On the solutions of the universal differential equation with three regular singularities (On solutions of $K{Z}_3$)

Hoang Ngoc Minh, Vincel

Confluentes Mathematici, Tome 11 (2019) no. 2, pp. 25-64.

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations ($K{Z}_3$) and our recent results on combinatorial aspects of zeta functions on several variables.

In particular, we describe the action of the differential Galois group of $K{Z}_3$ on the asymptotic expansions of its solutions leading to a group of associators which contains the unique Drinfel’d associator (or Drinfel’d series). Non trivial expressions of an associator with rational coefficients are also explicitly provided, based on the algebraic structure and the singularity analysis of the multi-indexed polylogarithms and harmonic sums.

Classification : 05E16, 11M32, 16T05, 20F10, 33F10, 44A20
Mots-clés : Algebraic Basis, Combinatorial Hopf Algebra, Harmonic Sum, Polylogarithm, Polyzeta

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