P(x) P(y) = P(P(x) y) + P(x P(y)) + \lambda P(x y)
Rota–Baxter algebras came historically from several sources, including enumerative combinatorics problems (Gian-Carlo Rota), -calculus (Jackson integral) and integrable systems (Baxter). More recently the application to the combinatorics of renormalization and Birkhoff decomposition have been found, showing that the essence of renormalization is not that crucially existing only in the presence of complex analysis.
A large bibliography on Rota–Baxter algebras can be found at Li Guo’s Rota–Baxter algebra page.
K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras and dendriform dialgebras, Jour. Pure Appl. Algebra 212 (2008), 320-339, arXiv:math/0503647
K. Ebrahimi-Fard, L. Guo, Rota–Baxter algebras in renormalization of perturbative quantum field theory, In Universality and Renormalization, I. Binder and D. Kreimer, editors, Fields Institute Communicatins, v. 50, AMS 2007, 47-105, arXiv:hep-th/0604116.
K. Ebrahimi-Fard, L. Guo, D. Manchon, Birkhoff type decompositions and the Baker–Campbell–Hausdorff recursion, Comm. Math. Physics 267 (2006) 821-845, arXiv:math-ph/0602004
There is also a categorification of the concept of Rota–Baxter algebra, namely the Rota–Baxter category, cf.
Among motivatring results, there are also Spitzer identities from