Is Weniger's Transformation Capable to Simulate Stieltjes Function Branch Cut?

The resummation of Stieltjes series remains a key challenge in mathematical physics, especially when Pad\'e approximants fail, as in the case of superfactorially divergent series. Weniger’s $\delta$-transformation, which incorporates a priori structural information on Stieltjes series, namely the inverse factorial series representation of their converging factors, offers a superior framework with respect to Pad\'e. Here, the problem of the pole distribution of the $\delta$-transformation is addressed. We show that the algebraic structure of the transformation, together with the intrinsic log-concavity of Stieltjes moments, satisfy the necessary conditions for having real poles. Moreover, by recasting the denominator of the $\delta$-transformation rational approximant as a high-order derivative of a log-concave polynomial and invoking the Gauss-Lucas theorem, a possible geometrical justification of the pole positioning along the negative real axis is proposed. While a fully rigorous proof remains an open challenge, our conjecture is substantiated by a comprehensive numerical investigation across an extensive catalog of Stieltjes series. In particular, our results provide systematic evidence that the mandatory branch cut conditions are respected even in the more delicate case of superfactorial growth, recently addressed from a converging factor perspective.