Abstract
We prove shuffle relations which relate a product of regularised integrals of classical symbols \( \int^{reg} \sigma_i\, d\xi_i, i=1, \ldots, k\) to regularised nested iterated integrals:
where Σ k is the group of permutations over k elements. We show that these shuffle relations hold if all the symbols σ i have vanishing residue; this is true of non-integer order symbols on which the regularised integrals have all the expected properties such as Stokes’ property [MMP]. In general the shuffle relations hold up to finite parts of corrective terms arising from a renormalisation on tensor products of classical symbols, a procedure adapted from renormalisation methods to compute Feynman diagrams familiar to physicists. We relate the shuffle relations for regularised integrals of symbols with shuffle relations for multiple zeta functions adapting the above constructions to the case of a symbol on the unit circle.
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Manchon, D., Paycha, S. Shuffle Relations for Regularised Integrals of Symbols. Commun. Math. Phys. 270, 13–51 (2007). https://doi.org/10.1007/s00220-006-0141-y
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DOI: https://doi.org/10.1007/s00220-006-0141-y