Mittag-Leffler series for generating continuum sum?

[mike3]

Hi.

I saw this:

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. This is interesting: because then perhaps maybe we could apply Faulhaber's formula to this to yield a continuum sum that works even for functions which are not entire: it appears the reason the original Faulhaber's formula (Faulhaber-on-Taylor formula) was not working is because continuum sum is a "global" operation (unlike derivative), as I mentioned in a recent post to the thread "Continuum sum formula rescued?", that is, the operation not only depends on the behavior of the function being put into it near the sum indices, but also very far away, and since a Taylor series of finite convergence radius only looks locally like the function they expand, we can explain the failure of the Faulhaber formula when applied to Taylor expansions of non-entire functions. Yet if the Mittag-Leffler series, on the other hand, converges over a whole cut plane, then this problem should not arise, as the global behavior will be correct.

If this could be done, it might then enable the usage of Ansus' sum formula for tetration to extend tetration to any complex base and height.

But the problem is I can't test it, since the description on the page is incomplete: what does the notation \( c_{\nu}^{(n)} \) mean? If \( c_{\nu} \) is the \( \nu \)-th Taylor coefficient, then what's \( c_{\nu}^{(n)} \)? What are the magic numbers \( k_n \)? I can't access the references, because I don't have access to a university library.

[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.

[mike3]

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. The reasoning being that the "reason it does not converge" (if you go and read the thread "Continuum sum formula rescued?", I mention this there) may be that the Taylor series doesn't look "globally" like the function it represents due to the limited convergence (and the partial sums of it don't approach, on a "global scale", the function), while th Mittag-Leffler extension would, and thus maybe the Faulhaber formula will succeed there, when it failed on the Taylor series. But I can't give it a shot, without being able to compute that formula. My first test would be to try finding the Mittag-Leffler expansion for log(1 + z) (Taylor series at 0 is the Mercator series), and then apply Faulhaber's formula and see if we get convergence to the log-factorial.

[mike3]

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,

\( f(z) = \int_{0}^{\infty} \exp(-\exp(t)) \sum_{n=0}^{\infty} a_n \frac{(tz)^n}{\mu(n)} dt \)

with

\( \mu(n) = \int_{0}^{\infty} \exp(-\exp(t)) t^n dt \).

But it the formulas don't seem to work when tested numerically. Try it with the reciprocal series for \( f(z) = \frac{1}{1 - z} \), which they mention in the paper, i.e. \( f(z) = \sum_{n=0}^{\infty} z^n \). Then try evaluating using these formulas at \( z = -2 \), 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 \( \frac{1}{1 - -2} = \frac{1}{3} \). 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 \( \mu(n) \) takes a long time.

[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 \( f(z) = \sum_{n=0}^{\infty} a_n z^n \) 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).

[mike3]

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).

Perhaps, but could it be used to sum the diverging coefficients of the Faulhaber applied to the Taylor series, though (which would have a slower growth), instead of the series itself (which describes the fast growing tetrational)?

Yet I still am wondering why it seems so tough to get it going for their example function.

Though regardless, I'm still curious about that Mittag-Leffler expansion thing. Anything on how to determine those magic constants? Do you have access to the references mentioned?

[mike3]

So where could I find more about this? As I can't access the references, so I can't find out how to get those numbers...

[bo198214]

So where could I find more about this? As I can't access the references, so I can't find out how to get those numbers...

Which reference do you need?
I found the 3 volumes of
A.I. Markushevich, "Theory of functions of a complex variable" , 2
as djvu in the internet however from 1965 or so and not the edition from 1977. It contains the Mittag-Leffler theorems but I can not find the their star expansion, I guess its only contained in the 1977 version.

The other references are in French and I am not so fluent in French, so which would you need?

I anyway think that the coefficients are not just simple fractions or so (otherwise they would be given in the springer online reference), perhaps they will depend on the singularities.

[bo198214]

I found the 3 volumes of
A.I. Markushevich, "Theory of functions of a complex variable" , 2
as djvu in the internet however from 1965 or so and not the edition from 1977. It contains the Mittag-Leffler theorems but I can not find the their star expansion, I guess its only contained in the 1977 version.

Oh now I found the description. It is in volume 3 (pp 273, theorem 8.7) and not in volume 2 like the citation suggests. But - bad news - in the 1965 edition the coefficients are not computed explicitly.

[mike3]

I found the 3 volumes of
A.I. Markushevich, "Theory of functions of a complex variable" , 2
as djvu in the internet however from 1965 or so and not the edition from 1977. It contains the Mittag-Leffler theorems but I can not find the their star expansion, I guess its only contained in the 1977 version.

Oh now I found the description. It is in volume 3 (pp 273, theorem 8.7) and not in volume 2 like the citation suggests. But - bad news - in the 1965 edition the coefficients are not computed explicitly.

Does it refer to some other book/paper/etc. within it when it discusses the formula and its coefficients? Could you chase after that?

What about the 1977 one?