Double shuffle relations for arborified zeta values

Abstract

Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multiple zeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborifed zeta values are algebra morphisms for these new products on trees.

Series representation of arborified zeta values

“…This main theorem justifies, still in the second section, the definition of tree zeta values (Definition 2.14). Theorem 2.20 is then a direct consequence of Theorem 2.12 together with the aforementioned results of [6] on arborified zeta values.…”

“…The next definition characterises the rooted forest to which we will be able to attach a iterated integral. It is taken from [6]. Definition 2.3.…”

“…We focus on these forests for two reasons. First, these convergent rooted forests are the image of the branched binarisation map s T : F N * −→ F {x,y} (see [6,Definition A.1]). Second, for these forests, the associated arborified zeta values admit a series representation, which is one of the main result of this paper.…”

“…AZVs appeared in the work of Ecalle [3] and much later in the work of Yamamoto [4]. Their extensive study started in [5] and was completed in [6]. The renormalisation of their divergent counterparts was performed in [7].…”

“…The renormalisation of their divergent counterparts was performed in [7]. In particular, it was shown in [6] that AZVs are rational sums of MZVs. It was also shown in the same paper that, while AZVs exists as iterated sums and iterated integrals like MZVs, these two objects are not related by the most natural generalisation of the binarisation map s.…”