On functional equations for Nielsen polylogarithms

Abstract

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Clean Single-Valued Polylogarithms

“…In this section, we carry out the requisite calculations necessary to explicitly write the singlevalued Nielsen polylogarithm S n,2 , and the clean version thereof. This provides the missing derivation for a formula for sv S n,2 (x) already stated in [15] (the main properties of which were however verified therein). We recall first the definition of the Nielsen polylogarithm.…”

“…Concurrently an inversion result valid for an MPL of arbitrary depth was given in [49], a clean single-valued version of which (up to depth 3) we provide in Section 5. Further reductions in weight 4 and 5, focusing on the so-called Grassmannian polylogarithm, are investigated in [14], whereas identities and reductions involving the so-called Nielsen polylogarithms in weights 5 through 8 are investigated in [15] (also using the clean single-valued version established in Section 6 below).…”

“…Indeed, it is often easier to find identities modulo product terms, e.g., by starting from identities that hold modulo shuffle products at the symbol level (cf. [13,15]). The combinatorics involved in R will restore all the product terms necessary to obtain a numerical identity between single-valued polylogarithms, up to a single constant of integration.…”

“…We also briefly repeat the calculation of ΠS dr n,2 (x), which was completed in detail in [15]. Here we proceed directly via the already calculated coproduct of S dr n,2 (x), whereas the calculation in [15] was reduced to an analysis of which terms contribute in the infinitesimal coproduct. From (4.3), we have…”

“…These clean single-valued functions (or more precisely the version without R n , which still satisfies Theorem 1.1) are used in [15] to obtain numerically verifiable identities and reductions for Nielsen polylogarithms, including numerical results on their special values.…”