Convergence of the Full Family of Conjectured Trajectories

Establish that for every rank r3 and every trajectory in the set of trajectories whose minimum lower-parameter value occurs with multiplicity at least two, the dual convergence theorem holds and the resulting dual CMF limit equals Euler’s constant .

Background

The paper proves dual convergence only for the rank-three subfamily , where the three lower parameters are equal to one and the upper parameters are nonpositive integers. It proposes the larger family for arbitrary rank 3, based on numerical evidence, but leaves the required convergence and consequent evaluation at unresolved.

References

For any r \geq 3 and for any \mathbf{t}\in \mathcal{T}_r (\cref{def:conjectured-trajectories}), \cref{thm:dual-convergence} and consequently \cref{thm:main} hold.

— An Infinite Family of Apéry-like Recurrences for Euler's Constant  (2610.13919 - Kalisch et al., 9 Oct 2026) in Conjecture 1, “Conjectured Apéry-like Approximations”; Definition 4, “Conjectured Trajectories”

A question that arises naturally is: can alternative parameterizations of the G-function also be used to construct holonomic sequences in a similar way? Moreover, could D-finite functions other than the Generalized Hypergeometric Function and the Meijer G-function be used for this purpose?

— An Infinite Family of Apéry-like Recurrences for Euler's Constant  (2610.13919 - Kalisch et al., 9 Oct 2026) in Section 9, “Future Work,” item 2