Construction of a Lax Pair for the $E_6^{(1)}$ $q$Painlevé System
Abstract
We construct a Lax pair for the $E^{(1)}_6 $ $q$Painlevé system from first principles by employing the general theory of semiclassical orthogonal polynomial systems characterised by divideddifference 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 byproduct of our construction we derive the coupled firstorder $q$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.
 Publication:

arXiv eprints
 Pub Date:
 June 2012
 arXiv:
 arXiv:1207.0041
 Bibcode:
 2012arXiv1207.0041W
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematical Physics;
 39A05;
 42C05;
 34M55;
 34M56;
 33C45;
 37K35
 EPrint:
 SIGMA 8 (2012), 097, 27 pages