(09/23/2009, 08:25 AM)Ansus Wrote: I already suggested using Euler-Maclaurin formula here: http://math.eretrandre.org/tetrationforu...79#pid3979
but Mike3 already showed that Faulhaber's formula can be summed up using special technique so I thought the issue is over.
Not sure if the E-M and Faulhaber converge the same way, but E-M has smaller coefficients than Bernoulli(?), so perhaps even if it doesn't converge directly maybe a laxer divergent sum could be applied.
The problem is that it is fussy to choose the right parameters with divergent sum, and it gets nasty when you go to complex bases or \( 0 < b < 1 \), which have negative \( \log(b) \) and so complex-valued \( \log(b)^x \) at real x. In that range I'm concentrating on b = 0.04 (repelling fixed point, attracting 2-cycle, for the integer towers).
The question then would be, how bad do the coefficients diverge? Does anyone have anything that could compute a possible power series expansion at 0 for tetration to, say base e out to a whole lot of terms, say to 64 or 128 terms, to get a "feel" for how they go? I've been toying around with the Cauchy integral but haven't yet gotten it to go (could post a thread on that). Then one can examine the rate of divergence of the terms in the continuum-sum-coefficient sums. Perhaps if they do not diverge too severely, one could use a more lightweight divergent sum method, perhaps that doesn't involve as many fussy parameters, and so extension to more bases would be possible.
Also, E-M has \( \frac{B_{2k}}{(2k)!} \) which looks pretty good in terms of the behavior as k goes to infinity, but the way the derivatives of tetration behave is the big problem. But perhaps, even if we also need a special divergent sum here too, maybe it does not need to be so heavyweight. Hence why I'd like to see those coefficients.