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

Abstract

We construct a Lax pair for the $E_6^{(1)}$ $q$-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 $E_6^{(1)}$ $q$-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.

References in corpus (4)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Lax forms of the $q$-Painlevé equations
• Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E^{(1)}_n
• Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy

Cited by in corpus (7)

• Geometric Aspects of Painlevé Equations
• Commutation Relations and Discrete Garnier Systems
• Lax pairs of discrete Painlevé equations: $(A_2+A_1{)}^{(1)}$ case
• Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation
• The Padé interpolation method applied to $q$-Painlevé equations
• Lattice equations arising from discrete Painlevé systems. II. $A_4^{(1)}$ case
• A $3\times 3$ Lax Form for the $q$-Painlevé Equation of Type $E_6$