Borel summation and other continuation/summability methods for continuum sums

[mike3]

There is always base \( \eta \), but of course, it can be solved by other means. The upper superexponential at base \( \eta \) is entire with no singularities, and with super-exponential growth. Perhaps if you can get the Borel summation to work for \( \eta \), you can figure out how to make it work for other bases.

And it's such superexponential growth that would be why it wouldn't work, at least without the Borel, etc. methods. And if we apply them, the presence of singularities should not be a problem, and so we should be able to solve for the lower superexponential as well. I'm not sure how having the upper would be helpful anyway, especially considering the failure of the "base change method" to produce an analytic function.