On the Riemann-Hilbert Problem for a q-Difference Painlevé Equation

On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem

“…This is the Jimbo-Sakai linear problem [21], which we have scaled to remove redundant parameters (see Section 3.1 for details). Throughout this paper we assume that the parameters κ and t 0 satisfy the non-resonance conditions (1.4), which ensure that the linear problem is fully non-resonant (see [22][Definition 1.1]).…”

“…Our second main result is the identification of the monodromy manifold with an explicit surface. To state this result, we define a set of coordinates on the monodromy manifold M(κ, t 0 ), using a construction introduced in our previous paper [22].…”

“…This mapping is an embedding, by the same argument as given in the proof of Lemma 5. which is commutative, because right multiplication by a diagonal matrix translates to scalar multiplication of ρ as shown in equation (2.18). We first determine the image of M (κ, t 0 ) under P , following the technique developed in our previous paper [22], and then obtain Theorem 2.15 by projecting this image into (P 1 ) 4 /C * via ι P .…”

“…By means of a lengthy calculation, one can check that both determinants, coincide, up to some nonzero scalar multipliers, with the equation T hom (ρ x 1 , ρ y 1 , ρ x 2 , ρ y 2 , ρ x 3 , ρ y 3 , ρ x 4 , ρ y 4 ) = 0, where T hom is defined in Definition 2.14. We refer the interested reader to our previous work [22][Appendix B] where an analogous computation is given.…”

On a class of q -orthogonal polynomials and the q -Riemann–Hilbert problem

“…More recently, the application of Galois theory has been used to derive an analytic approach to singular regular linear q-difference systems [32]. Isomonodromy conditions have led to RHPs corresponding to q-difference Painlevé equations q-P IV [33] and q-P VI [34].…”