One-parameter semigroups of analytic functions, fixed points and the Koenigs function

Analogues of the Berkson-Porta formula for the infinitesimal generator of a one-parameter semigroup of holomorphic maps of the unit disc into itself are obtained in the case when, along with a Denjoy-Wolff point, there also exist other fixed points. With each one-parameter semigroup a so-called Koenigs function is associated, which is a solution, common for all elements of the one-parameter semigroup, of a certain functional equation (Schroder's equation in the case of an interior Denjoy-Wolff point and Abel's equation in the case of a boundary Denjoy-Wolff point). A parametric representation for classes of Koenigs functions that takes account of the Denjoy-Wolff point and other fixed points of the maps in the one-parameter semigroup is presented.