Could be tetration if this integral converges

[JmsNxn]

Amazing. I only saw the first bit of the paper, I'm really excited right now because it seems there's a simple integral-transform representation for the "Weyl differintegral". This is really really really really really interesting to me, because of ... dum dum dum ... the continuum sum!

In particular, I am really interested in your Definition 3. How does that expression come from the original Weyl differintegral which was defined for Fourier series? I.e. how do you generalize the Fourier series definition to that definition?

Well Weyl defined the operator on periodic functions slightly differently but it's equivalent. I did not come up with definition 3. It's been long standing. It's also called the exponential differintegral, The Riemann liouville operator with lower limit negative infinity. Etc etc..

I am lacking proofs on convergence for the mellin transform of lots of interesting forms of functions. However, I have made progress on continuum sums. That is actually my next paper. Essentially I've shown that if is holomorphic in the strip and:

for

i can generate an indefinite sum of on that same strip> Im currently trying to show i can take as many indefinite sums as i want.

I can't explain now but its all involving the weyl differintegral.