closed form for regular superfunction expressed as a periodic function

[sheldonison]

Mike,

Thanks a lot for your links and your post! I'll have to program it into pari-gp, and verify that it matches the discreet Fourier series terms. Next goal; rewrite the equations for other bases (except \( \eta \)), especially for bases<sqrt(2).
- Sheldon

....
Doing some tests, it appears that

\( B_n(1, 2! a_2, ..., n! a_n) \)

has only one occurrence of \( n! a_n \), and no higher powers of it....
\( n! (L^{n-1} - 1) a_n = B_n(1, 2! a_2, 3! a_3, ..., (n-1)! a_{n-1}, 0) \)

\( a_n = \frac{B_n(1, 2! a_2, 3! a_3, ..., (n-1)! a_{n-1}, 0)}{n! (L^{n-1} - 1)} \).

And this is the recurrent formula for the general coefficients. Together with \( a_0 = L \) and \( a_1 = 1 \), this completes the as-close-to-explicit-as-possible-so-far formula for the coefficients of the Fourier series for the regular iteration.

EDIT: posts corrected to include factorials on terms \( a_n \) in Bell polynomials