Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System

We construct a Lax pair for the E₆⁽¹⁾ q-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lat...

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Datum:	2012
Hauptverfasser:	Witte, N.S., Ormerod, C.M.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2012
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1486942019-02-19T01:27:25Z Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System Witte, N.S. Ormerod, C.M. We construct a Lax pair for the E₆⁽¹⁾ q-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the q-linear lattice - through a natural generalisation of the big q-Jacobi weight. As a by-product of our construction we derive the coupled first-order q-difference equations for the E₆⁽¹⁾ q-Painlevé system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations. 2012 Article Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System / N.S. Witte, C.M. Ormerod // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 39A05; 42C05; 34M55; 34M56; 33C45; 37K35 DOI: http://dx.doi.org/10.3842/SIGMA.2012.097 http://dspace.nbuv.gov.ua/handle/123456789/148694 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	We construct a Lax pair for the E₆⁽¹⁾ q-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the q-linear lattice - through a natural generalisation of the big q-Jacobi weight. As a by-product of our construction we derive the coupled first-order q-difference equations for the E₆⁽¹⁾ q-Painlevé system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations.
format	Article
author	Witte, N.S. Ormerod, C.M.
spellingShingle	Witte, N.S. Ormerod, C.M. Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Witte, N.S. Ormerod, C.M.
author_sort	Witte, N.S.
title	Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System
title_short	Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System
title_full	Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System
title_fullStr	Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System
title_full_unstemmed	Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System
title_sort	construction of a lax pair for the e₆⁽¹⁾ q-painlevé system
publisher	Інститут математики НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/148694
citation_txt	Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System / N.S. Witte, C.M. Ormerod // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 18 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT wittens constructionofalaxpairforthee61qpainlevesystem AT ormerodcm constructionofalaxpairforthee61qpainlevesystem
first_indexed	2025-07-12T19:59:55Z
last_indexed	2025-07-12T19:59:55Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 097, 27 pages Construction of a Lax Pair for the E (1) 6 q-Painlevé System Nicholas S. WITTE † and Christopher M. ORMEROD ‡ † Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia E-mail: nsw@ms.unimelb.edu.au URL: http://www.ms.unimelb.edu.au/~nsw/ ‡ Department of Mathematics and Statistics, La Trobe University, Bundoora VIC 3086, Australia E-mail: C.Ormerod@latrobe.edu.au Received September 05, 2012, in final form November 29, 2012; Published online December 11, 2012 http://dx.doi.org/10.3842/SIGMA.2012.097 Abstract. We construct a Lax pair for the E (1) 6 q-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices – the q-linear lattice – through a natural generalisation of the big q-Jacobi weight. As a by-product of our construction we derive the coupled first- order q-difference equations for the E (1) 6 q-Painlevé system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations. Key words: non-uniform lattices; divided-difference operators; orthogonal polynomials; semi-classical weights; isomonodromic deformations; Askey table 2010 Mathematics Subject Classification: 39A05; 42C05; 34M55; 34M56; 33C45; 37K35 1 Background and motivation Since the recent discoveries of q-analogues of the Painlevé equations, see for example [4] and [13] which are of relevance to the present study, and their classification (of these and others) ac- cording to the theory of rational surfaces by Sakai [16] interest has grown in finding Lax pairs for these systems. This problem also has the independent interest as a search for discrete and q-analogues to the isomonodromic systems of the continuous Painlevé equations, and an appro- priate analogue to the concept of monodromy. Such interest, in fact, goes back to the period when the discrete analogues of the Painlevé equations were first discussed, as one can see in [12]. In this work we illustrate a general method for constructing Lax pairs for all the systems in the Sakai scheme, as given in the study [17], with the particular case of the E (1) 6 system. In this method all aspects of the Lax pairs are constructed, and in the end we verify the identification with the E (1) 6 system by deriving the appropriate coupled first-order q-difference equations. We will utilise the form of the E (1) 6 q-Painlevé system as given in [6] and [5] in terms of the variables f , g under the mapping (t, f, g) 7→ ( qt, f(qt) ≡ f̂ , g(qt) ≡ ĝ ) , and f(q−1t) ≡ f̌ , etc. In these variables the coupled first-order q-difference equations are (gf̌ − 1)(gf − 1) = t2 (b1g − 1)(b2g − 1)(b3g − 1)(b4g − 1) (g − b6t)(g − b−16 t) , (1.1) nsw@ms.unimelb.edu.au http://www.ms.unimelb.edu.au/~nsw/ C.Ormerod@latrobe.edu.au http://dx.doi.org/10.3842/SIGMA.2012.097 2 N.S. Witte and C.M. Ormerod (fĝ − 1)(fg − 1) = qt2 (f − b1)(f − b2)(f − b3)(f − b4) (f − b5qt)(f − b−15 t) , (1.2) with five independent parameters b1, . . . , b6 subject to the constraint b1b2b3b4 = 1. Our approach is to construct a sequence of τ -functions starting with a deformation of a specific weight in the Askey table of hypergeometric orthogonal polynomial systems [7]. However for the purposes of the present work we will not explicitly exhibit these τ -functions although one could do so easily. The weight that we will take is the big q-Jacobi weight1 given by equation (14.5.2) of [7] w(x) = ( a−1x, c−1x; q ) ∞( x, bc−1x; q ) ∞ . (1.3) The essential property of this weight, and the others in the Askey table, that we will utilise is that they possess the q-analogue of the semi-classical property with respect to x, namely that it satisfies the linear, first-order homogeneous q-difference equation w(qx) w(x) = a(1− x)(c− bx) (a− x)(c− x) , where the right-hand side is manifestly rational in x. Another feature of this weight is that the discrete lattice forming the support for the orthogonal polynomial system is the q-linear lattice, one of four discrete quadratic lattices. Consequently the perspective provided by our theoretical approach, then indicates that this case is the master case for the q-linear lattices (as opposed to the D (1) 5 system, for example) and all systems with such support will be degenerations of it. The weight (1.3) has to be generalised, or deformed, in order to become relevant to q-Painlevé systems, and such a generalisation turns out to introduce a new variable t and associated parameter so that it retains the semi-classical character with respect to this variable. Using such a sequence of τ -functions one employs arguments to construct three systems of linear divided-difference equations which in turn characterise these. One of these is the three-term recurrence relation of the polynomials orthogonal with respect to the deformed weight, which in the Painlevé theory context is a distinguished Schlesinger transformation, while the two others are our Lax pairs with respect to the spectral variable x and the deformation variable t. Our method constructs a specific sequence of classical solutions to the E (1) 6 system and thus is technically valid for integer values of a particular parameter, however we can simply analytically continue our results to the general case. Lax pairs have been found for the E (1) 6 system system using completely different techniques. In [15] Sakai used a particular degeneration of a two-variable Garnier extension to the Lax pairs for the D (1) 5 q-Painlevé system2 (see [14] for details on the multi-variable Garnier exten- sion). More recently Yamada [18] has reported Lax pairs for the E (1) 6 system by employing a degeneration starting from a Lax pair for the E (1) 8 q-Painlevé equation through a sequence of limits E (1) 8 → E (1) 7 → E (1) 6 . The plan of our study is as follows. In Section 2 we recount the notations, definitions and basic facts of the general theory [17] in a self-contained manner omitting proofs. We draw heavily upon this theory in Section 3 where we apply it to the q-linear lattice and a natural extension or deformation of the big q-Jacobi weight. Again, using techniques first expounded in [17], we find explicit forms for the Lax pairs and verify the identification with the E (1) 6 q-Painlevé system. At the conclusion of our study, in Section 4, we relate our Lax pairs with those of both Sakai and Yamada. 1However we will employ a different parameterisation of the big q-Jacobi weight from that of the conventional form (1.3) in order that our results conform to the the E (1) 6 q-Painlevé system as given by (1.1), (1.2); see (3.1). 2This later system is also known as the q-PVI system and its Lax pairs were constructed in [4]. Construction of a Lax Pair for the E (1) 6 q-Painlevé System 3 2 Deformed semi-classical OPS on quadratic lattices We begin by summarising the key results of [17], in particular Sections 2, 3, 4 and 6 of that work, which relate to semi-classical orthogonal polynomial systems with support on discrete, quadratic lattices. Let Πn[x] denote the linear space of polynomials in x over C with degree at most n ∈ Z≥0. We define the divided-difference operator (DDO) Dx by Dxf(x) := f(ι+(x))− f(ι−(x)) ι+(x)− ι−(x) , (2.1) and impose the condition that Dx : Πn[x] → Πn−1[x] for all n ∈ N. In consequence we deduce that ι±(x) are the two y-roots of the quadratic equation Ay2 + 2Bxy + Cx2 + 2Dy + 2Ex+ F = 0. (2.2) Assuming A 6= 0 the two y-roots y± := ι±(x) for a given x-value satisfy ι+(x) + ι−(x) = −2 Bx+D A , ι+(x)ι−(x) = Cx2 + 2Ex+ F A , and their inverse functions ι−1± are defined by ι−1± (ι±(x)) = x. For a given y-value the quadra- tic (2.2) also defines two x-roots, if C 6= 0, which are consecutive points on the x-lattice, xs, xs+1 parameterised by the variable s ∈ Z and therefore defines a map xs 7→ xs+1. Thus we have the sequence of x-values . . . , x−2, x−1, x0, x1, x2, . . . given by . . ., ι−(x0) = ι+(x−1), ι−(x1) = ι+(x0), . . . which we denote as the lattice or the direct lattice G, and the sequence of y-values . . . , y−2, y−1, y0, y1, y2, . . . given by . . ., y0 = ι−(x0), y1 = ι−(x1), y2 = ι−(x2), . . . as the dual lattice G̃ (and distinct from the former in general). A companion operator to the divided- difference operator Dx is the mean or average operator Mx defined by Mxf(x) = 1 2 [f(ι+(x)) + f(ι−(x))] , so that the property Mx : Πn[x] → Πn[x] is ensured by the condition we imposed upon Dx. The difference between consecutive points on the dual lattice is given a distinguished notation through the definition ∆y(x) := ι+(x)− ι−(x). We will also employ an operator notation for the mappings from points on the direct lattice to the dual lattice E±x f(x) := f(ι±(x)) so that (2.1) can be written Dxf(x) = E+ x f − E−x f E+ x x− E−x x , for arbitrary functions f(x). The inverse functions ι−1± (x) define operators (E±x )−1 which map points on the dual lattice to the direct lattice and also an adjoint to the divided-difference operator Dx D∗xf(x) := f ( ι−1+ (x) ) − f ( ι−1− (x) ) ι−1+ (x)− ι−1− (x) = (E+ x )−1f − (E−x )−1f (E+ x )−1x− (E−x )−1x . The composite operators Ex := (E−x )−1E+ x and E−1x = (E+ x )−1E−x map between consecutive points on the direct lattice3. 3However in the situation of a symmetric quadratic A = C and D = E , which entails no loss of generality, then we have (E+ x )−1 = E− x and (E− x )−1 = E+ x and consequently there is no distinction between the divided-difference operator and its adjoint. 4 N.S. Witte and C.M. Ormerod Assuming AC 6= 0 one can classify these non-uniform quadratic lattices (or SNUL, special non-uniform lattices) according to two parameters – the discriminant B2 −AC and Θ = det A B D B C E D E F  , or AΘ = (B2 − AC)(D2 − AF) − (BD − AE)2. The quadratic lattices are classified into four sub-cases [9, 10]: q-quadratic (B2 − AC 6= 0 and Θ < 0), quadratic (B2 − AC = 0 and Θ < 0), q-linear (B2−AC 6= 0 and Θ = 0) and linear (B2−AC = 0 and Θ = 0), as the conic sections are divided into the elliptic/hyperbolic, parabolic, intersecting straight lines and parallel straight lines respectively. The q-quadratic lattice, in its general non-symmetrical form, is the most general case and the other lattices can be found from this by limiting processes. For the quadratic class of lattices the parameterisation on s can be made explicit using trigonometric/hyperbolic functions or their degenerations so we can employ a parameterisation such that ys = ι−(xs) = xs−1/2 and ys+1 = ι+(xs) = xs+1/2. We denote the totality of lattice points by G[x0] := {xs : s ∈ Z} with the point x0 as the basal point, and of the dual lattice by G̃[x0] := {xs : s ∈ Z+ 1 2}. We define the D-integral of a function defined on the x-lattice f : G[x] → C with basal point x0 by the Riemann sum over the lattice points I[f ](x0) = ∫ G Dx f(x) := ∑ s∈Z ∆y(xs)f(xs), where the sum is either a finite subset of Z, namely {0, . . . ,N}, Z≥0, or Z. This definition reduces to the usual definition of the difference integral and the Thomae–Jackson q-integrals in the canonical forms of the linear and q-linear lattices respectively. Amongst a number of properties that flow from this definition we have an analog of the fundamental theorem of calculus∫ x0≤xs≤xN DxDxf(x) = f(E+ x xN)− f(E−x x0). (2.3) Central to our study are orthogonal polynomial systems (OPS) defined on G, and a general reference for a background on these and other considerations is the monograph by Ismail [3]. Our OPS is defined via orthogonality relations with support on G∫ G Dx w(x)pn(x) lm(x) = { 0, 0 ≤ m < n, hn, m = n, n ≥ 0, hn 6= 0, where {lm(x)}∞m=0 is any system of polynomial bases with exact degx(lm) = m. Such relations define a sequence of orthogonal polynomials {pn(x)}∞n=0 under suitable conditions (see [3]). An immediate consequence of orthogonality is that the orthogonal polynomials satisfy a three term recurrence relation of the form an+1pn+1(x) = (x− bn)pn(x)− anpn−1(x), n ≥ 0, an 6= 0, p−1 = 0, p0 = γ0. (2.4) However we require non-polynomial solutions to this linear second-order difference equation, which are linearly independent of the polynomial solutions. To this end we define the Stieltjes function f(x) ≡ ∫ G Dy w(y) x− y , x /∈ G. Construction of a Lax Pair for the E (1) 6 q-Painlevé System 5 A set of non-polynomial solutions to (2.4), termed associated functions or functions of the second kind, and which generalise the Stieltjes function, are given by qn(x) ≡ ∫ G Dy w(y) pn(y) x− y , n ≥ 0, x /∈ G. The associated function solutions differ from the orthogonal polynomial solutions in that they have the initial conditions q−1 = 1/a0γ0, q0 = γ0f . The utility and importance of the Stieltjes function lies in the fact that that it connects pn and qn whereby the difference fpn−qn is exactly a polynomial of degree n − 1 which itself satisfies (2.4) in place of pn. This relation is crucial for the arguments adopted in [17]. With the polynomial and non-polynomial solutions we form the 2× 2 matrix variable, which occupies a primary position in our theory: Yn(x) =  pn(x) qn(x) w(x) pn−1(x) qn−1(x) w(x)  . In this matrix variable the three-term recurrence relation takes the form Yn+1 = KnYn, Kn(x) = 1 an+1 ( x− bn −an an+1 0 ) , detKn = an an+1 . (2.5) A key result required in the analysis of OPS are the expansions of polynomial solutions about the fixed singularity at x =∞ pn(x) = γn xn −(n−1∑ i=0 bi ) xn−1 +  ∑ 0≤i<j<n bibj − n−1∑ i=1 a2i xn−2 + O ( xn−3 ) , (2.6) valid for n ≥ 1, while for the associated functions the expansions read qn(x) = γ−1n x−n−1 + ( n∑ i=0 bi ) x−n−2 +  ∑ 0≤i≤j≤n bibj + n+1∑ i=1 a2i x−n−3 + O ( x−n−4 ) , (2.7) valid for n ≥ 0. In order to proceed any further we need to impose some structure on the weight characterising our OPS – in particular its spectral characteristics – and this takes the form of the definition of a D-semi-classical weight [9]. Such a weight satisfies a first-order homogeneous divided-difference equation w(y+) w(y−) = W + ∆yV W −∆yV (x), (2.8) where W (x), V (x) are irreducible polynomials in the spectral variable x, which we call spectral polynomials. As a consequence of this, under reasonable assumptions on the parameters of the weight, the Stieltjes function satisfies an inhomogeneous form of (2.8) WDxf = 2VMxf + U, (2.9) where in addition U(x) is a polynomial of x. A generic or regular D-semi-classical weight has two properties: (i) strict inequalities in the degrees of the spectral data polynomials, i.e., degxW = M , degx V = M − 1 and degx U = M − 2, and 6 N.S. Witte and C.M. Ormerod (ii) the lattice generated by any zero of (W 2 − ∆y2V 2)(x), say x̃1, does not coincide with another zero, x̃2, i.e. if (W 2 −∆y2V 2)(x̃2) = 0 then x̃2 /∈ ι2Z± x̃1. Further consequences of semi-classical assumptions are a system of spectral divided-difference equations for the matrix variable Yn, i.e., the spectral divided-difference equation DxYn(x) := AnMxYn(x) = 1 Wn(x) ( Ωn(x) −anΘn(x) anΘn−1(x) −Ωn(x)− 2V (x) ) MxYn(x), n ≥ 0, (2.10) with An termed the spectral matrix. For the D-semi-classical class of weights the coefficients appearing in the spectral matrix, Wn, Ωn, Θn, are polynomials in x, with fixed degrees inde- pendent of the index n. These spectral coefficients have terminating expansions about x = ∞ with the leading order terms Wn(x) = 1 2W + 1 4 [W + ∆yV ] ( y+ y− )n + 1 4 [W −∆yV ] ( y− y+ )n + O ( xM−1 ) , n ≥ 0, (2.11) Θn(x) = 1 y−∆y [W + ∆yV ] ( y+ y− )n − 1 y+∆y [W −∆yV ] ( y− y+ )n + O ( xM−3 ) , n ≥ 0, (2.12) Ωn(x) + V (x) = 1 2∆y [W + ∆yV ] ( y+ y− )n − 1 2∆y [W −∆yV ] ( y− y+ )n + O ( xM−2 ) , n ≥ 0, (2.13) where M = degx(Wn). Compatibility of the spectral divided-difference equations (2.10) and recurrence relations (2.5) imply that the spectral matrix and the recurrence matrix satisfy Kn(y+) ( 1− 1 2∆yAn )−1 ( 1 + 1 2∆yAn ) = ( 1− 1 2∆yAn+1 )−1 ( 1 + 1 2∆yAn+1 ) Kn(y−), n ≥ 0. (2.14) These relations can be rewritten in terms of the spectral coefficients arising in (2.10) as recur- rence relations in n, Wn+1 = Wn + 1 4∆y2Θn, n ≥ 0, (2.15) Ωn+1 + Ωn + 2V = (Mxx− bn)Θn, n ≥ 0, (2.16) (WnΩn+1 −Wn+1Ωn)(Mxx− bn) = −1 4∆y2Ωn+1Ωn +WnWn+1 + a2n+1WnΘn+1 − a2nWn+1Θn−1, n ≥ 0. (2.17) Another important deduction from these relations is that the spectral coefficients satisfy a bi- linear relation Wn(Wn −W ) = −1 4∆y2 det ( Ωn −anΘn anΘn−1 −Ωn − 2V ) , n ≥ 0. (2.18) The matrix product appearing in (2.14), and recurring subsequently, is called the Cayley trans- form of An and it has the evaluation( 1− 1 2∆yAn )−1 ( 1 + 1 2∆yAn ) (2.19) Construction of a Lax Pair for the E (1) 6 q-Painlevé System 7 = 1 W + ∆yV ( 2Wn −W + ∆y(Ωn + V ) −∆yanΘn ∆yanΘn−1 2Wn −W −∆y(Ωn + V ) ) , n ≥ 0. This result motivates the following definitions W± := 2Wn −W ±∆y(Ωn + V ), T+ := ∆yanΘn, T− := ∆yanΘn−1, n ≥ 1, (2.20) whilst for n = 0 we have W±(n = 0) := W±∆yV , T+(n = 0) := −∆ya0γ 2 0U , and T−(n = 0) := 0. Thus we define A∗n := ( W+ −T+ T− W− ) . (2.21) In a scalar formulation of the matrix linear divided-difference equation (2.10) one of the com- ponents, pn say, satisfies a linear second-order divided-difference equation of the form E+ x ( W + ∆yV ∆yΘn ) (E+ x )2pn + E−x ( W −∆yV ∆yΘn ) (E−x )2pn − { E+ x ( W+ ∆yΘn ) + E−x ( W− ∆yΘn )} E+ x E − x pn = 0. (2.22) Thus far our theoretical construction can only account for the OPS occurring in the Askey table – the hypergeometric and basic hypergeometric orthogonal polynomial systems [7]. To step beyond these, and in particular to make contact with the discrete Painlevé systems, one has to introduce pairs of deformation variables and parameters into the OPS. We denote such a single deformation variable by t, defined on a quadratic lattice (and possibly distinct from that of the spectral variable), with advanced and retarded nodes at ι±(t) = u±, ∆u = ι+(t) − ι−(t). We introduce such deformations with imposed structures that are analogous to those of the spectral variable. Thus, corresponding to the definition (2.8), we deem that a deformed D-semi-classical weight w(x; t) satisfies the additional first-order homogeneous divided-difference equation w(x;u+) w(x;u−) = R+ ∆uS R−∆uS (x; t), (2.23) where the deformation data polynomials, R(x; t), S(x; t), are irreducible polynomials in x. The spectral data polynomials, W (x; t), V (x; t), and the deformation data polynomials, R(x; t), S(x; t), now must satisfy the compatibility relation W + ∆yV W −∆yV (x;u+) R+ ∆uS R−∆uS (y−; t) = W + ∆yV W −∆yV (x;u−) R+ ∆uS R−∆uS (y+; t). (2.24) The deformed D-semi-classical deformation condition that corresponds to (2.9) is that the Stielt- jes transform satisfies an inhomogeneous version of (2.23) RDtf = 2SMtf + T, with T (x; t) being an irreducible polynomial in x with respect to R and S. Compatibility of spectral and deformation divided-difference equations for f implies the following identity on U and T ∆y [ (W + ∆yV )(x;u+) (W + ∆yV )(x;u−) (R+ ∆uS)(y−; t)U(x;u−)− (R−∆uS)(y−; t)U(x;u+) ] = ∆u [ (W + ∆yV )(x;u+)T (y−; t)− (W −∆yV )(x;u+) (R−∆uS)(y−; t) (R−∆uS)(y+; t) T (y+; t) ] . 8 N.S. Witte and C.M. Ormerod Corresponding to the (2.10) the deformed D-semi-classical OPS satisfies the deformation divided- difference equation DtYn := BnMtYn = 1 Rn ( Γn Φn Ψn Ξn ) MtYn, n ≥ 0. (2.25) The deformation coefficients appearing in matrix Bn above satisfy a linear identity Ψn = − an an−1 Φn−1, n ≥ 1, (2.26) and a trace identity ∆u(Γn + Ξn) = 2Hn [ R+ ∆uS an(u−) − R−∆uS an(u+) ] , n ≥ 0, which means that only three of these are independent. Here Hn is a constant with respect to x and arises as a decoupling constant which will be set subsequently in applications to a convenient value. The deformation coefficients are all polynomials in x, with fixed degrees independent of the index n but with non-trivial t dependence. Let L = max(degxR,degx S). As x → ∞ we have the leading orders of the terminating expansions of the following deformation coefficients 2 Hn Rn = −(γn(u+) + γn(u−)) [ R−∆uS γn−1(u+) + R+ ∆uS γn−1(u−) ] + O ( xL−1 ) n ≥ 0, (2.27) ∆u 2Hn Φn = [ (R+ ∆uS) γn(u+) γn(u−) − (R−∆uS) γn(u−) γn(u+) ] x−1 + O ( xL−2 ) , n ≥ 0, (2.28) and ∆u Hn Γn = (γn(u−)− γn(u+)) [ R+ ∆uS γn−1(u−) + R−∆uS γn−1(u+) ] + O ( xL−1 ) , n ≥ 0. (2.29) Compatibility of the deformation divided-difference equation (2.25) and the recurrence rela- tion (2.5) implies the relation Kn(;u+) ( 1− 1 2∆uBn )−1 ( 1 + 1 2∆uBn ) = ( 1− 1 2∆uBn+1 )−1 ( 1 + 1 2∆uBn+1 ) Kn(;u−), n ≥ 0. (2.30) From this we can deduce that the deformation coefficients, Rn, Γn, Φn, satisfy recurrence relations in n in parallel to those of (2.15), (2.16) an+1(u−) Hn+1 (−2Rn+1 + ∆uΓn+1) + an(u−) Hn (2Rn + ∆uΓn) = −[x− bn(u−)] ∆u Hn Φn + 2an(u−) ( R+ ∆uS an(u−) − R−∆uS an(u+) ) , n ≥ 0, an+1(u+) Hn+1 (2Rn+1 + ∆uΓn+1) + an(u+) Hn (−2Rn + ∆uΓn) = −[x− bn(u+)] ∆u Hn Φn + 2an(u+) ( R+ ∆uS an(u−) − R−∆uS an(u+) ) , n ≥ 0. The deformation coefficients satisfy the bilinear or determinantal identity R2 n + 1 4∆u2 [ΓnΞn − ΦnΨn] = −HnRn [ R+ ∆uS an(u−) + R−∆uS an(u+) ] , n ≥ 0, Construction of a Lax Pair for the E (1) 6 q-Painlevé System 9 which is the analogue of (2.18). The matrix product given in (2.30) has the evaluation( 1− 1 2∆uBn )−1 ( 1 + 1 2∆uBn ) = an(u−) 2Hn(R+ ∆uS) × 2Rn + 2Hn R−∆uS an(u+) + ∆uΓn ∆uΦn ∆uΨn 2Rn + 2Hn R+ ∆uS an(u−) −∆uΓn.  , n ≥ 0. This again motivates the definitions R± := 2Rn + 2Hn R∓∆uS an(u±) ±∆uΓn, P+ := −∆uΦn, P− := ∆uΨn, n ≥ 1, (2.31) together with B∗n := ( R+ −P+ P− R− ) . Our final relation expresses the compatibility of the spectral and deformation divided-diffe- rence equations. The spectral matrix An(x; t) and the deformation matrix Bn(x; t) satisfy the D-Schlesinger equation( 1− 1 2∆yAn(;u+) )−1 ( 1 + 1 2∆yAn(;u+) ) ( 1− 1 2∆uBn(y−; ) )−1 ( 1 + 1 2∆uBn(y−; ) ) (2.32) = ( 1− 1 2∆uBn(y+; ) )−1 ( 1 + 1 2∆uBn(y+; ) ) ( 1− 1 2∆yAn(;u−) )−1 ( 1 + 1 2∆yAn(;u−) ) . Let us define the quotient χ ≡ (W + ∆yV )(x;u+) (W + ∆yV )(x;u−) (R+ ∆uS)(y−; t) (R+ ∆uS)(y+; t) = (W −∆yV )(x;u+) (W −∆yV )(x;u−) (R−∆uS)(y−; t) (R−∆uS)(y+; t) . The compatibility relation (2.32) can be rewritten as the matrix equation χB∗n(y+; t)A∗n(x;u−) = A∗n(x;u+)B∗n(y−; t), (2.33) or component-wise with the new variables in the more practical form as χ [W+(x;u−)R+(y+; t)− T−(x;u−)P+(y+; t)] = W+(x;u+)R+(y−; t)− T+(x;u+)P−(y−; t), (2.34) χ [T+(x;u−)R+(y+; t) + W−(x;u−)P+(y+; t)] = T+(x;u+)R−(y−; t) + W+(x;u+)P+(y−; t), (2.35) χ [T−(x;u−)R−(y+; t) + W+(x;u−)P−(y+; t)] = T−(x;u+)R+(y−; t) + W−(x;u+)P−(y−; t), (2.36) χ [W−(x;u−)R−(y+; t)− T+(x;u−)P−(y+; t)] = W−(x;u+)R−(y−; t)− T−(x;u+)P+(y−; t). (2.37) For a general quadratic lattice there exists two fixed points defined by ι+(x) = ι−(x), and let us denote these two points of the x-lattice by xL and xR. By analogy with the linear lattices we conjecture the existence of fundamental solutions to the spectral divided-difference equation about x = xL, xR which we denote by YL, YR respectively. Furthermore let us define the connection matrix P (x; t) := YR(x; t)−1YL(x; t). 10 N.S. Witte and C.M. Ormerod From the spectral divided-difference equation (2.10) it is clear that P is a D-constant function with respect to x, that is to say P (y+; t) = P (y−; t). In addition it is clear from the deformation divided-difference equation (2.25) that this type of deformation is also a connection preserving deformation in the sense that P (x;u+) = P (x;u−). This is our analogue of the monodromy matrix and generalises the connection matrix of Birkhoff and his school [1, 2], although we emphasise that we have made an empirical observation of this fact and not provided any rigorous statement of it. 3 Big q-Jacobi OPS As our central reference on the Askey table of basic hypergeometric orthogonal polynomial systems we employ [8], or its modern version [7]. We consider a sub-case of the quadratic lattices, in particular the q-linear lattice in both the spectral and deformation variables x and t in its standardised form, so that ι+(x) = qx, ι−(x) = x, ∆y(x) = (q − 1)x and ι+(t) = qt, ι−(t) = t, ∆u(t) = (q − 1)t. In [7] the big q-Jacobi weight given by equation (14.5.2) is w(x) = (a−1x, c−1x; q)∞ (x, bc−1x; q)∞ , subject to 0 < aq, bq < 1, c < 0 with respect to the Thomae–Jackson q-integral∫ aq bq dqx f(x). The q-shifted factorials have the standard definition (a; q)∞ = ∞∏ j=0 ( 1− aqj ) , \|q\| < 1, (a1, . . . , an; q)∞ = (a1; q)∞ · · · (an; q)∞. We deform this weight by introducing an extra q-shifted factorial into the numerator and de- nominator containing the deformation variable and parameter, and relabeling the big q-Jacobi parameters. We propose the following weight w(x; t) = ( b2x, b3x, b −1 6 xt−1; q ) ∞( b1x, b4x, b6xt−1; q ) ∞ . (3.1) A condition b1b2b3b4 = 1 will apply, so we have four free parameters. We do not need to specify the support for this weight for the purposes of our work, but suffice it to say that any Thomae– Jackson q-integral with terminals coinciding with any pair of zeros and poles of the weight would be suitable. The spectral data polynomials are computed to be W + ∆yV = b6 (1− b1x) (1− b4x) (t− b6x) , W −∆yV = (1− b2x) (1− b3x) (b6t− x) . (3.2) Clearly the regular M = 3 case is applicable and we seek solutions to the spectral coefficients with degxWn = 3, degx Ωn = 2, degx Θn = 1. Our procedure is to employ the following Construction of a Lax Pair for the E (1) 6 q-Painlevé System 11 algorithm, as detailed in [17]. Firstly we parameterise the spectral matrix in a minimal way; secondly we relate the parameterisation of the deformation matrix to that of the spectral matrix and thus close the system of unknowns; and finally utilise these parameterisations in the system of over-determined equations to derive evolution equations for our primary variables. What constitutes the primary variables will emerge from the calculations themselves. Proposition 1. Let us define a new parameter b5 replacing qn by qn = b5 b1b4b6 , n ∈ Z≥0. Let the parameters satisfy the conditions q 6= 1, b5 6= q−1/2,±1, q1/2, b1b4 6= 0,∞ and b2b3 6= 0,∞. Given the degrees of the spectral coefficients we parameterise these by 2Wn −W = w3x 3 + w2x 2 + w1x+ w0, Ωn + V = v2x 2 + v1x+ v0, Θn = u1(x− λn). Let λn be the unique zero of the (1, 2) component of A∗n, i.e., Θn(x) and define the further variables νn = (2Wn −W )(λn, t) and µn = (Ωn + V )(λn, t). Then the spectral coefficients are given by 2Wn −W = x2 νn λ2n + 1 2(x− λn) × [ −b6 ( b5 + b−15 ) x2 + 1 + (b1 + b2 + b3 + b4) b6t+ b26 λn x− 2tb6 x+ λn λ2n ] , (3.3) Ωn + V = µn + b6 2b5(1− q) ( 1− b25 ) λ2n (x− λn) { − ( 1− b25 )2 λ2nx − 2b25 [ b−11 + b−12 + b−13 + b−14 + ( b6 + b−16 ) t− 2λn ] λ2n + b5b −1 6 ( 1 + b25 ) [( 1 + (b1 + b2 + b3 + b4)b6t+ b26 ) λn + 2νn − 2tb6 ] } , (3.4) and Θn = − b6 ( 1− qb25 ) q(1− q)b5 (x− λn). (3.5) We note that λn, µn, νn satisfy the quadratic relation ν2n = (1− q)2λ2nµ2n + b6(b1λn − 1)(b2λn − 1)(b3λn − 1)(b4λn − 1)(λn − tb6)(b6λn − t). (3.6) Proof. Consistent with the known data, i.e., the degrees, from (2.11), (2.12), (2.13) we compute the leading coefficients to be u1 = − b6 ( 1− qb25 ) q(1− q)b5 , v2 = − b6 ( 1− b25 ) 2(1− q)b5 , w3 = − b6 ( 1 + b25 ) 2b5 , confirming the relation given by the coefficient of [x6] in (2.18), w2 3 = (q−1)2v22 +b26. In addition we identify the diagonal elements of the [x3] coefficient of A∗n κ+ ≡ w3 + (q − 1)v2 = −b5b6, κ− ≡ w3 − (q − 1)v2 = −b6 b5 . 12 N.S. Witte and C.M. Ormerod From the coefficient of [x0] in (2.18) we deduce (modulo a sign ambiguity) w0 = b6t, and from the coefficient of [x1] in (2.18) we similarly find w1 = −1 2 [ 1 + t(b1 + b2 + b3 + b4)b6 + b26 ] . Now utilising the condition νn = (2Wn −W )(λn, t) we invert this to compute w2 = 1 + t(b1 + b2 + b3 + b4)b6 + b26 2λn + 1 2b6 ( b5 + b−15 ) λn + νn − tb6 λ2n . Proceeding further we infer from the coefficient of [x5] in (2.18) that v1 = ( 1 + b25 ) w2 − ( b−11 + b−12 + b−13 + b−14 ) b5b6 − b5 ( 1 + b26 ) t (1− q) ( 1− b25 ) , and employing the previous result for w2 we derive (1− q) ( 1− b25 )( 1 + b25 )v1 = − b5b6 [ b−11 + b−12 + b−13 + b−14 + ( b6 + b−16 ) t ] 1 + b25 + 1 + t(b1 + b2 + b3 + b4)b6 + b26 2λn + b6 ( 1 + b25 ) 2b5 λn + νn − tb6 λ2n . This leaves v0 to be determined. Imposing the relation µn = (Ωn + V )(λn, t) we can invert this and find (1− q) ( 1− b25 ) v0 = (1− q) ( 1− b25 ) µn + b5b6 [ b−11 + b−12 + b−13 + b−14 + ( b6 + b−16 ) t ] λn − 2b5b6λ 2 n − 1 2 ( 1 + b25 )[ 1 + t(b1 + b2 + b3 + b4)b6 + b26 ] + ( 1 + b25 ) λn (b6t− νn). This concludes our proof. � Remark 1. We observe that the appearance of the quantity qnb1b4b6 with n ∈ Z≥0 and its replacement by the new parameter b5 constitutes a special condition. This condition is one of the necessary conditions for a member of our particular sequence of classical solutions to the E (1) 6 q-Painlevé equations, and is built-in by our construction. The other condition derives from the initial conditions n = 0 in our construction, see (2.4) and following (2.20). From our deformed weight (3.1) we compute the deformation data polynomials to be R+ ∆uS = 1 b6 (b6qt− x), R−∆uS = (qt− b6x). (3.7) We can verify that the compatibility relation (2.24) is identically satisfied by our spectral and deformation data polynomials. We see that this places us in the class L = 1. We will employ an abbreviation for the dependent variables evaluated at advanced or retarded times, e.g., λn(t) = λn, λn(qt) = λ̂n, λn ( q−1t ) = λ̌n. In the second stage of our algorithm we parameterise the Cayley transform of the deformation matrix B∗n = ( R+ −P+ P− R− ) , n ≥ 0, consistent with known degrees, i.e., degxR± = 1, degxP± = 0, so that R± = r1,±x+ r0,±, P± = p±. Construction of a Lax Pair for the E (1) 6 q-Painlevé System 13 Lemma 1. Let us assume b6 6= 0 and b5 6= q−1/2, q1/2. Then the off-diagonal components of the deformation matrix are given by p+ = −ânr1,− + anr1,+, (3.8) p− = −anr1,− + ânr1,+. (3.9) Proof. We resolve the A-B compatibility relation (2.33) into monomials of x. Examining the x7 coefficient of the (1, 2) and (2, 1) components yields (3.8) and (3.9) respectively. � Lemma 2. Let us assume b5 6= q1/2, an, ân 6= 0 and λn 6= b6t, b −1 6 t. Then the spectral and deformation matrices satisfy the following residue formulae R−(b6qt, t) + W+(b6qt, qt) T+(b6qt, qt) P+(b6qt, t) = 0, (3.10) R−(b−16 qt, t) + W+(b−16 qt, qt) T+(b−16 qt, qt) P+ ( b−16 qt, t ) = 0, (3.11) and R+(b6qt, t) + W−(b6t, t) T+(b6t, t) P+(b6qt, t) = 0, (3.12) R+(b−16 qt, t) + W−(b−16 t, t) T+(b−16 t, t) P+ ( b−16 qt, t ) = 0. (3.13) Proof. In this step we compute the residues of the A-B compatibility relation, with respect to x, at the zeros and poles of χ(x, t) = (x− qb6t) (b6x− qt) q (x− b6t) (b6x− t) . (3.14) From the residue of (2.34) at the zero x = b6qt we deduce (3.10), and from the same equation at the zero x = b−16 qt we deduce (3.11). From the residue of (2.37) at the pole x = b6t we deduce (3.12), and from the same equation at the pole x = b−16 t we deduce (3.13). � Remark 2. Although the above proof appealed to the vanishing of the right-hand side of one of the compatibility conditions, namely (2.34), at either of the two zeros of χ, in fact under these conditions the right-hand sides of all the other compatibility conditions, i.e. (2.35), (2.36), and (2.37), also vanish. This is because χ = 0 implies ( R2 − ∆u2S2 )( b±16 qt; t ) = 0 and( W 2 − ∆y2V 2 )( b±16 qt; qt ) = 0, and furthermore the spectral and deformation matrices satisfy the determinantal identities detA∗n = W+W− + T+T− = W 2 −∆y2V 2, detB∗n = R+R− + P+P− = an ân ( R2 −∆u2S2 ) . Therefore under the specialisations x = b±16 qt the right-hand sides of (2.35), (2.36), and (2.37) are proportional to the right-hand side of (2.34), and the vanishing of the latter implies the vanishing of the former. In this way we ensure that all components of the A-B compatibility vanish under the single condition. A similar observation applies to the left-hand sides of the compatibility relations at the zeros of χ−1, i.e. x = b±16 t. 14 N.S. Witte and C.M. Ormerod We introduce our first change of variables, µn, νn 7→ z±, via the relations νn = 1 2 λn[κ+z+ + κ−z−], (3.15) µn = 1 2(q − 1) [κ+z+ − κ−z−]. (3.16) The new variables satisfy an identity corresponding to (3.6) which reads κ+κ−z+z− = 1 λ2n [ W 2 −∆y2V 2 ] (λn, t). Next we subtract (3.11) from (3.10), in order to eliminate both z− and r0,−. This yields qp+( 1− qb25 ) ân [ − b5 ( b5qtλ̂n − 1 ) b6λ̂n + qb25t( λ̂n − b6qt )( b6λ̂n − qt ) ẑ+ ] + qt 1 b6 r1,− = 0. (3.17) This result motivates the definition of the new variable Z Z = − b5b6qt (λ̂n − b6qt)(b6λ̂n − qt) ẑ+ + b5qtλ̂n − 1 λ̂n ∣∣∣∣∣ t→q−1t . Definition 1. In terms of this new variable Z we have z+ = 1 b5b6t (λn − tb6)(b6λn − t)[(b5t− Z)λn − 1] λn , (3.18) z− = b5t (b1λn − 1)(b2λn − 1)(b3λn − 1)(b4λn − 1) λn[(b5t− Z)λn − 1] . (3.19) Our final rewrite of the dependent variables is λn(t)→ g(t), (3.20) Z(t)→ b5t− f(q−1t). (3.21) We are now in a position to undertake the third stage of our derivation. The first of the evolution equations is given in the following result. Proposition 2. Let us assume that q 6= 0, b5 6= q−1/2, t 6= 0, g 6= 0, b6t, b −1 6 t and an 6= 0. The variables f , g satisfy the first-order q-difference equation ( gf̌ − 1 ) (gf − 1) = t2 ( g − b−11 )( g − b−12 )( g − b−13 )( g − b−14 ) (g − b6t) ( g − b−16 t ) . (3.22) This evolution equation is identical to the second equation of equation (4.15) of Kajiwara et al. [6] and to the second equation of equation (3.23) of Kajiwara et al. [5]. Proof. Subtract (3.13) from (3.12) in order to eliminate both z+ and r0,+. This yields the relation qp+( −1 + qb25 ) an [ t (b6t− λn)(−t+ b6λn) z− − (b5 − tλn) b6λn ] + qt 1 b6 r1,+ = 0. (3.23) Thus we have two different ways of computing the ratio of r1,+ to r1,−; on the one hand we have from (3.17) r1,+ = − [b5(qb5t− Ẑ)− t]ân b5Ẑan r1,−, (3.24) Construction of a Lax Pair for the E (1) 6 q-Painlevé System 15 whereas using (3.23) we have b5an ân r1,+ r1,− = [ b5b6t 2 (b1λn − 1) (b2λn − 1) (b3λn − 1) (b4λn − 1) + (b5 − tλn) (λn − tb6) (b6λn − t) [λn (b5t− Z)− 1] ] ÷ [ t2b6 (b1λn − 1) (b2λn − 1) (b3λn − 1) (b4λn − 1) − (qb5tλn − 1) (λn − tb6) (b6λn − t) [λn (b5t− Z)− 1] ] . Equating these two forms gives (3.22). � The second evolution equation, to be paired with the first (3.22) as a coupled system, is given next. Proposition 3. Let us make the following assumptions: t 6= 0, b5 6= 1, q−1/2, q−1, f 6= 0, b5f 6= t, g 6= 0, ĝ 6= 0 and an 6= 0. In addition let us assume that the condition ĝ 6= 1− qb5tg − qb25 + qb25fg f − b5qt , holds. The variables f , g satisfy the first-order q-difference equation (fĝ − 1)(fg − 1) = qt2 (f − b1)(f − b2)(f − b3)(f − b4) (f − b5qt) ( f − b−15 t ) . (3.25) This evolution equation is the same as the first equation of equation (4.15) in Kajiwara et al. [6] and the first equation of equation (3.23) in Kajiwara et al. [5], both subject to typographical corrections. Proof. Cross multiplying the relations (3.10), (3.11), (3.12), (3.13) we can eliminate all refe- rence to the deformation matrix and deduce the identity W+(b6qt, qt) T+(b6qt, qt) W−(b6t, t) T+(b6t, t) = W+ ( qb−16 t, qt ) T+ ( qb−16 t, qt ) W− ( b−16 t, t ) T+ ( b−16 t, t ) . (3.26) Into this identity we employ the following evaluations for the advanced and retarded values of z± ẑ+ = q−1 b5b6t (fĝ − 1)(ĝ − b6qt)(ĝb6 − qt) ĝ , ẑ− = qb5t (ĝb1 − 1)(ĝb2 − 1)(ĝb3 − 1)(ĝb4 − 1) ĝ(fĝ − 1) , z+ = t b5 (gb1 − 1)(gb2 − 1)(gb3 − 1)(gb4 − 1t) g(fg − 1) , z− = b5 b6t (fg − 1)(g − b6t)(gb6 − t) g . We find that this relation factorises into two non-trivial factors, the first of which is proportional to ĝ − 1− qb5tg − qb25 + qb25fg f − qb5t . Assuming this is non-zero our evolution equation is then the remaining factor of (3.26) (fĝ − 1)(fg − 1) = qb5t 2 (f − b1)(f − b2)(f − b3)(f − b4) (f − qb5t)(b5f − t) , or alternatively (3.25). � 16 N.S. Witte and C.M. Ormerod Lastly we have an auxiliary evolution equation which controls the normalisation of the or- thogonal polynomial system. Proposition 4. Let us assume b6 6= 0, b5 6= q−1/2, f 6= b5qt, b −1 5 t and γn 6= 0 for n ≥ 0. The leading coefficient of the polynomials or second-kind solutions (see (2.6), (2.7)) satisfy the first-order q-difference equation( γ̂n b6γn )2 = f − b−15 t f − b5qt . (3.27) Proof. Using the leading order, i.e., the [x] terms, in the expansions (2.27), (2.29) with definitions (2.31) we can compute r1,+. However by using these same expansions to compute r1,− and the equation (3.24), which relates these two quantities, we have an alternative expression for r1,+. Equating these expressions then gives (3.27). � We conclude our discussion by summarising our results for the spectral and deformation matrices in terms of the fn, gn variables. Henceforth we will restore the index n on all our variables. The form of the spectral matrix is given in the following proposition. Proposition 5. Assume that \|q\| 6= 1, b6 6= 0, b25 6= q−1, 1, q and an 6= 0. The spectral matrix elements (2.19), (2.20), (2.21) are given by Tn,+ = b6 b5 ( q−1 − b25 ) anx(x− gn), Tn,− = b6 b5 ( 1− q−1b25 ) anx(x− gn−1), and Wn,+ x(x− gn) = −xb5b6 + b6( 1− b25 ) [−b25 t + b5 ( 1 b1 + 1 b2 + 1 b3 + 1 b4 )] − b6(b5fn − t) (1− b25)t [ gn(t− fnb5) fn + tb5 ( 1 + b26 ) b6 ] + b6t (1− b25) [ b25 g2n − 1− b25 xgn − (b1 + b2 + b3 + b4)b 2 5 1 gn + b25 fn gn − gn fn + (1− gnb1)(1− gnb2)(1− gnb3)(1− gnb4) ( gn − xb25 ) (1− fngn)g2n(x− gn) ] , (3.28) or Wn,+ x(x− gn) = − b6t( 1− b25 ) (fn − b1)(fn − b2)(fn − b3)(fn − b4) ( 1− b25fnx ) f2n(1− fngn)(1− fnx) + b6t(1− xb1)(1− xb2)(1− xb3)(1− xb4) x(1− xfn)(x− gn) − b6(b5fn − t) fn x − b6(b5fn − t) b5 ( 1− b25 ) t { b25 ( 1 + b26 ) t b6 + b5gn fn (t− b5fn) + b5 f2n [ t+ b5fn − tfn ( 1 b1 + 1 b2 + 1 b3 + 1 b4 )]} , and tWn,− x(x− gn) = (1− xfn)(x− b6t)(xb6 − t) x(x− gn) − b6 1− b25 (t− b1b5)(t− b2b5)(t− b3b5)(t− b4b5) b5(tgn − b5) Construction of a Lax Pair for the E (1) 6 q-Painlevé System 17 + b6(b5fn − t) b5 ( 1− b25 ) [(1− b25)x− t2 gn − gnb25 + b25 ( 1 + b26 ) t b6 − b5t 2(1− gnb1)(1− gnb2)(1− gnb3)(1− gnb4) gn(1− fngn)(tgn − b5) ] , (3.29) or tWn,− x(x− gn) = (1− xfn)(x− b6t)(b6x− t) x(x− gn) + b6t 2( 1− b25 ) (fn − b1)(fn − b2)(fn − b3)(fn − b4) f2n(1− fngn) + b6(b5fn − t) b5 ( 1− b25 ) {(1− b25)x+ b25 ( 1 + b26 ) t b6 + b5gn fn (t− b5fn) + b5 f2n [ t+ b5fn − tfn ( 1 b1 + 1 b2 + 1 b3 + 1 b4 )]} . Proof. This follows from applying the transformations (3.15), (3.16), (3.18), (3.19) and (3.20), (3.21) successively to (3.5), (3.3) and (3.4). The alternative forms arise from applying partial fraction expansions with respect to either of fn or gn. � The deformation matrix is summarised in the next result. Proposition 6. Assume that \|q\| 6= 1, b6 6= 0, b25 6= q−1, 1, q. The deformation matrix elements are given by Rn,+ = γ̂n b6γn { x+ 1 1− b25 [ − q ( t2b6 + g2nb 2 5b6 − tgnb25 ( 1 + b26 )) gnb6 + qt2b5(1− b1gn)(1− b2gn)(1− b3gn)(1− b4gn) gn(fngn − 1)(tgn − b5) − q(t− b1b5)(t− b2b5)(t− b3b5)(t− b4b5) (tgn − b5)(fnb5 − t) ]} , (3.30) and Rn,− = b6γn−1 γ̂n−1 { x+ 1 1− b25 [ q ( t2b6 + g2nb 2 5b6 − tgn ( 1 + b26 )) gnb6 − qt2b5(1− b1gn)(1− b2gn)(1− b3gn)(1− b4gn) gn(fngn − 1)(tgn − b5) + q(t− b1b5)(t− b2b5)(t− b3b5)(t− b4b5) (tgn − b5)(fnb5 − t) ]} . Furthermore Pn,+ = an [ γ̂n b6γn − b6γn γ̂n ] , Pn,− = an [ γ̂n−1 b6γn−1 − b6γn−1 γ̂n−1 ] . Proof. Using the leading orders in the expansions (2.27), (2.29), i.e., the [x] terms, with definitions (2.31) we deduce r1,+ = γ̂n b6γn , r1,− = b6γn−1 γ̂n−1 . 18 N.S. Witte and C.M. Ormerod Using the leading orders in the expansions (2.28), i.e., the [x0] terms, with definition (2.31) and (2.26) we deduce p+ = an [ γ̂n b6γn − b6γn γ̂n ] , p− = an [ γ̂n−1 b6γn−1 − b6γn−1 γ̂n−1 ] . Using the coefficient of the [x7] term in the (1, 1) element of the A-B compatibility rela- tions (2.32), along with the solution of (3.22) for fn(q−1t) and (3.25) for gn(qt) we deduce ( 1− b25 )r0,+ r1,+ = qb5 ( t fn − b5 ) gn + ( b5qt fn − 1 ) ĝn + [ 1 + qb25 − b5 ( b−11 + b−12 + b−13 + b−14 ) qt ] 1 fn + ( 1 + b26 ) b6 b25qt, whereas if we examine the [x7] term in the (2, 2) element of the A-B compatibility relations in the same way then we find ( 1− b25 )r0,− r1,− = qb5 ( b5 − t fn ) gn + ( 1− b5qt fn ) ĝn + [ b5 ( b−11 + b−12 + b−13 + b−14 ) qt− qb25 − 1 ] 1 fn − ( 1 + b26 ) b6 qt. Into both of these expressions we can employ (3.25) for ĝn and make a partial fraction expansion with respect to fn. � 4 Reconciliation with the Lax pairs of Sakai and Yamada 4.1 Sakai Lax Pair In [15] Sakai constructed a Lax pair for the E (1) 6 q-Painlevé equations using a degeneration of a two-variable case of the Garnier system based upon the Lax pairs for the D (1) 5 q-Painlevé system [14]. Subsequently Murata [11] gave more details for this Lax pair. We intend to establish a correspondence between our Lax pair and that of Sakai. We will carry this out in a sequence of simple steps rather than as a single step as this will reveal how similar they are. Our first step is to give a variation on the parameterisation of the spectral and deformation matrices to that given in Section 3. In this alternative formulation, we seek a spectral matrix Ã(x; t) (actually identical to the Cayley transform A∗n) with the specifications Ã(x; t) = A0 +A1x+A2x 2 +A3x 3, (4.1) and (i) the determinant is b6 (b1x− 1) (b2x− 1) (b3x− 1) (b4x− 1) (x− b6t) (b6x− t) , (ii) A3 is diagonal with entries κ1 = −b5b6 and κ2 = −b6/b5, (iii) A0 = b6t1, (iv) the root of the (1,2) entry of Ã(x; t) with respect to x is λ, (v) Ã(λ; t) is lower triangular with diagonal entries −b5b6λz+ and − b6λz− b5 where b26z−z+λ 2 = det Ã(λ; t). Construction of a Lax Pair for the E (1) 6 q-Painlevé System 19 Any such matrix is in the general form Ã(x; t) = tb6I − b5b6x[z1 + (x− α)(x− λ)] b6wx (x− λ) b5 b5b6x(xγ + δ) w b6x[z2 + (x− β)(x− λ)] b5  , where the properties specify the variables( 1− b25 ) α = 1 b1 + 1 b2 + 1 b3 + 1 b4 + ( 1 b6 + b6 ) t − (b1 + b2 + b3 + b4)b5 t λ − b5 ( 1 b6 + b6 ) 1 λ + b25z1 λ + z2 λ − 2λ, (4.2) ( 1− b25 ) β = − ( 1 b1 + 1 b2 + 1 b3 + 1 b4 ) b25 − ( 1 b6 + b6 ) b25t + (b1 + b2 + b3 + b4)b5 t λ + b5 ( 1 b6 + b6 ) 1 λ − b25z1 λ − z2 λ + 2b25λ, (4.3) γ = −(b3b4 + b2b3 + b2b4 + b1b2 + b1b3 + b1b4)− ( 1 b1 + 1 b2 + 1 b3 + 1 b4 )( 1 b6 + b6 ) t − t2 + αβ + z1 + z2 + 2(α+ β)λ+ λ2, (4.4) δ = b1 + b2 + b3 + b4 + [ (b3b4 + b2b3 + b2b4 + b1b2 + b1b3 + b1b4) ( 1 b6 + b6 ) − ( 1 b5 + b5 )] t + ( 1 b1 + 1 b2 + 1 b3 + 1 b4 ) t2 − z1(β + λ)− z2(α+ λ) + (−2αβ + γ)λ− (α+ β)λ2. (4.5) The z1 and z2 are related to z± by z1 = z+ + t b5λ and z2 = z− + tb5 λ . (4.6) In addition w = 1− qb25 q an. We seek a deformation matrix B̃(x; t) of the form B̃(x; t) = x (x− b6qt) ( x− b−16 qt )(x1 +B0), where B0 = [ r1,1 r1,2 r2,1 r2,2 ] . (4.7) This leads to the compatibility relation B̃(qx; t)Ã(x; t) = Ã(x; qt)B̃(x; t). (4.8) This relation is just a rewriting of (2.32) whereby all the factors of χ are placed into the denominator of B̃ by the above definition. Lemma 3. The overdetermined system (4.8), with (4.1) and (4.7) is satisfied if the coupled E (1) 6 q-Painlevé equations (3.22) and (3.25) are satisfied. Proof. Examining the coefficient of x6 in the numerator of the (1,2) entry of (4.8) we find r1,2 = q 1− qb25 (ŵ − w). (4.9) 20 N.S. Witte and C.M. Ormerod Now we seek two alternative expressions for r1,2 – one involving quantities at the advanced time qt and another involving those at the unshifted time t. The first of these is found from solving for the (1,2) entry of the residue of (4.8) at x = b6qt simultaneously with the (1,2) entry of the residue of (4.8) at x = b−16 qt. This yields r1,2 = −qtŵ ( qtb6 − λ̂ )( qt− b6λ̂ ) b5 { qt ( 1− b5b6ẑ1 ) + ( qtb6 − λ̂ )[ b6 + qtb5 ( qt− b6λ̂ )]} . (4.10) The second expression for r1,2 is found from solving for the (1,2) entry of the residue of (4.8) at x = b6t simultaneously with the (1,2) entry of the residue of (4.8) at x = b−16 t. This gives r1,2 = −qtw(b6t− λ)(t− b6λ) tb5 − tb6z2 + b5b6(b6t− λ) + t(b6t− λ)(t− b6λ) . (4.11) Combining (4.9) and (4.10) or (4.9) and (4.11), and employing the change of variables (3.18) and (3.19) with (3.20) and (3.21), we can solve for ŵ in two ways. Assuming w is non-zero it cancels out, leaving an expression for Ẑ in terms of Z and λ. This is equivalent to the first E (1) 6 q-Painlevé equation (3.22). To find the second equation we solve (4.8) for Ã(x; t) Ã(x; t) = B̃(qx; t)−1Ã(x; qt)B̃(x; t), and use this to find the zero of Ã(x; t)12, i.e., g(t). In addition to r1,2 we now require r2,2 (even though the denominator of Ã(x; t)12 depends on r1,1, r1,2, r2,1, r2,2 identities resulting from the compatibility conditions imply that this will trivialise – see the subsequent observation). The entry r2,2 has already been found, along with r1,2, from the arguments given earlier and this is r2,2 = λ̂− qt ( 1 + b26 ) b6 − ( b6qt− λ̂ )( qt− b6λ̂ ) × b6 + q2t2b5 ( 1 + b26 ) − b5b6qt ( α̂+ λ̂ ) b26 ( λ̂− b6qt ) − b6qt+ b5b26qtẑ1 − b5b6qt ( b6qt− λ̂ )( qt− b6λ̂ ) . (4.12) The numerator of Ã(x; t)12 appears to be a polynomial of degree 6 in x, however it has trivial zeros matching those of the denominator q2b5 b26 (x− b6t)(x− b6qt)(b6x− t)(b6x− qt)ŵ, so that their ratio is in fact polynomial of degree 2. Into Ã(x; t)12 we first substitute for r1,2 using (4.9), then for r2,2 using (4.12), and thirdly for α̂, β̂, γ̂, δ̂ using (4.2), (4.3), (4.4), (4.5) at up-shifted times, respectively. Into the resulting expression we employ (4.6) for ẑ1, ẑ2 along with (3.18), (3.19) at the up-shifted time to bring the whole expression in terms of λ̂ and Ẑ. The relevant zero of the ensuing expression (the other zero is x = 0) then gives λ = g in terms of λ̂ = ĝ and f , or equivalently by (3.25). � Now we recount the formulation given by Sakai [15] and Murata [11]. Their Lax pairs are Y (qx; t) = A(x, t)Y (x; t), Y (x; qt) = B(x, t)Y (x; t), satisfying the compatibility condition A(x, qt)B(x, t) = B(qx, t)A(x, t). The spectral matrix is parameterised in the following way A(x, t) = ( κ1W (x, t) κ2wL(x, t) κ1w −1X(x, t) κ2Z(x, t) ) = A0 +A1x+A2x 2 +A3x 3, subject to the key properties Construction of a Lax Pair for the E (1) 6 q-Painlevé System 21 (i) the determinant of A(x, t) is κ1κ2(x− a1)(x− a2)(x− a3)(x− a4)(x− a5t)(x− a6t), (ii) A3 is diagonal with entries κ1 and κ2 = qκ1, (iii) A0 has eigenvalues θ1t and θ2t, (iv) the single root of the (1,2) entry of A(x, t) in x is λ, (v) A(λ, t) is lower triangular with diagonal entries κ1µ1 and κ2µ2. Given these requirements, the entries of A(x, t) are specified by L(x, t) = x− λ, Z(x, t) = µ2 + (x− λ) [ δ2 + x2 + x(γ + λ) ] , W (x, t) = µ1 + (x− λ) [ δ1 + x2 + x(−γ − e1 + λ) ] , X(x, t) = [ WZ − (x− a1)(x− a2)(x− a3)(x− a4)(x− a5t)(x− a6t) ] L−1, where (κ1 − κ2)δ1 = λ−1[κ1µ1 + κ2µ2 − θ1t− θ2t]− κ2 [ γ(γ + e1) + 2λ2 − λe1 + e2 ] , (κ1 − κ2)δ2 = −λ−1[κ1µ1 + κ2µ2 − θ1t− θ2t] + κ1 [ γ(γ + e1) + 2λ2 − λe1 + e2 ] , µ1µ2 = (λ− a1)(λ− a2)(λ− a3)(λ− a4)(λ− a5t)(λ− a6t), θ1θ2 = a1a2a3a4a5a6qκ 2 1. Here ej is the jth elementary symmetric function of the indeterminates {a1, a2, a3, a4, a5t, a6t}. Despite the expression for X(x, t), it is a quadratic polynomial in x. In Murata’s notation we have µ̃ = µ1, µ = µ2, δ̃ = δ1 and δ = δ2. The deformation matrix B(x, t) is a rational function in x of the form B(x, t) = x(x1 +B0) (x− a5qt)(x− a6qt) . Next we consider the first transformation of the Sakai linear problem with the following definition: Y (x, t) = S(x, t)−1Y (x, t), and S = ( 1 0 s1 + s2x x ) . The transformed spectral linear problem is Y (qx, t) = A(x, t)Y (x, t), with a transformed spectral matrix A(x, t) = S(qx, t)−1A(x, t)S(x, t). We fix the parameters of the transformation by the requirement that the coefficient of x−1 in the (2,1) entry of S is zero (only the (2,1) entry is non-zero) and also that the coefficient of x0 in the (2,1) entry of S is zero. Thus we find s1 = 1 2qκ1wλ [ (θ2 − θ1)t+ κ1(µ1 − qµ2 + (qδ2 − δ1)λ) ] , 22 N.S. Witte and C.M. Ormerod q(qθ1 − θ2)ws2 = ( 2θ1θ2 κ1 t− qθ1µ2 − θ2µ1 ) 1 λ2 − qκ1e5t−1 1 λ − e1θ2 + (qθ1 − θ2)γ + (qθ1 + θ2)λ. The new spectral matrix can be parameterised by the polynomial A(x, t) = A0 + A1x+ A2x 2 + A3x 3, and possesses the following properties (i) the determinant of A(x, t) is κ21(x− a1)(x− a2)(x− a3)(x− a4)(x− a5t)(x− a6t), (ii) A3 = κ11, (iii) A0 is diagonal with entries θ1t and q−1θ2t, (iv) the roots of the (1,2) entry of A in x are 0 and λ, (v) A(λ, t) is lower triangular with diagonal entries κ1µ1 and κ2µ2. Any such matrix admits the general form A(x, t) = ( θ1t+ κ1x[(x− λ)(x− a) + ν1] qκ1wx(x− λ) κ1w −1x(xc + d ) q−1θ2t+ κ1x[(x− λ)(x− b) + ν2] ) , where the properties given above fix the introduced parameters as (qθ1 − θ2)a = [ θ2ν1 + qθ1ν2 + qκ1e5t −1]λ−1 + qθ1e1 − 2qθ1λ, (qθ1 − θ2)b = − [ θ2ν1 + qθ1ν2 + qκ1e5t −1]λ−1 − θ2e1 + 2θ2λ, qc = ab + 2(a + b)λ+ λ2 − e2 + ν1 + ν2, qd = −(a + b)λ2 − 2abλ− aν2 − bν1 + (qc − ν1 − ν2)λ+ e3 + qθ1 + θ2 qκ1 t. The variables, ν1 and ν2 are defined by ν1 = κ1µ1 − θ1t κ1λ and ν2 = qκ1µ2 − θ2t qκ1λ . The transformed deformation matrix B is computed using B(x, t) = S(x, qt)−1B(x, t)S(x, t), and has the form B = x(xB0 + 1) (x− a5qt)(x− a6qt) . We define a new variable ν using µ2 ≡ (λ− a1)(λ− a2)(λ− a3)(λ− a4) λ− ν̌ , and by implication µ1 ≡ (λ− a5t)(λ− a6t)(λ− ν̌). Construction of a Lax Pair for the E (1) 6 q-Painlevé System 23 Using identical techniques to those employed in the proof of Lemma 3, we can show that the compatibility relation leads to the evolution equations (λ− ν̌)(λ− ν) = (λ− a1)(λ− a2)(λ− a3)(λ− a4) (λ− a5t)(λ− a6t) ,( 1− ν λ̂ )( 1− ν λ ) = a5a6 q (ν − a1)(ν − a2)(ν − a3)(ν − a4) (a5a6tν + θ1/qκ1)(a5a6tν + θ2/qκ1) . To make the full correspondence between our system and this one we must consider a further transformation, given by the linear solution Y(x, t) = [ ϑq ( q−1x )]3 eq,t(x) Y ( x−1, t−1 ) . The prefactors are elliptic functions defined in terms of the q-factorial by ϑq(z) = ( q,−qz,−z−1; q ) ∞, eq,t(z) = ϑq(z)ϑq ( t−1 ) ϑq ( zt−1 ) , with properties ϑq(qz) = qzϑq(z), eq,t(qz) = teq,t(z), eq,qt(z) = zeq,t(z). This is the solution satisfying the linear equations Y ( q−1x, t ) = A(x, t)Y(x, t), Y ( x, q−1t ) = B(x, t)Y(x, t). The transformed spectral matrix A is given by A(x, t) = tx3A ( x−1, t−1 ) = A3 + A2x+ A1x 2 + A0x 3, (4.13) which swaps the roles of the leading matrices around x = 0 and x = ∞. This spectral matrix has the properties (i) the determinant of A(x, t) is κ21(1− a1x)(1− a2x)(1− a3x)(1− a4x)(t− a5x)(t− a6x), (ii) A3 = κ1t1, (iii) A0 is diagonal with entries θ1 and q−1θ2, (iv) the roots of the (1,2) entry of A(x, t) in x are 0 and λ−1, (v) A(λ, t) is lower triangular with diagonal entries κ1µ1tλ −3 and κ1µ2tλ −3. The transformed deformation matrix has the form B(x, t) = x (t− a5qx)(t− a6qx) (x1 + B0). (4.14) Since the compatibility relation between (4.13) and (4.14) is rationally equivalent to that for Y , the evolution equations are the same. It is clear that Y and Yn satisfy equivalent linear problems and that the following correspon- dences hold: q 7→ q−1, t 7→ t−1, λ(t) 7→ 1 g(t−1) , ν(t) 7→ f ( t−1 ) , κ1 7→ b6, ai 7→ bi i = 1, 2, 3, 4, a5 7→ 1 b6 , θ1 7→ −b5b6, q−1θ2 7→ − b6 b5 . 24 N.S. Witte and C.M. Ormerod 4.2 Reconciliation with the Lax pair of Yamada [18] In his derivation of a Lax pair for the E (1) 6 q-Painlevé system Yamada employed the degeneration limits of E (1) 8 q-Painlevé → E (1) 7 q-Painlevé → E (1) 6 q-Painlevé. In doing so he retained eight parameters b1, . . . , b8 constrained by qb1b2b3b4 = b5b6b7b8, and his E (1) 6 q-Painlevé equation was given by the mapping of the variables t 7→ q−1t, f, g 7→ f̄ , ḡ, subject to the coupled first-order system (see his (36)) (fg − 1)(f̄g − 1) ff̄ = q (b1g − 1)(b2g − 1)(b3g − 1)(b4g − 1) b5b6(b7g − t)(b8g − t) , (4.15) (fg − 1)(fg − 1) gg = (b1 − f)(b2 − f)(b3 − f)(b4 − f) (f − b5t)(f − b6t) . (4.16) The Lax pairs constructed by the degeneration limits were given as a coupled second-order q-difference equation in a scalar variable Y (z, t) (see his (37)) (b1q − z)(b2q − z)(b3q − z)(b4q − z)t2 q(qf − z)z4 [ Y (q−1z)− gz t2(gz − q) Y (z) ] + [ q(b1g − 1)(b2g − 1)(b3g − 1)(b4g − 1) g(fg − 1)z2(gz − q) − b5b6(b7g − t)(b8g − t) fgz3 ] Y (z) + (b5t− z)(b6t− z) t2z2(f − z) [ Y (qz)− t2(gz − 1) gz Y (z) ] = 0, (4.17) and a second-order, mixed q-difference equation, gz t2 Y (z) + (q − gz)Y ( q−1z ) − q−2gz(qf − z)Ȳ ( q−1z ) = 0. (4.18) In order to bring (4.15) and (4.16) into correspondence with our form of the E (1) 6 q-Painlevé system (see (1.1) and (1.2)) we will employ the following transformation of Yamada’s variables t 7→ t−1, z 7→ z−1, f, g 7→ g−1, f−1, Ỹ (z) = Y ( z−1 ) , and the specialisations of the parameters b5 7→ b−16 , b6 7→ b6, b7 7→ qb5, b8 7→ b−15 , so that b5b6 = 1 and b7b8 = q. Under these transformations we deduce that (4.15) becomes (1.2) and (4.16) becomes (1.1). Furthermore the pure second-order divided-difference equation (4.17) becomes 4∏ j=1 (1− bjz) t2z(z − g) Ỹ (z) + − 4∏ j=1 (1− bjz) z(z − g)(1− fz) + z 4∏ j=1 (bj − f) f(1− fg)(1− fz) − z(f − b5qt)(b5f − t) b5qt2f − (z − b6qt)(b6z − qt)(q − fz) b6qt2z(z − qg)  Ỹ ( q−1z ) + (z − b6qt)(b6z − qt) b6z(z − qg) Ỹ ( q−2z ) = 0, (4.19) Construction of a Lax Pair for the E (1) 6 q-Painlevé System 25 and the mixed divided-difference equation (4.18) becomes qt2 fz Ỹ ( q−1z; t ) − q (1− fz) fz Ỹ (z; t)− (z − g) fgz2 Ỹ (z; qt) = 0. (4.20) Having put Yamada’s Lax pairs into a suitable form we now seek to make a correspondence with our own theory and results. A single mixed divided-difference equation can be constructed from the matrix Lax pairs ((2.10) and (2.25)). For generic semi-classical systems on a q-lattice grid we can deduce either − 1 W + ∆yV 1 P+ pn(x; qt) + 1 (W + ∆yV )(R+ ∆uS) [ −W+ T+ + R+ P+ ] pn(x; t) + 1 R+ ∆uS 1 T+ pn(qx; t) = 0, or an alternative, − T+(x) (W −∆yV )(x) pn(qx; qt) + 1 (W −∆yV )(x)(R+ ∆uS)(qx) × [ T+(x)R+(qx) + P+(qx)W−(x) ] pn(qx; t)− P+(qx) (R+ ∆uS)(qx) pn(x; t) = 0, (4.21) which we will work with. Using the spectral and deformation data (3.2), (3.7) and the ex- plicit evaluations of the deformation matrix (3.30) and spectral matrix (3.29), we compute the coefficients of the above equation −(R+ ∆uS)(qx)T+(x) = an 1− qb25 b5 x(x− b6t)(x− g), −(W −∆yV )(x)P+(qx) = anγn γ̂n b6 ( 1− qb25 ) b5 t(1− b2x)(1− b3x)(x− b6t) b5qt− f , T+(x)R+(qx) + P+(qx)W−(x) = anγn γ̂n b6 ( 1− qb25 ) b5 (x− b6t)(b6x− t)(1− fx) b5qt− f . Now we set pn = FU where F is a gauge factor and U is the new independent variable, into (4.21) and make a direct comparison with (4.20). Comparing the coefficients of U(x; t) and U(qx; t) in this later equation we deduce that F (qx, t) F (x, t) = 1 t2 (1− b2x)(1− b3x) 1− b6xt−1 . A solution is given by F (x, t) = eq,t−2(x) ( b6xt −1; q ) ∞ (b2x, b3x; q)∞ C(x, t), where C is a q-constant function, C(qx, t) = C(x, t). Now comparing the coefficients of U(qx; qt) and U(qx; t) in the previous equation we find that F (qx, qt) F (qx, t) = γn γ̂n b6(b6x− t) qg(b5qt− f)x2 . Substituting our solution into this equation we find a complete cancellation of all the x dependent factors resulting in a pure q-difference equation in t γ̂nĈ γnC = b6qt g(f − b5qt) . 26 N.S. Witte and C.M. Ormerod Thus we just need a solution C(t) independent of x, however we only require the existence of a non-zero, bounded solution rather than knowledge of a specific solution. In conclusion we find that our new mixed, divided-difference equation is now t2U(x; t)− (1− fx)U(qx; t)− x− g qgx U(qx; qt) = 0, which is clearly proportional to (4.20) with the identification U(x; t) = Ỹ (q−1x; t). A second-order q-difference equation in the spectral variable x for one of the components, say pn, was given in (2.22), and for q-linear grids can be simplified as W + ∆yV T+ (x)pn(qx)− [ W+ T+ (x) + W− T+ ( q−1x )] pn(x) + W −∆yV T+ ( q−1x ) pn ( q−1x ) = 0. (4.22) From the explicit solution of the gauge factor we note F (qx, t) F (x, t) = (1− b2x)(1− b3x) t(t− b6x) , F ( q−1x, t ) F (x, t) = t ( t− b6q−1x )( 1− b2q−1x )( 1− b3q−1x ) . Substituting the change of variables into (4.22) we compute that W + ∆yV T+ (x) F (qx, t) F (x, t) = 1 (q − 1)u1an b6 t 4∏ j=1 (1− bjx) x(x− g) , W −∆yV T+ ( q−1x )F (q−1x, t) F (x, t) = 1 (q − 1)u1an t(x− b6qt)(b6x− qt) x(x− qg) . In addition, using the explicit representations of the diagonal elements of A∗n, i.e., W± (see (3.28), (3.29)) we compute that − 1 b6t [ W+ x(x− g) + W− x(x− g) ∣∣∣∣ q−1x ] = − 4∏ j=1 (1− bjx) x(x− g)(1− fx) + x 4∏ j=1 (f − bj) f(1− fg)(1− fx) − x(f − b5qt)(b5f − t) b5qt2f − (x− b6qt)(b6x− qt)(q − fx) b6qt2x(x− qg) . In summary we find 4∏ j=1 (1− bjx) t2x(x− g) U(qx) + − 4∏ j=1 (1− bjx) x(x− g)(1− fx) + x 4∏ j=1 (f − bj) f(1− fg)(1− fx) − x(f − b5qt)(b5f − t) b5qt2f − (x− b6qt)(b6x− qt)(q − fx) b6qt2x(x− qg) U(x) + (x− b6qt)(b6x− qt) b6x(x− qg) U ( q−1x ) = 0. (4.23) Thus we can see that (4.23) agrees exactly with (4.19) and the identification noted above. Construction of a Lax Pair for the E (1) 6 q-Painlevé System 27 Acknowledgements This research has been supported by the Australian Research Council’s Centre of Excellence for Mathematics and Statistics of Complex Systems. 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Not. 2011 (2011), no. 17, 3823–3838, arXiv:1004.1687. http://dx.doi.org/10.2307/1988577 http://www.jstor.org/stable/20025482 http://dx.doi.org/10.1007/BF00398316 http://arxiv.org/abs/chao-dyn/9507010 http://dx.doi.org/10.1155/IMRN.2005.1439 http://arxiv.org/abs/nlin.SI/0501051 http://dx.doi.org/10.1155/S1073792804140919 http://arxiv.org/abs/nlin.SI/0403036 http://dx.doi.org/10.1007/978-3-642-05014-5 http://aw.twi.tudelft.nl/~koekoek/askey/ http://dx.doi.org/10.1007/BFb0083366 http://dx.doi.org/10.1016/0377-0427(95)00114-X http://arxiv.org/abs/math.CA/9502228 http://dx.doi.org/10.1088/1751-8113/42/11/115201 http://arxiv.org/abs/0810.0058 http://dx.doi.org/10.1016/0375-9601(92)90905-2 http://dx.doi.org/10.1016/S0898-1221(01)00180-8 http://dx.doi.org/10.1619/fesi.48.273 http://dx.doi.org/10.1088/0305-4470/39/39/S13 http://dx.doi.org/10.1007/s002200100446 http://arxiv.org/abs/1204.2328 http://dx.doi.org/10.1093/imrn/rnq232 http://dx.doi.org/10.1093/imrn/rnq232 http://arxiv.org/abs/1004.1687 1 Background and motivation 2 Deformed semi-classical OPS on quadratic lattices 3 Big q-Jacobi OPS 4 Reconciliation with the Lax pairs of Sakai and Yamada 4.1 Sakai Lax Pair 4.2 Reconciliation with the Lax pair of Yamada Ya2011 References

Construction of a Lax Pair for the E₆⁽¹⁾ q-Painlevé System

Institution

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