Hello all! This notation is still a work in progress on my website but I would like it if you could help me compare it to FGH or some other notation. That way I would know if I should edit the function's expansion rate. Thank you!
Article Summary (but still read the passage below): Alpha notation, depending on the power level (eg. number of As) creates towers of Chain Notation in the same exact way that Chain notation creates towers of Arrow notation. Beta notation does the same with Alpha notation and so on through the Greek alphabet.
Compound Recursion and Greek Notation (excerpt from the website at https://sites.google.com/site/jonathansgoogology/theories-and-misc/compound-recursion)
Compounded recursion is one of the most important features of any googology function. Without it, we would still be using arrow notation. For example, we all know that multiplication can be represented in some ways as repeated addition. The same goes for exponentiation which is repeated multiplication. Repeated exponentiation is tetration, repeated tetration is pentation and so on and so forth. However, as much as you compound these functions, they will get nowhere- you would need to create a recursion of the compounding to achieve Graham’s number. So thus Conway Chain notation was invented. In a way, this notation is repeated arrow notation. The chain tells you the amount of arrows for the base and how many arrow towers are needed of this. For example, in Graham’s number, you have an arrow tower of 64 following the base of 3 and 3 separated by 4 up arrows. Just as you can compound the arrow notation to chain notation, you can compound the chain notation into something else. Sure, you can have levels of chain notation as well. Just as you have tetration, pentation, etc. in arrow notation you can have 3 chain arrow functions, 4 chain arrow functions, 5 chain arrow functions, etc. of Conway notation. A new function could tell you how to build powers of Conway chains in a recursion. It is here that the Alpha function comes in. You can have different levels of this: 4A4, 4AA4, 4AAA4, etc. But the beta function tells you how to compound and recurse this function. And on the notations go through the greek letters.