Extended Shuffle Product for Multiple Zeta Values

Abstract

The shuffle algebra on positive integers encodes the usual multiple zeta values (MZVs) (with positive arguments) thanks to the representations of MZVs by iterated Chen integrals of Kontsevich. Together with the quasi-shuffle (stuffle) algebra, it provides the algebraic framework to study relations among MZVs. This paper enlarges the shuffle algebra uniquely to what we call the extended shuffle algebra that encodes convergent multiple zeta series with arbitrary integer arguments, not just the positive ones in the usual case. To achieved this goal, we first replace the Rota-Baxter operator of weight zero (the integral operator) that characterizes the shuffle product by the differential operator which extends the shuffle product to the larger space. We then show that the subspace corresponding to the convergent MZVs with integer arguments becomes a subalgebra under this extended shuffle product. Furthermore, by lifting the extended shuffle algebra to the locality algebra of Chen symbols, we prove that taking summations of fractions from Chen symbols defines an algebra homomorphism from the above subalgebra to the subalgebra of real numbers spanned by convergent multiple zeta series.