Laplace transform of tetration

[bo198214]

Thus the Laplace transform of sexp_a is:

Isnt that the Fourier deveopment?

By equating the terms of the resulting Laplace series, we get the equation \( c_k e^{per(a) k} = \sum_{n \in N} \frac{(\log a)^n}{n!} \sum_{\Sigma k_i = k} \[\prod_{i=1}^n c_{k_i}\] \). The inner sum is over all integer sequences of length n which sum to k. The finitude of this sum is ensured by the fact that either all positive or all negative coefficients are zero.

And actually the \( c_k \) are the coefficients of the inverse Schröder powerseries.
Incidentally Dmitrii and I just finished an article about exactly that topic, which I append.