Fractional Integration

[tommy1729]

Also, I think the Master Theorem idea relates to some question I asked on MO a couple months ago-- you might find some of the answers others provided interesting: https://mathoverflow.net/questions/43549...nd-sum-n-1 

Yes, just reading this question I agree with Tom Copeland. He has answered questions of mine before on MO (I've had like 5 throwaways on MO for asking questions like yours ). I've interacted with him a good amount of times related to Integral Representations, and he's a good person to talk to about anything to do with integral representations. Ramanujan's work is actually a Fourier Transform result--which is at the root of all integral representations. Listen to Tom Copeland...

Im more into Laplace transforms.

I mentioned a few integral transforms at the fake function thread : 

https://math.eretrandre.org/tetrationfor...63&page=21

and other pages there.

Ofcourse we all did, but I wanted to point that out.

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I think in general for meromorphic (on C ) functions Ramanujan Master theorem gives you the right values.

regards

tommy1729

Analytic continuation and symmetry are essential see this related issue :

https://math.stackexchange.com/questions...0-ns-n-s-1

And likewise for its sum function.

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I would like to point out that all this probably relates to continuum sums and continuum products.

the continuum product is problematic around zero's of a function for a logical reason.
By taking log, this implies continu sums are problematic around log sing.

So I assume if a counterexample exists to Caleb question it relates to log singularities.

Natural boundaries are possible issues too because it means it is not defined to sum everywhere ( as the function is not everywhere analytic defined , even when defined there ) 

I think essential singularities are issues too.

Basically I am saying that only the non-analytic points matter, which makes sense since the techniques work for polynomials aka truncated taylor series.

Going around poles is easy in the riemann surface/analytic continuation sense, afterall there is only one path ( not multivalued ), hence this also shows why it works.


regards

tommy1729