Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 097, 27 pages      arXiv:1207.0041      https://doi.org/10.3842/SIGMA.2012.097

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

Nicholas S. Witte a and Christopher M. Ormerod b
a) Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia
b) Department of Mathematics and Statistics, La Trobe University, Bundoora VIC 3086, Australia

Received September 05, 2012, in final form November 29, 2012; Published online December 11, 2012

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.

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