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 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.