Consider the tetration of the function \( \\[15pt]
{e^x} \)
\( ^n(e^x)=(e^x)_1^{(e^x)_2^{(e^x)_2^{_..^{.e^x_n}}}} \)
For a natural number n, the taylor series of that function is
\( ^n(e^x)=\sum_{k=0}^{\infty}\frac{1}{k!}*T(n,k)*x^k \)
where \( \\[15pt]
{T(n,k)} \) is the OEIS A210725; When k<n, \( \\[15pt]
{T(n,k)=T(k,k)} \)
then, for \( \\[15pt]
{n\in\mathbb{N}} \):
\( ^n(x)=\sum_{k=0}^{\infty}\frac{1}{k!}*T(n,k)*(ln(x))^k \)
On the limit for \( \\[15pt]
{n\to\infty} \), 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
\( ^xb=\sum_{k=0}^{\infty}\frac{1}{k!}*T(x,k)*(ln(b))^k \)
The function T(x,k),according to OEIS is
\( T(x,k)=\sum_{j=1}^{x+1}{\binom{x-1}{j-1}*j*T({j-1},{k-1})*T({x-j},{k}) } \)
(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.
{e^x} \)
\( ^n(e^x)=(e^x)_1^{(e^x)_2^{(e^x)_2^{_..^{.e^x_n}}}} \)
For a natural number n, the taylor series of that function is
\( ^n(e^x)=\sum_{k=0}^{\infty}\frac{1}{k!}*T(n,k)*x^k \)
where \( \\[15pt]
{T(n,k)} \) is the OEIS A210725; When k<n, \( \\[15pt]
{T(n,k)=T(k,k)} \)
then, for \( \\[15pt]
{n\in\mathbb{N}} \):
\( ^n(x)=\sum_{k=0}^{\infty}\frac{1}{k!}*T(n,k)*(ln(x))^k \)
On the limit for \( \\[15pt]
{n\to\infty} \), 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
\( ^xb=\sum_{k=0}^{\infty}\frac{1}{k!}*T(x,k)*(ln(b))^k \)
The function T(x,k),according to OEIS is
\( T(x,k)=\sum_{j=1}^{x+1}{\binom{x-1}{j-1}*j*T({j-1},{k-1})*T({x-j},{k}) } \)
(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.
I have the result, but I do not yet know how to get it.
