Proof of the recursive formula for characteristic constants

Prove the conjectured recursion relation for the characteristic constants of the flat family of degree-one systems A^{(n)}(z,t)=A^{(n)}_0(t)+zA^{(n)}_1(t), and establish that the characteristic constants of a generic monodromy matrix in F_{\alpha,\beta}(x) are obtained by this recursion.

Background

The paper identifies the characteristic constants p_{ij} appearing in the entries of the Birkhoff connection matrix as constrained by the requirement that its determinant be constant. It proposes to determine these constants recursively by studying a degeneration of an n-dimensional family to an (n−1)-dimensional system at t→0.

The proposed recursion expresses p{(n)}_{ij} in terms of the lower-rank constants p{(n−1)}_{kj}, spectral data, entries of the limiting matrix, q-Pochhammer factors, and q-theta factors; the last column is given explicitly by a separate product formula. The authors state that proving the conjecture requires uniform q-Borel–Laplace analysis of the degeneration, where divergent series and q-Stokes phenomena arise.

References

Then by analyzing the q-Stokes phenomonon and matching it with the poles of the solutions, it is conjectured the following, whose main difficulty in its proof lies in a uniform q-Borel-Laplace analysis of the degeneration process of Q_\Omega(t)Y_\infty(z,t) as t\to 0, where the series itself becomes divergent and the q-Stokes phenomonon arises.

— Isomonodromic Deformations for Linear $q$-Difference Systems of Degree One  (2609.02952 - Ma, 2 Sep 2026) in Section 6, “Monodromy Problem”