Wednesday, July 4, 2012

1207.0041 (N. S. Witte et al.)

Construction of a Lax pair for the $ E_6^{(1)}$ $q$-Painlevé System    [PDF]

N. S. Witte, C. M. Ormerod
We construct a Lax pair for the $ E^{(1)}_6 $ $q$-Painlev\'e system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices \cite{Wi_2010a}. 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 $ $q$-Painlev\'e 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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