  We study generating functions in the context of Rota-Baxter algebras. We show
that exponential generating functions can be naturally viewed in a very special
case of complete free commutative Rota-Baxter algebras. This allows us to use
free Rota-Baxter algebras to give a broad class of algebraic structures in
which generalizations of generating functions can be studied. We generalize the
product formula and composition formula for exponential power series. We also
give generating functions both for known number families such as Stirling
numbers of the second kind and partition numbers, and for new number families
such as those from not necessarily disjoint partitions and partitions of
multisets.
