A simpler proof and explicit algorithm express multiple zeta values in terms of symmetric multiple zeta values and their products.
Glaisher, On the residues of the sums of products of the first p − 1 numbers and their powers, to modulus p2 or p3, Quarterly J
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An Explicit Expression for MZVs in Terms of Symmetric MZVs
A simpler proof and explicit algorithm express multiple zeta values in terms of symmetric multiple zeta values and their products.