Mittag-Leffler series for generating continuum sum?

[bo198214]

http://eom.springer.de/s/s087230.htm

Apparently, it seems there is a formula that can extend a Taylor series to a whole cut complex plane, called a Mittag-Leffler star of the function.

Without doubt this is a very interesting theorem.

However I didnt yet understand how you want to apply this to Faulhaber's formula.
I mean the resulting series has 0 convergence radius, but for the Mittag-Leffler star you need a powerseries with non-zero convergence radius, at least at one point, i.e. the center of the star.