Actually I wasn't thinking about Ramanujan. I was thinking about Euler and the integral representation for the Gamma function. I wanted to exploit integration by parts as beautifully as he did.

Why yes the difference operator is how I found the inverse, but this gives us an integral transform:

Is an expression for one of the inverses. We also have:

Each inverse operator works on different classes of functions. I'm having a little trouble finding the exact restrictions on the functions we can use.

Being more general, we can change the Riemann-liouville differintegral to work on different functions by changing the limits of integration in the integral expression. We can solve the following continuum sum using different limits:

Therefore:

I'm wondering how to apply this to tetration or hyper operators. This performs a fair amount of mathematical work and solves a nice iteration problem--maybe its related to hyper operators *fingers crossed*

Why yes the difference operator is how I found the inverse, but this gives us an integral transform:

Is an expression for one of the inverses. We also have:

Each inverse operator works on different classes of functions. I'm having a little trouble finding the exact restrictions on the functions we can use.

Being more general, we can change the Riemann-liouville differintegral to work on different functions by changing the limits of integration in the integral expression. We can solve the following continuum sum using different limits:

Therefore:

I'm wondering how to apply this to tetration or hyper operators. This performs a fair amount of mathematical work and solves a nice iteration problem--maybe its related to hyper operators *fingers crossed*