Geodesic
Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlevé System
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: non-uniform lattices; divided-difference operators; orthogonal polynomials; semi-classical weights; isomonodromic deformations; Askey table. 
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     title = {Construction of a {Lax} {Pair} for the $E_6^{(1)}$ $q${-Painlev\'e} {System}},
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}
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Nicholas S. Witte; Christopher M. Ormerod. Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlevé System. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a96/

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