Rigorous proof of the identity for the dominant Stokes multiplier K_0(q)

Prove rigorously that the dominant Stokes multiplier K_0(q) associated with the solution w(z)∼z as |z|→0 of the q-difference first Painlevé equation w(qz) w^2(z) w(z/q)=w(z)−z satisfies K_0(q)=(q;q)_∞^{−2}, where K_0(q) is defined as the residue at t=−1 of the q-Borel transform of the formal series solution and (q;q)_∞=∏_{n=0}^∞ (1−q^n).

Background

In Sections 6–8, the authors analyze the Stokes multipliers K_j(q) arising as residues of the poles of the q-Borel transform of the formal solution to the q-Painlevé I equation. Numerical evidence and coefficient analysis suggest that the dominant multiplier K_0(q) equals (q;q)_∞^{−2}, and further relations express K_j(q) in terms of K_0(q).

Appendix A provides rigorous steps showing the asymptotic role of K_0(q) in a bounded domain for q, and Appendix B establishes an analogous identity for a simpler q-difference equation via a Riccati-type ansatz, which supports but does not prove the desired identity for the original q-Painlevé I equation. The authors explicitly call for a rigorous proof of K_0(q)=(q;q)_∞^{−2).