11/14/2009, 10:05 PM
(11/14/2009, 08:24 PM)mike3 Wrote: 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.
(11/14/2009, 09:18 PM)mike3 Wrote: 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 \( f(z) = \sum_{n=0}^{\infty} a_n z^n \) at z = 0,
\( f(z) = \int_{0}^{\infty} \exp(-\exp(t)) \sum_{n=0}^{\infty} a_n \frac{(tz)^n}{\mu(n)} dt \)
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).
