic systems will be considered elsewhere.

Compatibility conditions
In this section, we consider the equation (3.6) forgetting about the connection with the interpolation problem. Namely, we restart with the following equations L 2 : F (z) k/z 2 y(z/q) − G(z)A(k/z)y(z/q) + G(k/z)A(z)y(z) = 0, Proposition 4.1. As the necessary conditions for the compatibility, the pair of equations L 2 , L 3 in (4.1) gives the following equations for λ i , C and ξ i = T −1 (ξ i ). Namely Proof . When z = ξ i , the terms in L 2 , L 3 with coefficient G(z) vanishes, and we obtain equation (4.2). Similarly, putting z = λ i in L 2 and L 3 : F (z) k/z 2 y(z) − G(z)B(k/z)y(z) + G(k/z)B(z)y(z/q) = 0, the terms with coefficient F (z) vanishes, and we have equation (4.3).
Moreover, once the coefficients of y(qz), y(z/q) in L 1 are fixed as in equation (4.4), the properties (i)-(iii) determine the coefficient R(z) uniquely. Similarly,R(z) in equation (4.4) is characterized by the following conditions: (i)R(z) is a degree 4N + 2 theta function of base p.
These characteristic properties show that R(z) =R(z), hence L 1 =L 1 as desired.