Generalizations of Abel’s and Hurwitz’s identities

In 1826 N. Abel found a generalization of the binomial formula. In 1902 Abel’s theorem was further generalized by A. Hurwitz. In this paper we describe constructions that provide infinitely many identities each being a generalization of a Hurwitz’s identity. Moreover, we give combinatorial interpretations of all these identities as the forest volumes of certain directed graphs.