Computations with the double-exp series

[mike3]

I was thinking of trying to make a code to compute with the double-exp series mentioned here:

http://math.eretrandre.org/tetrationforu...78#pid4678

wondering if it may be better than the current approach, but I have some problems. To do the Ansus formula iteration, we need several things, done in this order:

1. continuum summation from to

2. exponential function to base

3. multiply by

4. integration from -1 to

5. normalize (divide by the value at 0)

The first, of course, is easy. However it produces a non-exponential linear term. This is fine, we can save it and work it implicitly (same as I do with the periodic approximation in single-exp series). The next one is another challenge. We have linear + another double-exp series. Applying the exponential function should yield exponential * base-b exponential of the double-exp series. This latter should have the same "periodicity" characteristics, but how can it be computed efficiently and to good precision? We could then bring the exponential into the exp-series (just add to the exponents of the parts of the terms.).

Then we multiply by . We could add this into the exponents of the exp-series. Finally, we can do the integral. This requires mixing the multiplying exponential into the exponentials of the series and then integrating. But also leaves a nonexponential linear term, too. Thus the result here is no longer a series of the form given! We've added to the exponents and even worse, have a non-exponential linear term! What can be done about this?