Construction of a Lax Pair for the E 6 (1) q-Painlevé System

We construct a Lax pair for the $ E^{(1)}_6 $ $q$-Painlevé system from first principles by employing the general theory ofsemi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of...

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Bibliographic Details
Published in:Symmetry, integrability and geometry, methods and applications integrability and geometry, methods and applications, 2012-01, Vol.8, p.097
Main Author: Witte, Nicholas S.
Format: Article
Language:English
Subjects:
Askey table
divided-difference operators
isomonodromic deformations
non-uniform lattices
orthogonal polynomials
semi-classical weights
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Summary:We construct a Lax pair for the $ E^{(1)}_6 $ $q$-Painlevé system from first principles by employing the general theory ofsemi-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.
ISSN:1815-0659
1815-0659
DOI:10.3842/SIGMA.2012.097