Mittag-Leffler series for generating continuum sum?

[mike3]

But then my question would be whether the value depends on how the terms are rearranged (I mean not only giving different branches, but a continuum of solutions).

I.e. given two transseries representations for an analytic function, if both converge and so do their Faulhaber continuum sums, do those sums agree whenever both converge? It would be interesting to determine this, but I'm not sure what the proof would be like.

Another thing I've wondered about is, could there be some way to assign some sort of value to the divergent sums of Faulhaber coefficients given an arbitrary power series of nonzero convergence radius as input?

Failing the Faulhaber method, could there be some other way to define continuum sum that agrees with it but covers a lot more ground?