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arXiv · 1207.0041

Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlevé System

Nicholas S. Witte·Christopher M. Ormerod
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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.

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Nicholas S. Witte, Christopher M. Ormerod. 2012-12-11. Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlevé System. https://doi.org/10.3842/sigma.2012.097

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