Does the Mellin transform have a half-iterate ?

[tommy1729]

I notice that Im not so comfortable with many iteration of the Mellin transform.
Convergeance issues seem to pop up out of nowhere.

For instance if we start with f(x) = exp(x) or f(x) = exp(-x) we get in trouble before we reach the 3rd iteration of the mellin transform.

And if we pick an elementary function between exp(x) and exp(-x), I seem to have trouble finding a closed form for every n th mellin transform.

Maybe I need to consider using hypergeometrics ?

What are standard tables of n th mellin transforms (for s,x > 0) , if that has even been made yet ?

regards

tommy1729