Well I'm not certain, but your operators seem to be like a modified lower hyperoperators. These operators satisfy the recursion:

where

Check if this function works. I'm pulling it out of a hat but I have a lot of math behind it.:

where

This may or may not work. Depending on if 1/x[s]y is holomorphic or meromorphic and if (x[s]y)[s-1]x is holo as well with decent enough behaviour.

where

Check if this function works. I'm pulling it out of a hat but I have a lot of math behind it.:

where

This may or may not work. Depending on if 1/x[s]y is holomorphic or meromorphic and if (x[s]y)[s-1]x is holo as well with decent enough behaviour.