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The Bogoliubov recursion is a particular procedure appearing in the processof renormalization in perturbative quantum field theory. It provides convergentexpressions for otherwise divergent integrals. We develop here a theory offunctional identities for noncommutative Rota-Baxter algebras which is shown toencode, among others, this process in the context of Connes-Kreimer's Hopfalgebra of renormalization. Our results generalize the seminal Cartier-Rotatheory of classical Spitzer-type identities for commutative Rota-Baxteralgebras. In the classical, commutative, case, these identities can beunderstood as deriving from the theory of symmetric functions. Here, we showthat an analogous property holds for noncommutative Rota-Baxter algebras. Thatis, we show that functional identities in the noncommutative setting can bederived from the theory of noncommutative symmetric functions. Lie idempotents,and particularly the Dynkin idempotent play a crucial role in the process.Their action on the pro-unipotent groups such as those of perturbativerenormalization is described in detail along the way.

A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion