The question is whether there is an established word for the transformation that starts with an invertible formal power series over a field $k$, $u(x)=xg(x)=x(1+a_1x+a_2x^2+\cdots)$ and delivers the series $u_m(x)=x\bigl(g(x^m)\bigr)^{1/m}$. The resulting series has nonzero coefficients only in degrees $\equiv1\pmod m$; we always take $m$ to be indivisible by the characteristic of $k$. One sees that $u\mapsto u_m$ is a homomorphism on the group of series $u(x)$ with $u(0)=0$ and $u'(0)=1$. The transformation has been around for very long, but I don’t know whether anybody has named it.


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