Shuffle relations for regularised integrals of symbols

Abstract

We prove shuffle relations which relate a product of regularised integrals of classical symbols to regularised nested (Chen) iterated integrals, which hold if all the symbols involved have non-vanishing residue. This is true in particular for non-integer order symbols. In general the shuffle relations hold up to finite parts of corrective terms arising from renormalisation on tensor products of classical symbols, a procedure adapted from renormalisation procedures on Feynman diagrams familiar to physicists. We relate the shuffle relations for regularised integrals of symbols with shuffle relations for multizeta functions adapting the above constructions to the case of symbols on the unit circle.

40 pages,latex. Changes concern sections 4 and 5 : an error in section 4 has been corrected, and the link between section 5 and the previous ones has been precised

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Cited by in corpus (10)

• Rota-Baxter Algebras and Dendriform Algebras
• Free Rota-Baxter algebras and rooted trees
• Hopf algebra approach to Feynman diagram calculations
• Algebraic Birkhoff decomposition and its applications
• Renormalization of multiple zeta values
• Stochastic expansions and Hopf algebras
• Differential Birkhoff decomposition and the renormalization of multiple zeta values
• Renormalised Chen integrals for symbols on Rn and renormalised polyzeta functions
• Double shuffle relations and renormalization of multiple zeta values
• Renormalised iterated integrals of symbols with linear constraints