Explicit higher-order terms in the q→0 Riemann–Hilbert expansion for q-Painlevé VI

Derive explicit formulas for the higher-order coefficients R_k(f,g;t), for k ≥ 1, in the asymptotic power series expansion around q = 0 of the Riemann–Hilbert correspondence associated with the q-difference sixth Painlevé equation, namely RH_t(f,g) = RH_t^diamond(f,g) + sum_{k=1}^∞ q^k R_k(f,g;t), where RH_t^diamond: 𝔛_t → 𝔽_t^diamond is the crystal-limit isomorphism and (f,g) ranges over the open surface 𝔛_t. Provide expressions for R_k(f,g;t) in terms of the parameters κ = (κ_0, κ_t, κ_1, κ_∞), the independent variable t, and the geometric data of 𝔛_t and 𝔽_t^diamond.

Background

The paper proves that, in the crystal limit q → 0, the Riemann–Hilbert correspondence for the q-difference sixth Painlevé equation becomes an explicit bi-rational isomorphism RH_t^diamond between a subspace of the initial value surface and an affine Segre surface, and it provides closed-form formulas for RH_t^diamond.

Beyond this leading-order description, the authors establish analyticity of the full Riemann–Hilbert mapping in q near 0 and show it admits a convergent power series expansion RH_t(f,g) = RH_t^diamond(f,g) + sum_{k=1}^∞ q^k R_k(f,g;t). However, the coefficients R_k(f,g;t) for k ≥ 1 are not determined. Computing these terms would refine the crystal-limit analysis and connect to potential conformal field theory methods referenced in the paper.