The Abel equation for the slog was \( \text{slog}_b(b^x)=\text{slog}_b(x)+1 \).
Our original recurring for the super exponential (which I will call similarly) sexp is \( \text{sexp}_b(x+1)=b^{\text{sexp}_b(x)} \).
So if we develop it at 0, say \( \text{sexp}_b(x)=\sum_{n=0}^\infty \rho_n x^n \) satisfying the inverted Abel equation:
\( \sum_{n=0}^\infty \rho_n (x+1)^n = \exp\left(\log(b)\sum_{n=0}^\infty \rho_n x^n\right) \)
we get
\( \sum_{n=0}^\infty \rho_n \sum_{k=0}^n \left(n\\k\right) x^k = 1 + \sum_{n=0}^\infty B(\log(b)\rho_1,\dots,\log(b)\rho_n) \frac{x^n}{n!} \)
where \( B(x_1,\dots,x_n) \) is the complete Bell polynomial.
The left side develops to \( \sum_{k=0}^\infty x^k \sum_{n=k}^\infty \rho_n \left(n\\k\right) \) and so we have the infinite equation system
\( \sum_{i=k}^\infty \rho_i \left(i\\k\right) = B(\log(b)\rho_1,\dots,\log(b)\rho_k) \frac{1}{k!} \)
And I wonder if we solve it the natural way whether we get exactly the inverse of the slog (which I assume). Unfortunately there is no complete Bell polynomial in Maple (at least I didnt find it) and I am too lazy in the moment to program it myself
And yes, it is not a linear equation system. Perhaps it is despite solvable, who knows ...
Our original recurring for the super exponential (which I will call similarly) sexp is \( \text{sexp}_b(x+1)=b^{\text{sexp}_b(x)} \).
So if we develop it at 0, say \( \text{sexp}_b(x)=\sum_{n=0}^\infty \rho_n x^n \) satisfying the inverted Abel equation:
\( \sum_{n=0}^\infty \rho_n (x+1)^n = \exp\left(\log(b)\sum_{n=0}^\infty \rho_n x^n\right) \)
we get
\( \sum_{n=0}^\infty \rho_n \sum_{k=0}^n \left(n\\k\right) x^k = 1 + \sum_{n=0}^\infty B(\log(b)\rho_1,\dots,\log(b)\rho_n) \frac{x^n}{n!} \)
where \( B(x_1,\dots,x_n) \) is the complete Bell polynomial.
The left side develops to \( \sum_{k=0}^\infty x^k \sum_{n=k}^\infty \rho_n \left(n\\k\right) \) and so we have the infinite equation system
\( \sum_{i=k}^\infty \rho_i \left(i\\k\right) = B(\log(b)\rho_1,\dots,\log(b)\rho_k) \frac{1}{k!} \)
And I wonder if we solve it the natural way whether we get exactly the inverse of the slog (which I assume). Unfortunately there is no complete Bell polynomial in Maple (at least I didnt find it) and I am too lazy in the moment to program it myself
And yes, it is not a linear equation system. Perhaps it is despite solvable, who knows ...
