The Alternating Block Decomposition of Iterated Integrals and Cyclic Insertion on Multiple Zeta Values

Abstract

Abstract 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.