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 Painlevé equation.

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Дата:	2017
Автор:	Yamada, Y.
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Опубліковано:	Інститут математики НАН України 2017
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	An Elliptic Garnier System from Interpolation / Y. Yamada // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 15 назв. — англ.

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spelling	irk-123456789-1487492019-02-19T01:23:45Z An Elliptic Garnier System from Interpolation Yamada, Y. 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 Painlevé equation. 2017 Article An Elliptic Garnier System from Interpolation / Y. Yamada // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 39A13; 33E05; 33E17; 41A05 DOI:10.3842/SIGMA.2017.069 http://dspace.nbuv.gov.ua/handle/123456789/148749 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	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 Painlevé equation.
format	Article
author	Yamada, Y.
spellingShingle	Yamada, Y. An Elliptic Garnier System from Interpolation Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Yamada, Y.
author_sort	Yamada, Y.
title	An Elliptic Garnier System from Interpolation
title_short	An Elliptic Garnier System from Interpolation
title_full	An Elliptic Garnier System from Interpolation
title_fullStr	An Elliptic Garnier System from Interpolation
title_full_unstemmed	An Elliptic Garnier System from Interpolation
title_sort	elliptic garnier system from interpolation
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/148749
citation_txt	An Elliptic Garnier System from Interpolation / Y. Yamada // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 15 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT yamaday anellipticgarniersystemfrominterpolation AT yamaday ellipticgarniersystemfrominterpolation
first_indexed	2025-07-12T20:09:07Z
last_indexed	2025-07-12T20:09:07Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 069, 8 pages An Elliptic Garnier System from Interpolation Yasuhiko YAMADA Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan E-mail: yamaday@math.kobe-u.ac.jp Received June 20, 2017, in final form August 30, 2017; Published online September 02, 2017 https://doi.org/10.3842/SIGMA.2017.069 Abstract. Considering a certain interpolation problem, we derive a series of elliptic differen- ce isomonodromic systems together with their Lax forms. These systems give a multivariate extension of the elliptic Painlevé equation. Key words: elliptic difference; isomonodromic systems; Lax form; interpolation problem 2010 Mathematics Subject Classification: 39A13; 33E05; 33E17; 41A05 1 Introduction There is a simple way to derive isomonodromic equations by studying suitable Padé approxi- mation 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 cer- tain elliptic interpolation problems and derive a multivariate extension of the elliptic-difference1 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. 2 Set up of the interpolation problem Fix p, q ∈ C such that \|p\|, \|q\| < 1. The elliptic Gamma function Γp,q(z) [10] and the theta function [z] (of base p) are defined as Γp,q(z) = (pq/z; p, q)∞ (z; p, q)∞ = ∏ i,j≥0 1− z−1pi+1qj+1 1− zpiqj , 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 1The q-difference limit of the obtained system is expected to be the one considered in [6]. mailto:yamaday@math.kobe-u.ac.jp https://doi.org/10.3842/SIGMA.2017.069 https://www.emis.de/journals/SIGMA/EHF2017.html 2 Y. Yamada [z] = (z, p/z; p)∞ = ∞∏ i=0 ( 1− xpi )( 1− x−1pi+1 ) . They satisfy the following fundamental relations: Γp,q(qz) = [z]Γp,q(z), Γp,q(z/q) = [z/q]−1Γp,q(z), [pz] = −z−1[z], [z] = −z[1/z], [z] = [p/z]. We also use the following notations: [z]s = Γp,q(q sz) Γp,q(z) , [x1, . . . , xl]s = [x1]s · · · [xl]s. In particular, [z]s = s−1∏ i=0 [qiz] for s ∈ Z≥0. Fix N ∈ Z≥2. Let k, u1, . . . , u2N be complex parameters satisfying a constraint 2N∏ i=1 ui = kN , and define a function ψ(z) as2 ψ(z) = 2N∏ i=1 Γp,q(ui/z) Γp,q(k/uiz) . We also define a shift T : x 7→ x of parameters x = k, ui as k = k/q, ui = { ui, 1 ≤ i ≤ N, ui/q, N < i ≤ 2N. This action is naturally extended to any functions f = f(k, ui) of parameters by f = f(k, ui). We put µ1(z) = ψ(z/q) ψ(z) = 2N∏ i=1 [ui/z] [k/uiz] , µ2(z) = ψ(z/q) ψ(z) = N∏ i=1 [ui/z] [k/uN+iz] , µ3(z) = ψ(z) ψ(z) = N∏ i=1 [k/quiz] [uN+i/qz] . These functions are p-periodic: f(pz) = f(z), and satisfy µ1(k/z) = µ1(z) −1, µ3(k/qz) = µ2(z), (2.1) due to the constraint 2N∏ i=1 ui = kN . 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) θden(z) , where θ∗(z) (∗ = num,den) are h- symmetric entire function with common quasi periodicity: θ∗(pz) = (h/pz2)dθ∗(z). The totality of such functions f(z) form a linear space of dimension d+ 1. For m,n ∈ Z≥0, consider the interpolation problem3 ψ(q−s) = P (q−s) Q(q−s) , s = 0, 1, . . . , N = m+ n, (2.2) 2Throughout the paper, any expression a · · · b/c · · · d means the long fraction a···b c···d . 3This is a kind of PPZ (prescribed poles and zeros) interpolation [15]. An Elliptic Garnier System from Interpolation 3 where P (z) (resp. Q(z)) are k/q-symmetric and p-periodic elliptic functions of order 2m (resp. 2n), with specified denominators Pden(z) (resp. Qden(z)). For convenience, we will choose them as Pden(z) = [u1/z, u1qz/k]m, Qden(z) = [k/u2z, qz/u2]n. 3 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): L2 : D3(z/q)w(z)−D2(z)w(z/q) +D1(z)w(z/q) = 0, L3 : D1(z)w(z)−D2(z)w(z) +D3(z)w(z/q) = 0. (3.1) The coefficients are determined by the Casorati determinants as D1(z) = \|u(z),u(z/q)\|, D2(z) = \|u(z),u(z/q)\|, D3(z) = \|u(z),u(z)\|, u(z) = [ P (z) ψ(z)Q(z) ] . Certain explicit formulas for P (z), Q(z) are available (see Remark 3.4), however, we do not need them for the computations here. Lemma 3.1. We have D1(z) = ψ(z) [z, k/z]m+n [ k/z2 ] X1,den(z) F (z), where X1,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 F (z) = Cz N−2∏ i=1 [z/λi, k/zλi], (3.2) where C, λ1, . . . , λN−2 are some constants independent of z. Proof. We put X1(z) = D1(z) ψ(z) = µ1(z)P (z)Q(z/q)− P (z/q)Q(z). (i) Obviously X1(z) is a p-periodic function. Due to the cancellations of the factors [u1/z][u2/z], the denominator X1,den(z) of X1(z) consists of 2(N + m + n) theta factors, hence X1(z) is of degree 2(N +m+ n). We choose the normalization of X1,den(z) as X1,den(z) = ( N∏ i=1 [k/uiz][uN+iz/k] ) [qu1/z, qu1z/k]m[qk/u2z, qz/u2]n, (3.3) so that X1,den(k/z) = µ1(z) −1X1,den(z). (ii) Due to the k/q-symmetry of P (z), Q(z) and equation (2.1), we have X1(k/z) = µ1(k/z)P (k/z)Q(k/qz)− P (k/qz)Q(k/z) = µ1(z) −1P (z/q)Q(z)− P (z/q)Q(z) = −µ1(z)−1X1(z). 4 Y. Yamada Combining this and X1,den(k/z) = µ1(z) −1X1,den(z), we see that the numerator X1,num(z) is k-antisymmetric: X1,num(k/z) = −X1,num(z). (iii) By the Padé interpolation condition, we have D1(q −s) = 0 for s = 0, 1, . . . ,m + n − 1. Hence X1,num(z) is divisible by [z, k/z]m+n [ k/z2 ] . From (i)–(iii), one obtain the desired result. � Lemma 3.2. We have D2(z) = ψ(z) [z]m+n[k/qz]m+n+1 X2,den(z) G(z), D3(z) = ψ(z) [z]m+n+1[k/qz]m+n X3,den(z) G(k/qz), where X2,den(z), X3,den(z) is given in equation (3.5) below, C ′ is a constant, and G(z) is a p- quasi periodic function of degree N − 1 which can be written as G(z) = C ′z N−1∏ i=1 [z/ξi], N−1∏ i=1 ξi = kqn−1 N∏ i=1 ui . (3.4) Proof. We put X2(z) = D2(z) ψ(z) = µ2(z)P (z)Q(z/q)− P (z/q)Q(z), X3(x) = D3(z) ψ(z) = µ3(z)P (z)Q(z)− P (z)Q(z). (i) Obviously X2(z), X3(z) are p-periodic elliptic functions. The denominators can be written as X2,den(z) = 2N∏ i=N+1 [uiz/k][qu1/z, qu1z/k]m[qz/u2, k/u2z]n, X3,den(z) = 2N∏ i=N+1 [ui/qz] [ u1/z, q 2u1z/k ] m [qz/u2, k/u2z]n. (3.5) Hence X2(z), X3(z) are both of degree N + 2m+ 2n. We note that X3,den(k/qz) = X2,den(z). (ii) From P (k/qz) = P (z), we have P (k/qz) = P (z/q) and similarly we have Q(k/qz) = Q(z), Q(k/qz) = Q(z/q). Using these relations and equation (2.1), we have X3(k/qz) = µ3(k/qz)P (k/qz)Q(k/qz)− P (k/qz)Q(k/qz) = µ2(z)P (z)Q(z/q)− P (z/q)Q(z) = X2(z). (iii) Due to the Padé interpolation condition we have D2(q −s) = 0 for s = 0, . . . ,m + n − 1 and D3(q −s) = 0 for s = 0, . . . ,m+ n. From (i)–(iii), we obtain the desired results. � Theorem 3.3. By a suitable gauge transformation y(z) = g(z)w(z), the L2, L3 equations take the following forms L2 : F (z) [ k/z2 ] y(z/q)−G(z)A(k/z)y(z/q) +G(k/z)A(z)y(z) = 0, L3 : F (z) [ k/qz2 ] y(z)−G(z)B(k/qz)y(z) +G(k/qz)B(z)y(z/q) = 0, (3.6) An Elliptic Garnier System from Interpolation 5 where F (z), G(z) are given by equations (3.2), (3.4), and A(z) = N+1∏ i=1 [z/ai], {ai}N+1 i=1 = { u1q m, u2q −n, u3, . . . , uN , q } , B(z) = N+1∏ i=1 [z/bi], {bi}N+1 i=1 = { k/uN+1, . . . , k/u2N , q −m−n}. Proof. First, using Lemmas 3.1 and 3.2, we rewrite the equations (3.1) as L2 : F (z) [ k/z2 ] w(z/q)−G(z)[k/qz] [kqn/u2z] [k/u2z] N∏ i=1 [k/uiz]w(z/q) +G(k/z)[z/q] [zqn/u2] [z/u2] N∏ i=1 [z/ui]w(z) = 0, L3 : N∏ i=1 [uN+i/q] [k/qui] F (z) [ k/qz2 ] w(z)−G(z) [ kqm+n−1/z ] [zu1qm+1z/k] [zu1qz/k] 2N∏ i=N+1 [ui/qz]w(z) +G(k/qz) [ qm+nz ] qm [z/u1q m] [z/u1] 2N∏ i=N+1 [zui/k]w(z/q) = 0. Then, by the gauge transformation y(z) = [u1/z, u1qz/k]mw(z), we obtain L2 : q−mF (z) [ k/z2 ] y(z/q)−G(z)A(k/z)y(z/q) +G(k/z)A(z)y(z) = 0, L3 : N∏ i=1 [uN+i/q] [k/qui] F (z) [ k/qz2 ] y(z)−G(z)B(k/qz)y(z) +G(k/qz)B(z)y(z/q) = 0. 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 re- sults (3.6). � Remark 3.4. An explicit expression of the Padé interpolants P (z), Q(z) is given by the deter- minant as follows P (z) = c ∣∣∣∣∣∣∣∣∣ µP0,0 · · · µP0,m ... . . . ... µPm−1,0 · · · µPm−1,m ψ0(z) · · · ψm(z) ∣∣∣∣∣∣∣∣∣ , Q(z) = c ∣∣∣∣∣∣∣∣∣ µQ0,0 · · · µQ0,n ... . . . ... µQn−1,0 · · · µQn−1,n φ0(z) · · · φn(z) ∣∣∣∣∣∣∣∣∣ , where c is a constant and ψj(z) = [u1, qu1/k, k/u3z, qz/u3]j [u1/z, qu1/z, k/u3, q/u3]j , φj(z) = [k/u2, q/u2, u4/z, qu4z/k]j [k/u2z, qz/u2, u4, qu4/k]j , µPi,j = 2N+6V2N+5 ( k q , q−m−n, k u1 q−j , k u2 qm+n−i−1, k u3 qj , k u4 qi, k u5 , . . . , k u2N ; q ) , µQi,j = 2N+6V2N+5 ( k q , q−m−n, u1q m+n−i−1, u2q −j , u3q i, u4q j , u5, . . . , u2N ; q ) , and nVn−1 is the elliptic hypergeometric series [12, 15] defined by n+5Vn+4(a0; a1, . . . , an; z) = ∞∑ s=0 [ a0q 2s ] [a0] n∏ i=0 [ai]s [qa0/ai]s zs. (3.7) 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. 6 Y. Yamada 4 Compatibility conditions In this section, we consider the equation (3.6) forgetting about the connection with the interpo- lation problem. Namely, we restart with the following equations L2 : F (z) [ k/z2 ] y(z/q)−G(z)A(k/z)y(z/q) +G(k/z)A(z)y(z) = 0, L3 : F (z) [ k/qz2 ] y(z)−G(z)B(k/qz)y(z) +G(k/qz)B(z)y(z/q) = 0, F (z) = Cz N−2∏ i=1 [z/λi, k/zλi], G(z) = z N−1∏ i=1 [z/ξi], A(z) = N+1∏ i=1 [z/ai], B(z) = N+1∏ i=1 [z/bi], (4.1) where {ai, bi}N+1 i=1 , k, ` are parameters and C, {λi}N−2i=1 , {ξi}N−1i=1 are variables such as ai = ai, bi = bi, k = k/q, ` = q`, k2`2 = q N+1∏ i=1 aibi, N−1∏ i=1 ξi = `. Proposition 4.1. As the necessary conditions for the compatibility, the pair of equations L2, L3 in (4.1) gives the following equations for λi, C and ξi = T−1(ξi). Namely F (z)F (z) [ k/z2 ][ k/qz2 ] = G(k/z)G(k/qz)U(z), (4.2) for z = ξi (1 ≤ i ≤ N − 1), and G(z)G(z) G(k/z)G(k/z) = U(z) U(k/z) , (4.3) for z = λi (1 ≤ i ≤ N − 2), where U(z) = A(z)B(z). Proof. When z = ξi, the terms in L2, L3 with coefficient G(z) vanishes, and we obtain equa- tion (4.2). Similarly, putting z = λi in L2 and L3 : F (z) [ k/z2 ] 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). � The equations (4.2), (4.3) give the evolution equation for 2(N − 2) variables {λi, ξi}N−2i=1 . In N = 3 case, it can be written in a symmetric way as ξ21 ξ22 2∏ j=1 [ξ1/λj ][ξ1/λj ] [ξ2/λj ][ξ2/λj ] = U(ξ1) U(ξ2) , λ21 λ22 2∏ j=1 [λ1/ξj ][λ1/ξj ] [λ2/ξj ][λ2/ξj ] = U(λ1) U(λ2) , where λ1λ2 = k, ξ1ξ2 = `. This is the elliptic Painlevé equation [5, 11] in factorized form [1, 4]. Its Lax pair is obtained in [13], and the higher-order analogues are also given in [8]. Theorem 4.2. The equations (4.2), (4.3) are sufficient for the compatibility of the Lax pair (4.1). Proof. Combining the equations L2 and L3 as the following diagrams: y( zq ) y(z) y( zq ) y(z) y(qz) @ @ @ @ @ @ @ @ @ @ @ @ L2(z) L3(z) @ @ @ @ @ @ L2(qz) L1: y( zq ) y(z) y(qz) y(z) y(qz) @ @ @ @ @ @ @ @ @ @ @ @ L3(qz)L3(z) @ @ @ @ @ @ L2(qz) L̃1: An Elliptic Garnier System from Interpolation 7 we obtain the following three term relations for y(z) or y(z), L1 : A(k/z)B(z)F (qz) [ k/q2z2 ] y(z/q)−R(z)y(z) +A(qz)B(k/qz)F (z) [ k/z2 ] y(qz) = 0, L̃1 : A(k/qz)B(z)F (qz) [ k/q3z2 ] y(z/q)− R̃(z)y(z) +A(qz)B ( k/q2z ) F (z) [ k/qz2 ] y(qz) = 0, (4.4) where R(z) = U(z)F (qz)G ( k z )[ k q2z2 ] G(z) + U ( k qz ) F (z)G(qz) [ k z2 ] G ( k qz ) − F (z)F (qz)F (z) [ k z2 ][ k qz2 ][ k q2z2 ] G(z)G ( k qz ) , R̃(z) = U(qz)F (z)G ( k q2z )[ k q2z2 ] G(qz) + U ( k qz ) F (qz)G(z) [ k q3z2 ] G ( k qz ) − F (z)F (qz)F (z) [ k z2 ][ k qz2 ][ k q2z2 ] G(z)G ( k qz ) . (4.5) Then the compatibility means the consistency between triangle relations on the following 7 points of the 3× 3 grid: @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ which is equivalently written as L1 = L̃1. It is easy to check the condition L1 = L̃1 for the coefficients of y(qz), y(z/q) and y(qz), y(z/q). So the problem is to show R(z) = R̃(z) under the equations (4.2), (4.3). For L1 given in equation (4.4), (4.5), one can check the following properties: (i) R(z) is holomorphic due to equation (4.2), and it is a degree 4N + 2 theta function of base p. (ii) R(k/qz) = −R(z), and hence R(z) is divisible by [ k/qz2 ] . (iii) The equation L1 holds when{[ k/z2 ] = 0, y(z) = y(z/q) } or {[ k/q2z2 ] = 0, y(z) = y(qz) } or {F (z) = 0, A(z)G(k/z)y(z) = A(k/z)G(z)y(z/q)} or {F (qz) = 0, A(qz)G(k/qz)y(qz) = A(k/qz)G(qz)y(z)}. Moreover, once the coefficients of y(qz), y(z/q) in L1 are fixed as in equation (4.4), the properties (i)–(iii) determine the coefficient R(z) uniquely. Similarly, R̃(z) in equation (4.4) is characterized by the following conditions: (i) R̃(z) is a degree 4N + 2 theta function of base p. (ii) R̃ ( k/q2z ) = −R̃(z), and hence R̃(z) is divisible by [ k/q2z2 ] . 8 Y. Yamada (iii) The equation L̃1 holds when{[ k/qz2 ] = 0, y(z) = y(z/q) } or {[ k/q3z2 ] = 0, y(z) = y(qz) } or{ F (z) = 0, A(z)G(k/qz)y(z) = A(k/qz)G(z)y(z/q) } or{ F (qz) = 0, A(qz)G ( k/q2z ) y(qz) = A(k/q2z)G(qz)y(z) } , where we used the equation (4.3) to rewrite the last two equations. These characteristic properties show that R(z) = R̃(z), hence L1 = L̃1 as desired. � Acknowledgments The author is grateful to the organizers and participants of the lecture series at the university of Sydney (November 28–30, 2016) and the ESI workshop “Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics” (Vienna, March 20–24, 2017) for their in- terests and discussions. He also thanks to referees for valuable comments and Dr. H. Nagao for discussions. This work is partially supported by JSPS KAKENHI (26287018). References [1] Kajiwara K., Noumi M., Yamada Y., Geometric aspects of Painlevé equations, J. Phys. A: Math. Theor. 50 (2017), 073001, 164 pages, arXiv:1509.08186. [2] Nagao H., Yamada Y., Study of q-Garnier system by Padé method, Funkcial. Ekvac., to appear, arXiv:1601.01099. [3] Nijhoff F., Delice N., On elliptic Lax pairs and isomonodromic deformation systems for elliptic lattice equations, arXiv:1605.00829. [4] Noumi M., Tsujimoto S., Yamada Y., Padé interpolation for elliptic Painlevé equation, in Symmetries, Integrable Systems and Representations, Springer Proc. Math. Stat., Vol. 40, Springer, Heidelberg, 2013, 463–482, arXiv:1204.0294. [5] Ohta Y., Ramani A., Grammaticos B., An affine Weyl group approach to the eight-parameter discrete Painlevé equation, J. Phys. A: Math. Gen. 34 (2001), 10523–10532. [6] Ormerod C.M., Rains E.M., Commutation relations and discrete Garnier systems, SIGMA 12 (2016), 110, 50 pages, arXiv:1601.06179. [7] Ormerod C.M., Rains E.M., An elliptic Garnier system, Comm. Math. Phys. 355 (2017), 741–766, arXiv:1607.07831. [8] Rains E.M., An isomonodromy interpretation of the hypergeometric solution of the elliptic Painlevé equation (and generalizations), SIGMA 7 (2011), 088, 24 pages, arXiv:0807.0258. [9] Rains E.M., The noncommutative geometry of elliptic difference equations, arXiv:1607.08876. [10] Ruijsenaars S.N.M., First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069–1146. [11] Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165–229. [12] Spiridonov V.P., Essays on the theory of elliptic hypergeometric functions, Russian Math. Surveys 63 (2008), 405–472, arXiv:0805.3135. [13] Yamada Y., A Lax formalism for the elliptic difference Painlevé equation, SIGMA 5 (2009), 042, 15 pages, arXiv:0811.1796. [14] Yamada Y., Padé method to Painlevé equations, Funkcial. Ekvac. 52 (2009), 83–92. [15] Zhedanov A.S., Padé interpolation table and biorthogonal rational functions, in Elliptic Integrable Systems, Rokko Lectures in Mathematics, Vol. 18, Kobe University, 2005, 323–363. https://doi.org/10.1088/1751-8121/50/7/073001 https://arxiv.org/abs/1509.08186 https://arxiv.org/abs/1601.01099 https://arxiv.org/abs/1605.00829 https://doi.org/10.1007/978-1-4471-4863-0_18 https://arxiv.org/abs/1204.0294 https://doi.org/10.1088/0305-4470/34/48/316 https://doi.org/10.3842/SIGMA.2016.110 https://arxiv.org/abs/1601.06179 https://doi.org/10.1007/s00220-017-2934-6 https://arxiv.org/abs/1607.07831 https://doi.org/10.3842/SIGMA.2011.088 https://arxiv.org/abs/0807.0258 https://arxiv.org/abs/1607.08876 https://doi.org/10.1063/1.531809 https://doi.org/10.1063/1.531809 https://doi.org/10.1007/s002200100446 https://doi.org/10.1070/RM2008v063n03ABEH004533 https://arxiv.org/abs/0805.3135 https://doi.org/10.3842/SIGMA.2009.042 https://arxiv.org/abs/0811.1796 https://doi.org/10.1619/fesi.52.83 1 Introduction 2 Set up of the interpolation problem 3 Derivation of the contiguous relations 4 Compatibility conditions References

An Elliptic Garnier System from Interpolation

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