base holomorphic tetration

[bo198214]

By a "graph that shows the convergence of the series" you mean like a graph of partial sums?

Yes, yes, x-axis: n, y-axis: n-th partial sum.

Also, I could try a root test or ratio test but I think to make it really good I'll need more terms...

With root test the same: x-axis: n, y-axis: \( 1/\sqrt[n]{|a_n|} \). The root-test should converge to something bigger than the distance from the development point to the perhaps-singularity.

I just saw the paper and there appears to be a recurrence formula on page 8 but am a little confused by the notation. Specifically, what's \( {g^m}_n \) supposed to be? nth Taylor coefficient of m iterations of g? But we have not yet determined g, so how can we iterate?

In this draft integer-iteration is always denoted by \( f^{[n]} \), while power is as usual \( f^{n} \). \( {g^m}_n \) depends only on values \( g_k \), \( k\le n \). You can take the polynomial of g truncated to n, then take the m-th power and then get the coefficient at n.