11/22/2007, 08:32 PM

Hmm, I didn't follow your post, but it's early and I've been thinking about the singularity at 0 that I only just noticed. Actually, I haven't quite been following the maths in the Imaginary zeros of f(z)= z^(1/z) discussion. (Not because you didn't explain the maths well, but because I haven't devoted the time yet.)

As for the numerators, I already found a Sloane series (A009306), which by changing every other sign and removing the first couple terms gives me the series I found. A separate Sloane series probably isn't required, merely the simple rules for how to get the numerators from the existing series.

The calculation of the terms in the Sloane series is done with basic functional composition of power series with rational coefficients, so determining an arbitrary term should be straightforward, and from the matrices involved in the composition we could derive a (not necessarily simple) formula.

As for the numerators, I already found a Sloane series (A009306), which by changing every other sign and removing the first couple terms gives me the series I found. A separate Sloane series probably isn't required, merely the simple rules for how to get the numerators from the existing series.

The calculation of the terms in the Sloane series is done with basic functional composition of power series with rational coefficients, so determining an arbitrary term should be straightforward, and from the matrices involved in the composition we could derive a (not necessarily simple) formula.

~ Jay Daniel Fox