Alternative Joyal integrals and differential Rota–Baxter structures

Determine whether any Joyal integral operator associated with a differential tower other than the analytic integral operator expcum can endow the ring of virtual species vir with a differential Rota–Baxter structure.

Background

The paper proves that the Joyal integral operator associated with a differential tower yields an integro-differential structure on the ring of virtual species only when the differential tower is the canonical analytic exponential e^X. Consequently, every other Joyal integral fails to produce an integro-differential ring. The authors then ask whether the weaker differential Rota–Baxter axioms might nevertheless hold for some of these alternative differential towers.

Resolving this question would determine whether differential Rota–Baxter structures exist beyond the unique integro-differential structure established in Theorem thm:intd, and would clarify the range of algebraic structures supported by noncanonical Joyal integrals.