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,

with

.

But it the formulas don't seem to work when tested numerically. Try it with the reciprocal series for , which they mention in the paper, i.e. . Then try evaluating using these formulas at , which outside the convergence radius for the series, but inside the Mittag-Leffler star. It seems to give huge values. Of course (and I highly suspect this is the case), I've missed something here... what might it be?

ADDENDUM: I see now, this does work... it's just that the terms "hump up" for reasonably large z-values before they get smaller and the thing converges... it needed 768 terms to converge to a few places for z = -2, but got ~0.3333 like I'd expect for . I think it gets better after that point since once you're over the "hump" the terms get small fairly quick, I suppose 1024 terms would get much more accuracy but calculating takes a long time.

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,

with

.

But it the formulas don't seem to work when tested numerically. Try it with the reciprocal series for , which they mention in the paper, i.e. . Then try evaluating using these formulas at , which outside the convergence radius for the series, but inside the Mittag-Leffler star. It seems to give huge values. Of course (and I highly suspect this is the case), I've missed something here... what might it be?

ADDENDUM: I see now, this does work... it's just that the terms "hump up" for reasonably large z-values before they get smaller and the thing converges... it needed 768 terms to converge to a few places for z = -2, but got ~0.3333 like I'd expect for . I think it gets better after that point since once you're over the "hump" the terms get small fairly quick, I suppose 1024 terms would get much more accuracy but calculating takes a long time.