# fractional iteration of $xe^x$ has zero convergence radius?

The equation $f(f(x))=xe^x=x+x^2+\frac{x^3}{2}+\frac{x^4}{6}+\dots$ has a unique formal powerseries solution. Is its convergence radius 0 as was shown by Baker for the equation $f(f(x))=e^x-1$? Or more generally: Let $F(x)=xe^x$ does the unique formal powerseries solution of $F(f_t(x))=f_t(F(x))$ with $f_t(x)=x+t x^2 +\dots$ have convergence radius 0 for all non-integer numbers $t$?

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Also look at the differences of that values; this is even more suggestive because of the typical pattern which suggests, that the differences of the logs grow asymptotic linearly with the logs of the index, which is just another expression for hypergeometric growthrate. So the convergence-radius for the half-iterate seems to be zero as is for the half-iterate of $exp(x)-1$ .