Typed Angularly Decorated Planar Rooted Trees and O-Rota-Baxter Algebras

As a generalization of Rota-Baxter algebras, the concept of an omega-Rota-Baxter could also be regarded as an algebraic abstraction of the integral analysis. In this paper, we introduce the concept of an omega-dendriform algebra and show the relationship between omega-Rota-Baxter algebras and omega-dendriform algebras. Then, we provide a multiplication recursion definition of typed, angularly decorated rooted trees. Finally, we construct the free omega-Rota-Baxter algebra by typed, angularly decorated rooted trees.