A one-parameter family of dendriform identities

The classical Spitzer and Bohnenblust–Spitzer identities [17, 1, 15] from probability theory can be formulated in terms of certain algebraic structures known as commutative Rota-Baxter algebras. Recently, Ebrahimi-Fard et al. [3] have extended these identities to noncommutative Rota-Baxter algebras. Their results can in fact be formulated in terms of dendriform dialgebras [4], a class of associative algebras whose multiplication split into two operations satisfying certain compatibility relations [10]. Here, we exploit a natural embedding of free dendriform dialgebras into free colored quasisymmetric functions in order to simplify the calculations, and to obtain a q-analog of the main formulas of [3, 4].