One very important formula

[Ansus]

So how do you extend it to real values of x, and is this solution real valued for bases \( b > e^{1/e} \)?

It is derived from regular iteration, so it diverges for higher bases, but my aim was to find an expression for tetration that does not refer to taetration itself, thus allowing to derive its properties.

David Knuth referred to the following operation calling it 'binomial convolution':

\( f(n)\star g(n)=\sum_{k=0}^n \left(n \\ k\right)f(n-k)g(k) \)

If we use such operator, we can write:

\( B_{n+1}^x=\sum_{k=0}^{x-1} B_n^x\star B_n^k \)

And \( B_1^x \) is always 1.

Thus the result of the convolution is a polynomial of x and k of degree n-1 and we can take indefinite sum of it symbolically.

Note also that binomial convolution corresponds to the product of exponential generating functions. This means product of tetrations corresponds to binomial convolution of Bell's numbers of higher orders.