Convergence classification of the Stieltjes bootstrap

Determine whether the observed relationship between the singularities of homogeneous solutions of the differential equations and the convergence type of the Stieltjes bootstrap holds in general, including exponential, algebraic, or absent convergence.

Background

The Stieltjes bootstrap bounds master integrals by imposing Hankel-matrix positivity constraints on derivatives generated by differential equations. In the examples studied, the authors observe different convergence behaviors depending on whether homogeneous solutions are Stieltjes functions or possess singularities on or away from the Stieltjes branch cut. They explicitly leave unresolved whether this observed pattern is universal.