Shuffle relations for regularised integrals of symbols

TL;DR

This paper establishes shuffle relations connecting regularised integrals of classical symbols with nested Chen integrals, incorporating renormalisation techniques and extending to multizeta functions.

Contribution

It introduces new shuffle relations for regularised integrals of symbols, including correction terms from renormalisation, and relates these to multizeta function identities.

Findings

Shuffle relations hold for non-integer order symbols.

Renormalisation introduces correction terms in the shuffle relations.

Connections to multizeta functions are established.

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.