The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.

We define a variant of real-analytic polylogarithms that are single-valued and that satisfy ''clean'' functional relations that do not involve any products of lower weight functions. We discuss the basic properties of these functions and, for depths one and two, we present some explicit formulas and results. We also give explicit formulas for the single-valued and clean single-valued version attached to the Nielsen polylogarithms Sn,2(x), and we show how the clean single-valued functions give new evaluations of multiple polylogarithms at certain algebraic points.

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lisoněk states that by inserting all cyclic permutations of some initial blocks of 2’s into the multiple zeta value ζ(1, 3, … , 1, 3) and summing, one obtains an explicit rational multiple of a power of π. Hoffman gives a conjectural identity of a similar flavour concerning $ 2 \zeta(3,3,\{2\}^m) - \zeta(3,\{2\}^m,1,2) $. In this paper, we introduce the ‘generalized cyclic insertion conjecture’, which we describe using a new combinatorial structure on iterated integrals—the so-called alternating block decomposition. We see that both the original BBBL cyclic insertion conjecture, and the Hoffman’s conjectural identity, are special cases of this generalized cyclic insertion conjecture. By using Brown’s motivic MZV framework, we establish that some symmetrized version of the generalized cyclic insertion conjecture always holds, up to a rational; this provides some evidence for the generalized conjecture.

We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi-Trudi formulae and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulae we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values.Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.

No abstract