MPHLEE You're taking the words out of my mouth. I use the super function operator which makes the whole thing come together. The trick is constructing an iterate of this operator and then interpolating it in the manner I posted. This requires generalizing Koenigs linearization theorem considerably. It does come out to this series and checking the convergence is a lot easier, which upon the recursive identity DOES come out using Ramanujan's master theorem, but the trick is hidden. The goal is to do it in a much more general setting. I am working very hard on cleaning it up and making sure all the proofs pop out like clock work from the more general schema. There are tons of operators that can be iterated through my methods, which is quite beguiling, but each requires a special treatment on their own. The key word, contraction operators will appear as objects that are easy to iterate. Just having a knowledge of how these operators iterate proves to be very valuable towards the super function operator. I'll get it out soon. Everything you're saying appears but there is much trickery to the solution. I'm just trying to rigorize everything now. The skeleton is there, I just need to prop it up with some muscular rigor.
\( \mathcal{C}_\phi \) is a contraction operator, this proves invaluable to the generalization. But we must be guaranteed a form of koenigs linearization for the interpolation method to work, and then we can scrap the construction and evaluation of the quasi schroder functions (built for arbitrary operators) and just use the fractional calculus interpolation. The recursion is the simple part, its proving there exists an exponentially bounded function, "factorable," that interpolates hyper operators. That's the difficult part.
