(08/31/2010, 06:51 AM)mike3 Wrote: The product of exp is the exponential of a formal power series. This can be expressed using the Bell polynomials:

.

This then becomes

.

Thus the equations to solve are

.

Since and , this is

.

This is derived from Faà di Bruno's formula, see

http://en.wikipedia.org/wiki/Fa%C3%A0_di...7s_formula

for details.

Doing some tests, it appears that

has only one occurrence of , and no higher powers of it, and it never seems to be multiplied by any sort of n-dependent coefficient. This means that . I don't have a proof at this point, but we can then solve this:

.

And this is the recurrent formula for the general coefficients. Together with and , 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 in Bell polynomials