Intrinsic characterization of the evaluation associated with the analytic integral

Characterize the evaluation operator associated with the analytic integral operator expcum on the ring of virtual species by a natural construction analogous to initialization at a quasi-zero species.

Background

For the analytic integral operator expcum determined by the canonical analytic exponential e^X, the paper gives the explicit evaluation formula $\E(\Phi)=\sum_{n\geq0}(-X)^n\Phi^{(n)}/n!$. The authors show through examples that this evaluation is not simply evaluation at the zero species, unlike the corresponding evaluation for linear species.

The paper notes that the analytic evaluation is nevertheless close to zero-species initialization: their difference is a differential constant. The unresolved problem is to find a more natural or intrinsic interpretation, specifically one resembling initialization at a suitable quasi-zero species.