12/17/2020, 03:48 PM

Consider the tetration of the function

For a natural number n, the taylor series of that function is

where is the OEIS A210725; When k<n,

then, for :

On the limit for , the T(n,k)=T(k,k) are the coefficients of the Lambert w function.

Now, I switch the variables names, because we are interested on constant base, and variable exponent. x=b, and n=x

The function T(x,k),according to OEIS is

(Note that T(x,k)=1 if x=0 or k=0)

So, the question is, there is an explicit, non recursive, formula for T(x,k)?

We can easily replace the binomial with the Gamma function, and the recursive formula probably has a fractal structure. A fractal structure means that it can be extended to non integer values of x by using the self similarity.

For a natural number n, the taylor series of that function is

where is the OEIS A210725; When k<n,

then, for :

On the limit for , the T(n,k)=T(k,k) are the coefficients of the Lambert w function.

Now, I switch the variables names, because we are interested on constant base, and variable exponent. x=b, and n=x

The function T(x,k),according to OEIS is

(Note that T(x,k)=1 if x=0 or k=0)

So, the question is, there is an explicit, non recursive, formula for T(x,k)?

We can easily replace the binomial with the Gamma function, and the recursive formula probably has a fractal structure. A fractal structure means that it can be extended to non integer values of x by using the self similarity.