Mittag-Leffler series for generating continuum sum?

[bo198214]

The idea is that maybe if Faulhaber's formula does not yield a convergent formula when applied directly to a Taylor series with finite convergence radius, perhaps it would if we could apply it to a Mittag-Leffler series or some other extension of the Taylor series to a cut plane.

ah, ok, understand.

I also stumbled upon this very interesting paper:

http://arxiv.org/pdf/hep-th/9206074

It mentions methods that sum power series in the Mittag-Leffler star. One formula it gives, is this: given a principal branch of an analytic function, represented by its power series at z = 0,

I am skeptical about those Borel-summation. Usually it requires the summable function to be of at most exponential type (or perhaps even fixed nested exponential type).