Holomorphic extension of the logistic sequence

The logistic problem is formulated in terms of the Superfunction and Abelfunction of the quadratic transfer function $H(z)=uz({1-z})$. The Superfunction $F$ as holomorphic solution of equation $H(F(z))=F({z+1})$ generalizes the logistic sequence to the complex values of the argument $z$. The efficient algorithm for the evaluation of function $F$ and its inverse function, id est, the Abelfunction G are suggested;$F\big(G(z)\big)=z$. The halfiteration $h(z)=F({1/2+G(z)})$ is constructed; in wide range of values $z$, the relation ${h(h(z))=H(z)}$ holds. For the special case ${u=4}$, the Superfunction $F$ and the Abelfunction $G$ are expressed in terms of elementary functions.