Transseries, nest-series, and other exotic series representations for tetration

[bo198214]

I can compute a nested series for the fractional iterates of \( e^x-1 \), but I don't claim the series converges. I think the series is a formal power series. It is interesting to know that the series is Borel-summable.

So your Schröder sums compute the regular iteration, is that true? I think it is very important to know those equalities. For example it took a while until I realized that the matrix approach introduced by Gottfried is actually equal to the regular iteration.
As a test, e.g. the regular half-iterate of \( e^x-1 \) has as the first 10 coefficients:
\( 0 \), \( 1 \), \( \frac{1}{4} \), \( \frac{1}{48} \), \( 0 \), \( \frac{1}{3840} \), \( -\frac{7}{92160} \), \( \frac{1}{645120} \), \( \frac{53}{3440640} \), \( -\frac{281}{30965760} \)

Or generally the \( t \)-th iterate has as the first 10 coefficients

\( 0 \),
\( 1 \),
\( \frac{1}{2} t \),
\( \frac{1}{4} t^{2} - \frac{1}{12} t \),
\( \frac{1}{8} t^{3} - \frac{5}{48} t^{2} + \frac{1}{48} t \),
\( \frac{1}{16} t^{4} - \frac{13}{144} t^{3} + \frac{1}{24} t^{2} - \frac{1}{180} t \),
\( \frac{1}{32} t^{5} - \frac{77}{1152} t^{4} + \frac{89}{1728} t^{3} - \frac{91}{5760} t^{2} + \frac{11}{8640} t \),
\( \frac{1}{64} t^{6} - \frac{29}{640} t^{5} + \frac{175}{3456} t^{4} - \frac{149}{5760} t^{3} + \frac{91}{17280} t^{2} - \frac{1}{6720} t \),
\( \frac{1}{128} t^{7} - \frac{223}{7680} t^{6} + \frac{1501}{34560} t^{5} - \frac{37}{1152} t^{4} + \frac{391}{34560} t^{3} - \frac{43}{32256} t^{2} - \frac{11}{241920} t \),
\( \frac{1}{256} t^{8} - \frac{481}{26880} t^{7} + \frac{2821}{82944} t^{6} - \frac{13943}{414720} t^{5} + \frac{725}{41472} t^{4} - \frac{2357}{580608} t^{3} + \frac{17}{107520} t^{2} + \frac{29}{1451520} t \),

Does that match your findings?
I think these formulas are completely derivable from integer-iteration. If one knows that each coefficient is just a polynomial then this polynomial is determined by the number of degree plus 1 values for \( t \) and these can be gained by just so many consecutive integer values. So this sounds really like your Schröder summation.

However an alternative approach is just to solve the equation \( f^{\circ t}\circ f=f\circ f^{\circ t} \) for \( f^{\circ t} \), where \( f \) and \( f^{\circ t} \) are treated as formal powerseries.