User blog:MilkyWay90/Generalized Factorial Function

My Generalized Factorial Function!

Here is the definition:

Take any two-parameter function F.

F(n, !) = F(n, F(n - 1, !))

F(1, !) = 1

For example

4 + ! = 4 + (3 + !) = 4 + 6 = 10

3 + ! = 3 + (2 + !) = 3 + 3 = 6

2 + ! = 2 + (1 + !) = 2 + 1 = 3

1 + ! = 1

Therefore, 4 + ! = 10

Another example (Ackermann Function):

A(3, !) = A(3, A(2, !)) = A(3, 5) = 253

A(2, !) = A(2, A(1, !)) = A(2, 1) = 5

A(1, !) = 1

Therefore, A(3, !) = 253

And you could even do it for the original factorial (multiplication):

4 * ! (or 4!) = 4 * (3 * !) = 4 * 6 = 24

3! = 3(2!) = 3 * 2 = 6

2! = 2(1!) = 2 * 1 = 2

1! = 1

My questions are:

How can I extend it to more arguments/parameters?

What is the growth rate of this function?

Has this been discovered before?

Is this even a valid function (Because in a function F(x, !), ! isn't a fixed number, but is based on what you put in for x)?

Feel free to leave any comments and answers.

Function made by MilkyWay90